#  IAC-23,D1,6,1,x78669
Multi-Agent 3D Map Reconstruction and Change Detection in Microgravity with
Free-Flying Robots

Holly Dinkel 0000-0002-7510-2066 Julia Di 0000-0001-5872-5694 Jamie Santos
Department of Physics at Chalmers University of Technology, Gothenburg, Sweden
0009-0007-7923-8482 Keenan Albee Maritime and Multi-Agent Autonomy Group at
the Jet Propulsion Laboratory, California Institute of Technology, Pasadena,
CA, USA 0000-0002-9655-2429 Paulo Borges Robotics and Autonomous Systems Group
at CSIRO Data61, Brisbane, QLD, Australia 0000-0001-8137-7245
Marina Moreira Intelligent Robotics Group at the NASA Ames Research Center,
Moffett Field, CA, USA 0000-0002-4180-473X Oleg Alexandrov Intelligent
Robotics Group at the NASA Ames Research Center, Moffett Field, CA, USA
0000-0001-7567-493X Brian Coltin Intelligent Robotics Group at the NASA Ames
Research Center, Moffett Field, CA, USA 0000-0003-2228-6815 Trey Smith
Intelligent Robotics Group at the NASA Ames Research Center, Moffett Field,
CA, USA 0000-0001-8650-8566

###### Abstract

Assistive free-flyer robots autonomously caring for future crewed
outposts—such as NASA’s Astrobee robots on the International Space Station
(ISS)—must be able to detect day-to-day interior changes to track inventory,
detect and diagnose faults, and monitor the outpost status. This work presents
a framework for multi-agent cooperative mapping and change detection to enable
robotic maintenance of space outposts. One agent is used to reconstruct a 3D
model of the environment from sequences of images and corresponding depth
information. Another agent is used to periodically scan the environment for
inconsistencies against the 3D model. Change detection is validated after
completing the surveys using real image and pose data collected by Astrobee
robots in a ground testing environment and from microgravity aboard the ISS.
This work outlines the objectives, requirements, and algorithmic modules for
the multi-agent reconstruction system, including recommendations for its use
by assistive free-flyers aboard future microgravity outposts.

§§footnotetext: Equal contribution

Keywords: Change Detection, Microgravity, Robotics, 3D Reconstruction,
Mapping, Artemis

##  1\. Introduction

Figure 1: FastCD detects scene changes against a pre-built 3D model of the
environment by computing inconsistencies among a set of RGB images projected
onto the 3D model using their corresponding poses, $\bm{\mathrm{T}}^{i}$.

On the International Space Station (ISS), astronaut time is expensive and
limited. Future orbital outposts, including Gateway, will likely have limited
crews or even be uncrewed for extended periods of time. Routine tasks on these
outposts provide compelling opportunities for robotic assistance [9] [17]. A
number of free-flying assistants have been proposed to address this need
including the SPHERES [24], CIMON, Int-Ball [4], and most recently Astrobee
[31] robots. Astrobee is a next-generation free-flying robot onboard the ISS
capable of deploying a variety of future research payloads and guest science
software intended to serve as an astronaut assistant to alleviate valuable
crew time [31, 6, 5, 21]. One area where Astrobee has the potential to reduce
astronaut workload is as a mobile sensor conducting and recording surveys.
Astronauts currently conduct manual regular visual surveys of the entire space
station, which Astrobee could alleviate. Free-flying robots, including
Astrobee, will be increasingly useful assistants, managing and maintaining
future microgravity outposts.

However, one challenge for deploying robots in human habitats is the interiors
of these environments are ever-changing: objects frequently disappear or are
reintroduced. A current map of the environment is vital for maintenance and
navigation, but creating a new map and computing between-map changes is
resource-intensive as robots could take hours to complete an environment scan.
Rather than repeat the mapping process for the entire environment, it is
advantageous to only recompute local changes—Fast Image-Based Geometric Change
Detection (FastCD) enables robots to quickly locate changes in a known world
model based on geometric projection operations performed on a small ($<10$)
batch of images as shown in Figure 1 [25]. This work develops FastCD for
operation on assistive free-flyers in microgravity. This work contributes:

  1. 1.

Discussion of a multi-agent scene change detection framework for free-flyer
robots. This application can also be extended for object discovery.

  2. 2.

Discussion of scene change detection considerations for resource-constrained
mobile robots in a space environment.

  3. 3.

Demonstration and evaluation of FastCD on data collected with Astrobee robots
at the NASA Ames Research Center Granite Lab microgravity research facility
and aboard the ISS.

  4. 4.

Release of a FastCD-compatible dataset of Granite Lab and ISS activity data at
https://bit.ly/astrobee_fastcd_data.

##  2\. Related Work

Because environment models should be up-to-date for most applications, a
common task in robotic surveying is environment reconstruction. For larger
environments, obtaining a new scan and generating 3D models of the environment
can be expensive, requiring dedicated robot mapping time, sensors, and
significant data processing. This cost can be reduced through 3D scene change
detection [27]. By identifying only local changes in an environment based on
existing information, which can be generated directly from raw imagery, a
mobile robot may direct its surveying to target change regions.

There are several types of change detection methods categorized according to
the target scene. Classical change detection is commonly formulated as a 2D or
2.5D problem, comparing new image data to old background images, and is
adopted in several satellite remote sensing and surveillance use cases [1, 8,
28]. Pairwise image comparison is straightforward, but may not scale
proportionally to map size. For traversing through a 3D environment in robotic
outpost maintenance, a 3D change detection approach is better suited.

Within 3D change detection, several works have investigated probabilistic
modeling. One recent unsupervised method of scene change detection uses
Gaussian Mixture Model (GMM) Clustering to identify changes between complete
point cloud maps of the environment [30]. This work processes a complete point
cloud map from one survey into region-based GMM clusters through split-and-
merge expectation-maximization [18], and detects between-scene changes by
comparing the cluster sets with the Earth Mover’s Distance [29]. Since this
method requires complete point cloud maps of the environment, it cannot be
deployed online on a mobile platform and is better suited for situational
awareness applications rather than anomaly response.

Another recent method repurposed map-based localization for change detection
[15]. This method does not perform well if self-localization methods are poor,
which may be the case in a space environment. For example, free-flying
assistive robots in microgravity do not have the availability of a gravity
vector for localization. Furthermore, in dense environments like the ISS, map
landmarks are not always available when there are occlusions from cargo bags
or human activity. Maps are also expensive to build online for a mobile robot
with limited computing power.

Other hybrid methods augment a 2D method with 3D information for remote
sensing applications. Changes are detected by comparing unordered 2D images
collected by a mobile agent to a 3D building information model which contains
information about the geometry of the environment [14]. This method may work
well for construction applications where environments are developed according
to a blueprint, but is not well suited for change detection in human habitats
which are frequently significantly reconfigured and for which a blueprint may
be unavailable.

Finally, other geometric treatments of 3D change detection create
inconsistency maps from RGB images or depth projections when compared to 3D
models [25, 35, 36]. These methods are fast and lightweight, suitable for
operation on a mobile robot when given a 3D world model of the environment,
but are sensitive to localization accuracy.

##  3\. Methodology

This work applies FastCD to sequences of RGB images to detect scene changes
and discover novel objects. The FastCD system is illustrated in Figure 3. and
discussed in the following sections. First, raw image and pose data are
collected with the Astrobee robot (Section 3.1) and processed to remove low-
movement images (Section 3.2). A camera pose is estimated for each image
(Section 3.3). A baseline 3D model of the environment is created (Section
3.4). Finally, candidate inconsistencies are computed between the images used
for change detection, and the resulting 3D regions of change are output
(Section 3.5).

Figure 2: From an image sequence, the change detection system removes low-
movement images, estimates the camera pose for remaining images, builds a 3D
environment map, computes between-image inconsistencies, and outputs detected
changes.

Figure 3: FastCD first back-projects images captured from different poses,
$\bm{\mathrm{T}}^{i}$ and $\bm{\mathrm{T}}^{i+1}$, onto a 3D model. Next, it
re-projects the 3D points from the model onto a common image plane for
comparison. (Inset) The Granite Lab is an environment which mimics
microgravity for testing the ground Astrobee units.

Figure 3. provides geometric intuition for FastCD, summarized in the following
steps:

  1. 1.

Let $i$ represent the index of an image-pose pair in the data sequence.

  2. 2.

Back-project (pixel-to-point) image $\mathcal{I}^{i}$ from pose
$\bm{\mathrm{T}}^{i}$ into the World frame of the 3D model using the pose and
projection matrix of the camera.

  3. 3.

Re-project (point-to-pixel) image $\mathcal{I}^{i}$ onto the image plane
$\mathcal{I}^{i\rightarrow i+1}$ of the camera at pose $\bm{\mathrm{T}}^{i+1}$
to create a new image containing the content from the image captured at
$\bm{\mathrm{T}}^{i}$ at pose $\bm{\mathrm{T}}^{i+1}$.

If there is no change in the 3D model between when the image was acquired and
when the model was created, all pixels taken from pose $\bm{\mathrm{T}}^{i}$
should correctly re-project onto $\mathcal{I}^{i\rightarrow i+1}$. In other
words, the image $\mathcal{I}^{i+1}$ at $\bm{\mathrm{T}}^{i+1}$ and the image
$\mathcal{I}^{i}$ from $\bm{\mathrm{T}}^{i}$ re-projected onto
$\mathcal{I}^{i\rightarrow i+1}$ should be approximately identical. If there
is a change from what is conveyed in the model, pixels corresponding to the
change will re-project onto $\mathcal{I}^{i\rightarrow i+1}$ incorrectly.

###  3.1 Data Collection

Data in this work were collected with Astrobee units in the Granite Lab and on
the ISS. The Granite Lab (inset in Figure 3.) is a facility that replicates
visual features of the ISS and mimics 3 DOF microgravity by placing Astrobee
on a near-frictionless air bearing.

The Navigation Camera (NavCam) on Astrobee is a fixed-focus RGB camera with a
wide field of view. This sensor collects Bayer images (at $1280\times 960$
resolution) at 5 Hz [31, 21]. When Astrobee surveys its environment, it flies
to waypoint stations where it may hold its pose to capture clear, high-
resolution ($5344\times 4008$) images with its Scientific Camera (SciCam).
Images collected with the SciCam cannot be streamed due to their large sizes,
so this work uses NavCam imagery by default.

Two types of paths were designed for Astrobee. In the surveys of the Granite
Lab, the paths were designed to emphasize translation. In the surveys of the
ISS, the paths were designed to emphasize 360o rotation, approximately fixing
the position of Astrobee. Figure 3.2 plots two paths against the corresponding
model, one for a survey in the Granite Lab and one for a survey in the ISS.
Throughout this work, different Astrobee units are referenced by their
nicknames (”BSharp”, ”Queen”, and ”Bumble”), as used in the Granite Lab and
ISS deployments.

###  3.2 Low-Movement Image Removal

Because Astrobee’s flight speed is low compared to the NavCam sensor rate, the
robot may capture sequential NavCam images which exhibit little to no movement
between-frame. Data are processed by removing these

Figure 4: The paths of the Granite Lab surveys emphasize translational wall-
scanning and the paths of the ISS surveys emphasize rotation.

low-movement images through feature matching and optical flow. Sparse features
are detected and matched between two images using the pyramidal implementation
of the Lucas-Kanade optical flow feature tracker [3]. If the distance between
these sets of features is less than a maximum distance threshold, low movement
is detected between the images and one of the images is discarded. Low-
movement image removal reduces the number of image candidates for change
detection from thousands to tens [21].

###  3.3 Camera Pose Estimation

FastCD requires an estimate of the pose of the camera in the World coordinate
frame defined by the 3D model. The camera pose is estimated as a homogeneous
transformation matrix, $\bm{\mathrm{T}}^{i}_{4\times 4}$, where $i$ represents
the index of the image-pose pair in the data sequence. The pose
$\bm{\mathrm{T}}^{i}$ can be computed from the body pose of the Astrobee robot
in the World frame, $\bm{\mathrm{T}}^{i}_{\text{Astrobee}}$, and the pose of
the NavCam with respect to the Astrobee robot,
${}^{\text{Astrobee}}\bm{\mathrm{T}}_{\text{NavCam}}$. The pose of the
Astrobee robot is estimated at a rate of approximately 5 Hz with a factor
graph-based localization system using an inertial measurement unit on the
robot [10, 7, 33]. The extrinsic calibration of the frame of the NavCam with
respect to the body frame of the Astrobee robot was performed using Kalibr
[16, 12, 13, 19]. The pose is computed as

$\displaystyle\bm{\mathrm{T}}^{i}=^{\text{World}}\bm{\mathrm{T}}_{\text{Astrobee}}^{i}$
${}^{\text{Astrobee}}\bm{\mathrm{T}}_{\text{NavCam}}.$ (1)

###  3.4 3D World Model Reconstruction

The NASA Ames Stereo Pipeline package generates a 3D map of the environment
from Astrobee Navigation Camera (NavCam), Hazard Camera (HazCam), and
localization data [32, 23]. It registers image data from the NavCam with depth
information from the HazCam using Theia structure-from-motion [34], and then
fuses the depth point clouds into a mesh [2]. This method was used to
reconstruct a 3D model of the Granite Lab environment included with the
contributed dataset and used throughout this work.

###  3.5 Computing Inconsistencies

Figure 5: A laptop and lights are discovered in the ISS from a set of $n=7$
Astrobee NavCam images. (Left) FastCD inputs include a sequence of $n=7$
images, $\mathcal{I}^{i}$, and the pose at which each image was captured,
$\bm{\mathrm{T}}^{i}$. (Center Top) FastCD computes inconsistencies between
images given the 3D model and filters the inconsistency image. (Center Bottom)
Images are projected onto a 3D model given the path and pose of the camera to
estimate 3D change regions. (Right) The 3D change regions are projected into
the 3D model as change ellipses.

Given the camera projection matrix, $\bm{\mathrm{P}}_{3\times 4}^{i}$, the
projection of an arbitrary 3D point in homogeneous coordinates,
$\bm{\mathrm{X}}^{i}_{4\times 1}$, onto the plane of image $i$ as a 2D pixel
in homogeneous coordinates, $\bm{\mathrm{x}}^{i}_{3\times 1}$, is

$\bm{\mathrm{x}}^{i}=\bm{\mathrm{P}}^{i}\bm{\mathrm{X}}^{i},$ (2)

where the projection matrix

$\bm{\mathrm{P}}^{i}=\bm{\mathrm{K}}\bm{\mathrm{R}}^{i}\left[\bm{\mathrm{I}}_{3\times
3}\mid-\bm{\mathrm{t}}^{i}\right]$ (3)

is computed from the camera intrinsic calibration matrix
$\bm{\mathrm{K}}_{3\times 3}$ and the rotation $\bm{\mathrm{R}}_{3\times
3}^{i}$ and translation $\bm{\mathrm{t}}_{3\times 1}^{i}$ components of
$\left(\bm{\mathrm{T}}^{i}\right)^{-1}$ that transform the World coordinates
into NavCam coordinates. Inverting Eq. (2) gives

$\bm{\mathrm{r}}^{i}=\left(\bm{\mathrm{R}}^{i}\right)^{\intercal}\bm{\mathrm{K}}^{-1}\bm{\mathrm{x}}^{i},$
(4)

the ray from the projection center of the camera through pixel
$\bm{\mathrm{x}}^{i}$ to the World. Each pixel of image
$\mathcal{I}^{i}_{1280\times 960}$ can then be back-projected onto the 3D
model.

Inconsistencies are detected between a pair of images captured from poses
$\bm{\mathrm{T}}^{i}$ and $\bm{\mathrm{T}}^{i+1}$ by creating a new intensity
image, $\mathcal{I}^{i\rightarrow i+1}_{1280\times 960}$, to store the content
of $\mathcal{I}^{i}$ from pose $\bm{\mathrm{T}}^{i+1}$ given the 3D model. The
intersection, $\bm{\mathrm{X}}^{i\rightarrow i+1}$, between a ray
$\bm{\mathrm{r}}^{i}$ from Eq. (4) and the 3D model is computed and and
projected onto $\mathcal{I}^{i\rightarrow i+1}$ as

$\bm{\mathrm{x}}^{i\rightarrow
i+1}=\bm{\mathrm{P}}^{i+1}\bm{\mathrm{X}}^{i\rightarrow i+1},$ (5)

where $\bm{\mathrm{P}}^{i}$ is the camera projection matrix corresponding to
pose $\bm{\mathrm{T}}^{i}$. Since image $\mathcal{I}^{i}$ and image
$\mathcal{I}^{i\rightarrow i+1}$ are at the same perspective,
$\mathcal{I}^{i\rightarrow i+1}$ can be used to segment $\mathcal{I}^{i+1}$.

An accurate camera pose is unavailable for most robotic systems and the 3D
model may be noisy. The pixel $\bm{\mathrm{x}}^{i\rightarrow i+1}$ therefore
has an associated uncertainty $\bm{\mathrm{\Sigma}}_{3\times 3}$ defined as

$\bm{\mathrm{\Sigma}}:=\bm{\mathrm{\Sigma}}_{\bm{\mathrm{x}}^{i\rightarrow
i+1}\bm{\mathrm{x}}^{i\rightarrow i+1}}.$ (6)

To account for this uncertainty, for each
$\bm{\mathrm{x}}^{i+1}\in\mathcal{I}^{i+1}$ compute the minimum Euclidean norm
of the intensity difference to each pixel $\bm{\mathrm{z}}$ of
$\mathcal{I}^{i\rightarrow i+1}$ in a region, $\mathcal{R}$, around
$\bm{\mathrm{x}}^{i+1}$ as

$d^{i\rightarrow
i+1}(\bm{\mathrm{x}}^{i+1})=\min_{\bm{\mathrm{z}}\in\mathcal{R}}\|\mathcal{I}^{i+1}(\bm{\mathrm{x}}^{i+1})-\mathcal{I}^{i\rightarrow
i+1}(\bm{\mathrm{z}})\|.$ (7)

For a distance threshold $\tau^{2}=11.82$, the critical value of the
$\chi_{2}^{2}$ distribution corresponding to the $3\sigma$ bound on the
standard normal distribution, the region $\mathcal{R}$ is defined as

$\small\mathcal{R}(\bm{\mathrm{z}})=\begin{cases}\left(\bm{\mathrm{x}}^{i+1}-\bm{\mathrm{z}}\right)^{\intercal}\bm{\mathrm{\Sigma}}^{-1}\left(\bm{\mathrm{x}}^{i+1}-\bm{\mathrm{z}}\right)<\tau^{2}&\forall\bm{\mathrm{z}}\in\mathcal{I}^{i\rightarrow
i+1}\\\ 0&\text{otherwise}\end{cases}.$ (8)

If no change is detected between the 3D model and images $\mathcal{I}^{i}$ and
$\mathcal{I}^{i+1}$, each $\bm{\mathrm{x}}^{i}\in\mathcal{I}^{i}$ should re-
project onto $\mathcal{I}^{i+1}$, images $\mathcal{I}^{i\rightarrow i+1}$ and
$\mathcal{I}^{i+1}$ should be identical, and
$\forall\bm{\mathrm{x}}^{i+1}\in{\mathcal{I}^{i+1}}$, $d^{i\rightarrow
i+1}(\bm{\mathrm{x}}^{i+1})\approx 0$. If there is a mismatch between the 3D
model and images due to a change in the environment, pixels corresponding to
the change will re-project onto the wrong place in $\mathcal{I}^{i+1}$ and the
change will be detected from a large $d^{i\rightarrow
i+1}(\bm{\mathrm{x}}^{i+1})$ value.

As originally highlighted in FastCD, this method has ambiguities when only two
images are compared. A point $\bm{\mathrm{X}}_{c}^{i}$ corresponding to a
change in the 3D model generates two pixel locations on
$\mathcal{I}^{i\rightarrow i+1}$. FastCD resolves this ambiguity using
multiple pair-wise image comparisons [26]. Two pixels $\bm{\mathrm{x}}^{i+1}$
and $\bm{\mathrm{x}}^{i+2}$ belonging to the same change will project onto
$\mathcal{I}^{i}$ at the same location. Therefore, pixels from different
images which re-project onto the same region of $\mathcal{I}^{i}$ represent
the real locations of change. Change regions are estimated in 2D by comparing
each image with $m$ neighboring images. The default value of $m=4$ is used in
this work unless otherwise noted. An example illustrating how inconsistencies
are computed is included in Figure 5. After an inconsistency image with the 2D
regions of change is computed, the inconsistency image is filtered through a
sequence of image processing steps including erosion-dilation, contour
filtering, and change region association.

###  3.6 Regions of Change

Once the regions containing changes are identified in 2D, the corresponding
mean location of change, $\overline{\bm{\mathrm{X}}}_{c}^{i}$, is estimated in
3D [26, 11]. Every region identified as a change has mean pixel location
$\overline{\bm{\mathrm{x}}}^{i}$ and covariance $\bm{\mathrm{\Sigma}}^{i}$.
The corresponding 3D point $\overline{\bm{\mathrm{X}}}_{c}^{i}$ is computed in
World coordinates by triangulating the mean location of change in each image.
The 3D locations of change are estimated by solving

$\begin{matrix}\bm{\mathrm{A}}\overline{\bm{\mathrm{X}}}_{c}=0,\end{matrix}$
(9)

In Eq. (9),

$\bm{\mathrm{A}}_{3n\times
4}=\begin{bmatrix}\mathrm{S}\left(\overline{\bm{\mathrm{x}}}^{1}\right)\bm{\mathrm{P}}^{1},&\ldots,&\mathrm{S}\left(\overline{\bm{\mathrm{x}}}^{n}\right)\bm{\mathrm{P}}^{n}\end{bmatrix}^{\intercal},$
(10)

$n$ is the number of images, $\bm{\mathrm{P}}^{i}$ is the projection matrix
corresponding to pose $\bm{\mathrm{T}}^{i}$, and
$\mathrm{S}\left(\overline{\bm{\mathrm{x}}}^{i}\right)$ is a skew-symmetric
matrix generated from the elements of
$\overline{\bm{\mathrm{x}}}^{i}=\left[x^{i},y^{i},w^{i}\right]^{\intercal}$
such that

$\mathrm{S}(\overline{\bm{\mathrm{x}}}^{i})=\begin{bmatrix}0&-w^{i}&y^{i}\\\
w^{i}&0&-x^{i}\\\ -y^{i}&x^{i}&0\end{bmatrix}.$ (11)

Eq. (9) is solved using singular value decomposition [26]. This procedure is
repeated for the sigma points from every mean and covariance of the same
region in every image to efficiently estimate 3D change regions without
requiring a dense reconstruction of the environment. The mean of the points
$\overline{\bm{\mathrm{X}}}_{c}$ represent the regions in 3D where changes
occurred. The covariances associated with $\overline{\bm{\mathrm{X}}}_{c}$ are
used to draw covariance ellipses to characterize the uncertainty of the
estimate.

##  4\. Results and Limitations

FastCD was applied to data collected by Astrobee units in two environments.
The ”Bsharp” Astrobee unit surveyed the controlled Granite Lab environment
(Section 4.1) and the ”Bumble” and ”Queen” Astrobee units surveyed the
uncontrolled ISS environment (Section 4.2). In this qualitative study, FastCD
detects large objects appearing in multiple sequential image frames.

###  4.1 Granite Lab Results

The ”Bsharp” Astrobee unit surveyed the Granite Lab five times. The robot
followed mostly translational paths, scanning each wall of the Granite Lab.
The first four surveys (20230419_bsharp_survey1, 20230419_bsharp_survey2,
20230419_bsharp_survey3, and 20230419_bsharp_survey4) are included in the
released dataset. For each of these surveys, one or more confounding objects
realistic to the ISS environment are introduced, including a cargo bag, a
crate, a cable, and a static Astrobee unit, shown in Figure 4.1. The fifth
survey is not included in the released dataset. This survey was used to build
the 3D model (model.obj) included with each Granite Lab survey, following the
framework described in Section 3.4. Object discovery in Granite Lab data is
shown in Figure 4.1.

Figure 6: Objects are introduced at known locations to the Granite Lab
environment for each of four surveys.

Figure 7: FastCD detects several objects added to the Granite Lab environment.
(Top Row) Cable detection from Survey 1. (Middle Row) Cable and cargo bag
detection from Survey 3. (Bottom Row) Cable and crate detection from Survey 4.

For each survey, the maximum number of between-image comparisons, $m$, was
varied from 2 to 6, using the FastCD default $m=4$ as a center point. Results
were consistent with those found in [26]: with $m<4$, the results were much
noisier with increased false-positive detections. However, with each Granite
Lab survey, the quality of the results did not increase with $m\geq 4$. False
positives were occasionally found near the camera position, although less
frequently, with $m\geq 4$. While the robot itself is technically a changing
object between images in the scene, this result could be mitigated by
filtering results within a certain proximity to the camera. After removing
artifacts, these early FastCD results appear promising.

To analyze the accuracy of the results, the 3D coordinates of the known
changes and of the detected changes are necessary ground truth data. The
latter data are difficult to evaluate directly. Furthermore, the output model
provides an overview of the area where changes are detected as a cluster of
ellipses, and therefore a single item may be highlighted as several changes.
However, for all surveys, the majority of ellipses are clustered around the
location of the added object.

Compute-constrained processors, including the 4-core, 2.5 [GHz] Snapdragon
805-based IFC6501s used by the Astrobee units, must meet real-time rates for
autonomy software while running multiple processes concurrently [31]. Runtime
must be verified to show, at the very least, standalone real-time performance
of any autonomy algorithm. Computational timing data were collected for the
first Granite Lab survey data on a computer with a 4-core, 3.20 [GHz] Intel
i5-4570 CPU. Table 4.1 reports runtimes for three FastCD computational
processes based on the number of images, $n$, in the batch used to compute
changes. All other FastCD parameters were kept at defaults. Data loading
accounts for most of the runtime. Data are loaded as .png images and .xml
poses and camera intrinsic parameters. For online change detection, these data
could be retrieved from queues and made available to other processes such as
localization.

Table 1: FastCD Runtime [s] $n$ | Data Loading | Inconsistencies | 3D Change
---|---|---|---
2 | 0.844 | 0.311 | 0.054
3 | 0.800 | 0.606 | 0.076
4 | 0.843 | 1.156 | 0.134
5 | 0.835 | 1.891 | 0.161
6 | 0.842 | 2.280 | 0.266
Per Image | – |  $\approx$ 0.281 | –

The most computationally intensive component of FastCD is computing
inconsistencies, with computation time scaling with $n$. For $n<7$, FastCD
localizes scene changes in $<2.5\mathrm{s}$, fast enough to present actionable
information about the environment to a mobile robot. However, for a change
detection algorithm running on Astrobee, a $10\times$ slowdown is expected due
to the computational load of other processes. Real-time performance of FastCD
on Astrobee units is expected to require further optimization.

###  4.2 ISS Activity Results

The ”Bumble” and ”Queen” Astrobee units each performed three surveys of the
ISS. The robots followed mostly rotational paths to scan the ISS Japanese
Experiment Module (JEM) autonomously. All six of these surveys are included
inh released dataset (20220608_bumble_survey1, 20220608_bumble_survey2,
20220608_bumble_survey3, 20220608_queen_survey1, 20220608_queen_survey2, and
20220608_queen_survey3). Unlike in the Granite Lab, the environment was not
modified for the purpose of this work: the ISS was surveyed as-is. The
complexity of the ISS environment precludes reconstruction of a clean map from
robot survey data, so this work uses the Astrobee simulation JEM model [22].
Samples of FastCD object discovery on the ISS data are shown in Figure 4.2.

The results in Figure 4.2 highlight the importance of noise filtering steps on
the computed 2D inconsistencies. The ISS, like many human habitats, is
cluttered. This makes it more difficult to detect changes at both local and
full-scene scales. The FastCD algorithm uses image processing functions to
remove insignificant regions estimated as change regions. Increasing the
number of geometric projection operations by increasing $m$ also reduces
noise. Future work using semantic labeling could further reduce noise prior to
3D location of change estimation. It is also important to note that the 3D
Model used for the ISS Activities is a simulated model of the JEM and was not
generated from image and pose data in the same way the Granite Lab 3D model
was generated. Since the model was not reconstructed from real data, it
introduces additional uncertainties due to the Sim2Real gap.

Figure 8: As the maximum number of between-image comparisons, $m$, increases,
the number of detected changes decreases, removing spurious detections.

###  4.3 Limitations

FastCD assumes a good estimate of the camera pose. One way this limitation can
be addressed for Astrobee is to perform bundle adjustment against a known 3D
map of the environment to obtain an accurate ground truth localization. For
offline applications, this method of obtaining the ground truth pose may be
acceptable. However, for online surveying and change detection, Astrobee would
not have access to ground truth pose information and an inaccurate camera pose
could lead to false positive or undetected changes.

Astrobee localization accuracy suffers most when the movement between
consecutive images is low and when the number of features shared between
consecutive images is low. This is usually due to rotation-only movement with
little or no translation, introducing cheirality issues [33]. As illustrated
in Fig. 3.2, the Granite Lab surveys used T-shaped paths designed to increase
translation in the survey. The ISS Activity surveys used rotate-in-place
paths. These rotation-only surveys suffer the most from localization
inaccuracy, increasing change detection errors. Other hardware- and
environment-specific requirements include good lighting, prominent visual
features, and a sufficient field of view for features to overlap between
frames.

##  5\. Discussion and Future Work

Motivated by the promise of assistive robotic technology in space
environments, this work applied FastCD, a fast scene change detection
algorithm, to object discovery for habitat maintenance. After discussing the
specific considerations for scene change detection on a resource-constrained
mobile robot in a space environment, FastCD was demonstrated on data collected
with Astrobee units in the Granite Lab and aboard the ISS. The dataset used in
this work is publicly released at https://bit.ly/astrobee_fastcd_data.

Future work could perform geometric scene change detection in real-time with
multiple cameras, use images with semantic annotations to improve change
localization, or feedback detected changes in real-time into an anomaly
recovery architecture [20]. Future work could also improve on elements
presented here. Currently, the limitations of FastCD for anomaly detection
include sensitivity to object size, sensitivity to camera pose accuracy, and
sensitivity to camera resolution. Robustness could also be further improved
through detection pruning.

Future work may also include runtime analysis on embedded processors. This
work found computing inconsistencies is the most resource-consuming process
and is related to $n$, the number of images in a batch. Computation time
should also depend on $m$, the maximum number of comparisons used for
detection in the FastCD algorithm. Investigating the impact of $m$ on timing
will be important to keep computational expense within the resource budget of
free-flyers such as Astrobee. Interesting extensions which may improve the
accuracy of the FastCD algorithm include change detection labeling and
semantics as well as robustness to illumination change.

Finally, this type of work unlocks numerous benefits for assistive robotics.
Because FastCD estimates locations of change with uncertainty, when an
environment map needs updating, only the local changes need to be identified
and remapped. This feature is applicable to any robotic system, whether in
space or terrestrial. Moreover, methods like FastCD allow Astrobee or other
free-flyers to feasibly work alongside astronauts in space habitats, thus
reducing astronaut workload. Furthermore, a great benefit of FastCD is its
speed, which is especially useful for a resource-constrained mobile platform.
As the world increasingly turns to automation to meet its needs, it is more
important than ever to enable robots with the capability of responding to
dynamic, human environments.

## Acknowledgements

The NASA Space Technology Graduate Research Opportunities 80NSSC21K1292 and
80NSSC18K1197 supported Holly Dinkel and Julia Di, respectively. Holly Dinkel
was additionally supported by a P.E.O. Scholar Award and the Zonta
International Amelia Earhart Fellowship. Julia Di was additionally supported
by the Soffen Memorial Fund, Stanford JEDI Initiative, and Future Space
Leaders Foundation. A portion of this research was carried out at the Jet
Propulsion Laboratory, California Institute of Technology, under a contract
with the National Aeronautics and Space Administration (80NM0018D0004). The
CSIRO Data61 and the Winston Churchill Fellowship supported Paulo V.K. Borges.
The NASA Game Changing Development (GCD) Program supported Jamie Santos,
Marina Moreira, Oleg Alexandrov, Brian Coltin, and Trey Smith. The authors
thank Ryan Soussan and the Astrobee Facilities team for supporting this work.

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## Authors

Holly Dinkel is pursuing a Ph.D. in aerospace engineering at the University of
Illinois Urbana-Champaign where she researches robotic caretaking as a NASA
Space Technology Graduate Research Fellow with the NASA Ames Research Center
Intelligent Robotics Group and the NASA Johnson Space Center Dexterous
Robotics Laboratory.

Julia Di is pursuing a Ph.D. in mechanical engineering at Stanford University
where she researches tactile sensing and perception. She was a NASA Space
Technology Graduate Research Fellow with the NASA Ames Research Center
Intelligent Robotics Group and NASA Jet Propulsion Laboratory Mobility and
Robotic Systems Section.

Jamie Santos completed a M.S. in Complex and Adaptive Systems at Chalmers
University in Gothenburg, Sweden. She researches change detection to enable
NASA’s free-flying Astrobee robots to detect anomalies on the International
Space Station with the NASA Ames Research Center Intelligent Robotics Group.

Keenan Albee is a Robotics Technologist in the Maritime and Multi-Agent
Autonomy group at the Jet Propulsion Laboratory, California Institute of
Technology. He received a Ph.D. in Aeronautics and Astronautics from the
Massachusetts Institute of Technology. He is interested in motion planning
under uncertainty, information-aware planning, and real-time model learning.

Paulo Borges is a Principal Research Scientist in the Robotics and Autonomous
Systems Group at CSIRO in Brisbane, Australia. He completed a Ph.D. in
Electronic Engineering from Queen Mary University of London, London, United
Kingdom. His research interests include robotic automation for the
manufacturing, energy, and agriculture industries, bridging industry,
innovation, and research.

Marina Moreira is a research engineer in the NASA Ames Research Center
Intelligent Robotics Group. She completed a M.S. in aerospace engineering from
Instituto Superior Técnico, Lisbon, Portugal. She develops and maintains open-
source software for the ISAAC problem and is interested in problems related to
robotic systems and control.

Oleg Alexandrov is a Research Scientist in the NASA Ames Research Center
Intelligent Robotics Group. He received a Ph.D. in applied mathematics from
the University of Minnesota. His research focuses on mapping using satellite
and robot images.

Brian Coltin is a Computer Scientist in the NASA Ames Research Center
Intelligent Robotics Group where he works on the Astrobee robot, the VIPER
rover, and flood mapping. He earned a Ph.D. in Robotics from Carnegie Mellon
University. His research interests include planning, scheduling, multi-robot
coordination, localization, and computer vision.

Trey Smith is a Computer Scientist in the NASA Ames Research Center
Intelligent Robotics Group where he works on the Astrobee robot and the VIPER
rover. He earned a Ph.D. in Robotics from Carnegie Mellon University. His
research interests include planning, scheduling, multi-robot coordination,
mapping, and computer vision.


Partial and Boundary Regularity for Uniformly Elliptic Equations: Armstrong,
Silvestre, and Smart applied this method to develop partial regularity results
for fully non-linear equations in [ASS12]. One should keep in mind that in
general, uniformly elliptic problems may develop singularities as the one
exhibited by [NV13]. However, the interesting result proved in [ASS12], shows
that those singularities must have positive Hausdorff co-dimension (and hence
zero measure). Silvestre and Sirakov proved in [SS14] that the singular set
can only be found in the interior of the domain of the equation, once again
relying on [Sav07].

Optimal Regularity for Degenerate Equations: Colombo and Figalli developed
regularity estimates for degenerate equations arising from traffic congestion
models in [CF14]. In collaboration with Pimentel, we demonstrated in [CLP21]
the continuity of $|Du|$, where $u$ solves the gradient-constrained problem
$\max(1-|Du|,\Delta u+1)=0$.

In [Le18], the paraboloid method is implemented to show a Harnack inequality
for singular elliptic equations which are $(D^{2}u)^{-1}$-like. These arise as
the linearizations of the Monge-Ampère equation.

Non-Local Minimal Surfaces: Caffarelli, Roquejoffre, and Savin established
regularity estimates for non-local minimal surfaces in [CRS10].

Parabolic and Non-Local Counterparts: The technique was extended to parabolic
problems by Wang in [Wan13], and for non-local equation by Yu in [Yu17].

##  5\. Open Problems

One of the most famous open problems in the field is to characterize the
dimension for which the translation invariant equation $F(D^{2}u)=0$ has
$C^{2}$ estimates.

It was shown by Nirenberg using complex analysis tools in [Nir53], that
$C^{2,\alpha}$ estimates are available in dimension 2. Recall that in Problem
20 we saw that $C^{2,\alpha}$ estimates are available for convex/concave
operators in any dimension. Some further generalizations have been studied in
[CC03, CY00]. However, it is known that in general we may find singularities
starting in dimension $5$, [NV13].

Do solutions of the uniformly elliptic equation $F(D^{2}u)=0$ have interior
$C^{2}$ estimates or do singularities may also develop in dimension 3 or 4?

Whenever singularities develop for the equation $F(D^{2}u)=0$, it was shown in
[ASS12] that they have positive Hausdorff co-dimension. This co-dimension is
at least the exponent $\varepsilon$ in the measure estimate for the Hessian
(Problem 15), which yields the question of how large could $\varepsilon$
actually be. It was observed in the same article that the in dimension 2, one
must have that $\varepsilon\leq C(\Lambda/\lambda)^{-1}$ and conjectured that
this is actually optimal.

In a recent approach to the Krylov-Safonov Theory by Mooney [Moo19], it has
been shown that one can take $\varepsilon\geq C_{n}(\Lambda/\lambda)^{1-n}$,
settling the previous conjecture in dimension 2; see also [Le20]. It was
conjectured in [Moo19] that this estimate is actually the optimal one in any
dimension, however its verification remains open.

One may also wonder about the optimal integrability exponent $p$ allowed as a
forcing term in a uniformly elliptic equation in order to get interior Hölder
estimates. As we know from the Sobolev and Morrey embeddings, this exponent
should be strictly greater than $n/2$. For uniformly elliptic operators, we
just saw that the theory holds for $p\geq n$. Due to fundamental estimates
arising from harmonic analysis in [FS84], it was shown in [Esc93] that the
optimal $p$ can always be taken slightly smaller in the interval $(n/2,n)$.
The actual dependence of the optimal integrability exponent $p$ on the
dimension and the ellipticity constants remains open.

A quite challenging question related to the integrability exponent for non-
local equations of order $2\sigma\in(0,2)$ is the following one: In contrast
to the second order theory, the non-local ABP estimate proved in [CS09]
depends on the $L^{\infty}$-norm for the forcing term. It is expected that a
maximum principle also holds in terms of the $L^{p}$-norm for the forcing
term, for some $p\in(n/(2\sigma),\infty)$. For the fractional Laplacians this
is a consequence of the $L^{p}$ bounds for the Riesz transform and the Sobolev
embedding. In the uniformly elliptic setting, the proof of this conjecture is
only known under restrictive assumptions on the kernel [GS12, Kit20].

Similar questions can also be posed regarding the optimal Hölder exponent. So
far, it seems that new techniques may be required in dimension greater or
equal than 3. See [Tei20] for some conjectures in the case of the
$p$-Laplacian.

The article [IS16], which studied Hölder estimates for elliptic equations that
hold whenever the gradient is large (Theorem 4.1), posed two interesting
problems. The first one was to extend the estimate for equations with drift
term. This was solved in [Moo15], whose strategy was presented in this note
(in the absence of drift).

The second problem remains open: Is it possible to get a Hölder estimate on a
solution of the parabolic problem

$\begin{cases}\partial_{t}u-\mathcal{P}^{-}_{\lambda,\Lambda}(D^{2}u)\geq-|f|\text{
in }\\{|Du|>\gamma\\}\cap B_{1}\times(-1,0],\\\
\partial_{t}u-\mathcal{P}^{+}_{\lambda,\Lambda}(D^{2}u)\leq|f|\text{ in
}\\{|Du|>\gamma\\}\cap B_{1}\times(-1,0].\end{cases}$

Notice that any $u=u(t)$, independent of $x$, is automatically a solution for
any forcing term $f$. This shows that we can not expect a regularity estimate
in time. Perhaps a Hölder estimate exclusively in space should be true.

Other reasonable extension in the static case would be to replace the constant
$\gamma$ by some non-negative function $\gamma\in L^{p}$. For $p>n$, we have
by Morrey’s inequality that $|Du|\leq\gamma$ implies $u\in C^{0,1-n/p}$. Would
the Theorem 4.1 hold under the assumption $\gamma\in L^{p}$ for some $p>n$?

##  6\. Ideas and References for the Problems

Morrey’s Inequality

1. Use the fundamental theorem of calculus and Hölder’s inequality.

2. Let $\rho\in(0,r)$ and $\theta\in\partial B_{1}$. By the fundamental theorem of calculus

$u(\rho\theta)-u(0)=\int_{0}^{\rho}Du(t\theta)\cdot\theta dt.$

Integrating over $\theta\in\partial B_{1}$

$\frac{1}{\rho^{n-1}}\int_{\partial
B_{\rho}}|u-u(0)|\leq\int_{0}^{\rho}\left(\frac{1}{t^{n-1}}\int_{\partial
B_{t}}|Du|\right)dt\leq\int_{B_{r}}\frac{|Du(x)|}{|x|^{n-1}}dx$

A final integration from $\rho=0$ to $\rho=r$, after having multiplied by
$\rho^{n-1}$, gives the desired identity.

3. By the triangle inequality $\operatorname*{osc}_{B_{r}}u\leq 2\sup_{B_{r}}|u-u(0)|$. Once again by the triangle inequality, we get that for $x\in B_{r}$ with $r\in(0,1/2)$

$|u(x)-u(0)|\leq\fint_{B_{r}}|u-u(0)|+\fint_{B_{r}(x)}|u-u(x)|.$

Hence the desired result follows by Hölder’s inequality using that
$1/|x|^{n-1}\in L^{q}(B_{2})$ for $q>n/(n-1)$.

Harnack’s Inequality

1. Use that for $r\in(0,1/3)$ and $x,y\in B_{r}$

$B_{1-3r}(y)\subseteq B_{1-r}(x)\subseteq B_{1}.$

2. Follows by using $u(0)$ instead of $\fint_{B_{1/3}}u$ in the proof of Lemma 2.6. We also replace $B_{1/3}$ by $B_{1/4}$.

Concave/Convex Functions

1. Consider an upper supporting plane for $u$ at $x_{0}\in\operatorname*{argmin}_{\overline{B_{r}}}u\subseteq\partial B_{r}$.

2. The set $\\{u\geq\sup_{B_{r}}u\\}$ must contain a ball of diameter $(1-r)$.

Weyl’s Lemma

1. Use polar coordinates for the integration defininf the convolution.

2. Using the Taylor expansion around zero for $x=\varepsilon\theta$ and $\theta\in B_{1}$

$u(x)-u(0)=\varepsilon Du(0)\cdot\theta+\frac{\varepsilon^{2}}{2}\theta\cdot
D^{2}u(0)\theta+o(\varepsilon^{2}).$

The first term is odd and its integral over $B_{\varepsilon}$ vanishes. The
last term is an error that vanishes faster than $\varepsilon^{2}$ as
$\varepsilon\to 0$. Hence we just keep the quadratic terms in the limit. Once
again, we notice the odd symmetry of $\theta_{i}\theta_{j}$ for $i\neq j$ to
keep just the quadratic terms of the form $\theta_{i}^{2}$ in the integration.

By using the spherical symmetry once again

$\int_{B_{1}}\theta_{i}^{2}d\theta=\frac{1}{n}\int_{B_{1}}|\theta|^{2}d\theta=\frac{|\partial
B_{1}|}{n}\int_{0}^{1}r^{n+1}dr=\frac{|B_{1}|}{n+1}.$

Details can be consulted in [HL11, Theorem 1.8 in Section 1.2].

Boundary Regularity

1. Show first that the following variation of the mean value property for $v=\min(u,0)$

$\fint_{B_{1}}v\leq v(0)+C\|(\Delta u)_{+}\|_{L^{p}(\\{u<0\\}\cap B_{1})}.$

Then the Hölder modulus of continuity will follow from the diminish of
oscillation approach, just like it was done for the interior Hölder estimate.

For $\varepsilon>0$, consider $\phi_{\varepsilon}\in
C^{\infty}({\mathbb{R}})$, non-decreasing and concave, such that

$\phi_{\varepsilon}(z)=\begin{cases}-z\text{ if }z\leq-\varepsilon,\\\ 0\text{
if }z\geq 0.\end{cases}$

Then we apply the mean value property to
$v_{\varepsilon}:=\varphi_{\varepsilon}(u)$ and conclude after sending
$\varepsilon\to 0^{+}$. Notice that

$(\Delta v_{\varepsilon})_{+}=(\phi_{\varepsilon}^{\prime}\Delta
u+\phi_{\varepsilon}^{\prime\prime}|Du|^{2})_{+}\leq\mathbbm{1}_{\\{u<0\\}}(\Delta
u)_{+}.$

Local Maximum Principle

1. Let $\theta$ to be fixed sufficiently small and assume by contradiction that $u<(1+\theta)u(0)$ in $B_{1}$. The idea is that $\|u_{+}\|_{L^{\varepsilon}(B_{1})}\leq 1$ and the mean value property for $-u$ compete in opposite directions when we estimate the measure of $\\{u\geq u(0)/2\\}\cap B_{1/3}$.

Let $M\geq 1$ be sufficiently large such that by Markov’s inequality

(6.11) $\displaystyle\frac{|\\{u\geq u(0)/2\\}\cap
B_{1/3}|}{|B_{1/3}|}\leq\frac{(u(0)/2)^{-\varepsilon}}{|B_{1/3}|}\|u_{+}\|_{L^{\varepsilon}(B_{1})}^{\varepsilon}\leq
CM^{-\varepsilon}<\frac{1}{2}.$

By a combination of the Markov inequality and the mean value property applied
to $v=(1+\theta)u(0)-u>0$

$\displaystyle\frac{|\\{u<u(0)/2\\}\cap B_{1/3}|}{|B_{1/3}|}$
$\displaystyle\leq\frac{|\\{v>u(0)/2\\}\cap B_{1/3}|}{|B_{1/3}|}$
$\displaystyle\leq 2u(0)^{-1}\fint_{B_{1/3}}v$ $\displaystyle\leq
2u(0)^{-1}(\theta u(0)+C),$ $\displaystyle\leq 2\theta+CM^{-1}.$

By taking $M$ sufficiently large $\theta$ sufficiently small we enforce the
right-hand side to be less than $1/2$, contradicting (6.11).

2. The idea proceeds once again by contradiction. We will use the previous part to obtain a sequence of points $\\{x_{k}\\}\subseteq B_{3/4}$ where $u(x_{k})\geq(1+\theta)u(x_{k-1})\geq\ldots\geq(1+\theta)^{k}M\to\infty$, contradicting the continuity of $u$. To prevent that the points escape from $B_{3/4}$, we will also show that we can make this choice with $|x_{k}-x_{k-1}|$ controlled by a geometric vanishing sequence.

The same argument as in the previous part shows that for some $\sigma\geq 1$
sufficiently large, it holds that if $B_{3r}(x_{0})\subseteq B_{1}$ with
$r=\sigma u(x_{0})^{-\varepsilon/n}$, then

$\begin{cases}\|u_{+}\|_{L^{\varepsilon}(B_{1})}\leq 1,\\\ \|(\Delta
u)_{-}\|_{L^{p}(B_{1})}\leq 1,\\\ u(x_{0})\geq
M,\end{cases}\qquad\Rightarrow\qquad\sup_{B_{r}(x_{0})}u\geq(1+\theta)u(x_{0}).$

Indeed, we only need to modify the upper estimate on the measure of $\\{u\geq
u(x_{0})/2\\}\cap B_{r}(x_{0})$ by

$\frac{|\\{u\geq u(x_{0})/2\\}\cap B_{r}(x_{0})|}{|B_{r}(x_{0})|}\leq
C\sigma^{-n}<\frac{1}{2}.$

By choosing $M$ even larger we can enforce that

$\sigma
M^{-\varepsilon/n}\sum_{k=0}^{\infty}(1+\theta)^{-k\varepsilon/n}\leq\frac{1}{4}.$

Then, if we assume that $u(x_{0})\geq M$ for some $x_{0}\in B_{1/2}$ we can
then construct a sequence of points $x_{1},x_{2},\ldots$ inductively such that

$x_{k+1}\in\operatorname*{argmax}_{\overline{B_{r_{k}}(x_{k})}}u,\qquad
r_{k}:=\sigma u(x_{k})^{-\varepsilon/n}.$

This is the desired sequence which leads to the expected contradiction.

Details can be consulted in [CC95, Lemma 4.7 and the proof of Lemma 4.4].

Heat Equation

1. (Mean Value Property) The main idea is to consider $v=\eta u$, for some non-negative and compactly supported cut-off function $\eta\in C^{\infty}_{c}(B_{1}\times(-1,0])$, so that $v$ would now satisfy the heat equation in ${\mathbb{R}}^{n}\times(-\infty,0]$.

Given that

$\partial_{t}v-\Delta v=\eta f+(\partial_{t}\eta-\Delta\eta)u-2D\eta\cdot Du.$

Then, if $\eta(0,0)=1$,

$\displaystyle v(0,0)=u(0,0)$
$\displaystyle=\iint(f\eta+(\partial_{t}\eta-\Delta\eta)u-2D\eta\cdot
Du)H_{-t}dxdt$ $\displaystyle=\iint(f\eta
H_{-t}+[(\partial_{t}\eta+\Delta\eta)H_{-t}+2D\eta\cdot DH_{-t}]u)dxdt$
$\displaystyle=\iint(f\eta H_{-t}+[(\partial_{t}\eta+\Delta\eta)+t^{-1}x\cdot
D\eta]H_{-t}u)dxdt.$

Hence we take

$\begin{cases}K_{2}=\eta H_{-t},\\\
K_{1}=[(\partial_{t}\eta+\Delta\eta)+t^{-1}x\cdot D\eta]H_{-t}.\end{cases}$

We can easily see that $K_{2}\geq 0$, now we need to choose $\eta$ such that
$(\partial_{t}\eta+\Delta\eta)+t^{-1}x\cdot D\eta\geq 0$. The function
$\eta=(1-2|x|^{2}+4nt)_{+}$ does the work, but it is not smooth and the
computations above can not be directly applied. Instead we may consider a
modification of this candidate by approximating the positive part with a
smooth function as was done for example in the solution of Problem 5.

See [Eva10, Chapter 2.3] for an alternative way to construct the mean value
property for the heat equation.

2. (Weak Harnack) In the previous construction we have that for some $\rho\in(0,1/2)$ sufficiently small, $\min_{\overline{B_{\rho}}\times[-\rho,-2\rho]}K_{1}>0$. The singularity of the heat kernel at the origin gives that

$\iint fK_{2}\leq C\|f\|_{L^{q}_{t}((-1,0]\to L^{p}_{x}(B_{1}))}.$

Then

$\iint_{B_{\rho}\times(-\rho,-2\rho]}u\leq
C(u(0,0)+\|f_{-}\|_{L^{q}_{t}((-1,0]\to L^{p}_{x}(B_{1}))}).$

The conclusion can finally be obtained from this estimate by a geometric
argument, similar to the one illustrated in Figure 3 and adapted to the
parabolic setting.

Sun Rising Lemma

1 and 2. Details can be consulted in [SS05, Section 3.3].

Maximum Principles

Figure 22. For the ABP maximum principle we consider the hyper-planes that
support the graph of $u$ from below. Any plane with slope less than $m/2$ can
be slid from below until it touches the graph of $u$.

1. The geometric idea is illustrated in Figure 22. For $m=\max_{\overline{B_{1}}}u_{-}$ and $T=Du\colon A\to{\mathbb{R}}^{n}$ we get that $B_{m/2}\subseteq T(A)$. The Jacobian of $T$ for $x_{0}\in A$ can be estimated using that if $e\in\operatorname{eig}(D^{2}u(x_{0}))$ then $0\leq\lambda e\leq\mathcal{P}_{\lambda,\Lambda}^{-}(D^{2}u(x_{0}))$. Therefore $0\leq\det(DT(x_{0}))\leq C(\mathcal{P}^{-}_{\lambda,\Lambda}(D^{2}u(x_{0})))^{n}$.

Details can be consulted in [CC95, Chapter 3].

2. Consider the mapping as above using the set of slopes in the convex envelope of

$B_{|u(x_{0})|/\operatorname*{diam}(\Omega)}\cup\\{|u(x_{0})|/\operatorname*{dist}(x_{0},\partial\Omega)\\}.$

Details can be consulted in [Fig17, Theorem 2.8].

Inf/Sup-Convolutions.

1. Follows directly from the definition.

2. We focus on a compact set $\overline{B_{r}(x_{0})}\subseteq\Omega$ where $u$ is bounded and uniformly continuous. Let us say that $M:=\operatorname*{osc}_{B_{r}(x_{0})}u<\infty$, and $\omega$ is an admissible modulus of continuity for $u$ over $B_{r}(x_{0})$.

Then we have that for $\rho:=\sqrt{2M\varepsilon}<r$ and $y_{0}\in
B_{r-\rho}(x_{0})$, the infimum that defines $u_{\varepsilon}$ is actually a
minimum that is achieved over $\overline{B_{\rho}(y_{0})}$. Let

$x_{0}\in\operatorname*{argmin}_{\overline{B_{\rho}(y_{0})}}(u-\varphi_{y_{0}}^{1/\varepsilon}).$

Then

$0\leq
u(y_{0})-u_{\varepsilon}(y_{0})=u(y_{0})-u(x_{0})-\frac{1}{2\varepsilon}|x_{0}-y_{0}|^{2}\leq
u(y_{0})-u(x_{0})\leq\omega(\rho)=\omega(\sqrt{2M\varepsilon}).$

3. One can notice that $x\mapsto u_{\varepsilon}(x)+|x|^{2}/(2\varepsilon)$ is the infimum of affine functions. Alternatively one can show that every point in the graph of $u_{\varepsilon}$ has a supporting paraboloid of the form $\varphi_{x_{0}}^{-1/\varepsilon}+C$ from above, see Figure 23.

4. Follows from a direct computation.

5. Prove that for every $y_{0}\in{\mathbb{R}}^{n}$, $(\varphi_{y_{0}}^{1/\varepsilon}+u_{\varepsilon}(y_{0}))$ is the largest paraboloid centered at $y_{0}$, and with Hessian $-\varepsilon^{-1}I$, that remains below $u$ in $\Omega$.

Measure Estimates for Operators with a Drift Term

1\. and 2. (Measure Estimate and Localization with Small Drift) Notice that in
the proof of Lemma 3.8 and Lemma 3.10 we get that the contact set is contained
in a set where $|Du|$ is universally bounded. Hence, the drift term can be
absorbed once we assume that $\|b\|_{L^{n}(B_{1})}$ is sufficiently small.

Figure 23. The paraboloid $\varphi_{x_{0}}^{-1/\varepsilon}$ must be tangent
to $u_{\varepsilon}$ from above. Otherwise we would be able to find at least
one vertex of the form $(y_{1},u_{\varepsilon}(y_{1}))$ strictly on top of the
graph of $\varphi_{x_{0}}^{-1/\varepsilon}$. However, by symmetry of the
paraboloids we observe that in such case the paraboloid
$\varphi_{y_{1}}^{1/\varepsilon}+u_{\varepsilon}(y_{1})$ must be larger than
$u$ at $x_{0}$, which contradicts the definition of $u_{\varepsilon}(y_{1})$.

3. (ABP Maximum Principle) Let $m=\max_{\overline{B_{1}}}u_{-}$ and consider for $r\in(0,1/2]$, the contact sets

$A_{r}:=\bigcup_{p\in B_{rm}\setminus
B_{rm/2}}\operatorname*{argmin}(u-\varphi_{p}).$

Show then that for some $C>1$, independent of $r$,

$1\leq C(\|b\|^{n}_{L^{n}(A_{r})}+(rm)^{-n}\|f\|_{L^{n}(A_{r})}^{n}).$

Add then the results for $r=1/2,1/4,\ldots,1/2^{k}$ such that $k=\lfloor
2C\Lambda^{n}\rfloor+1$, and use that $\|b\|_{L^{n}(B_{1})}\leq\Lambda$.

The proof in [HL11, Theorem 2.21] also relies on giving different weights to
the contact sets $A_{r}$

4. (Localized Measure Estimate) The idea is to localize the minimum to some $B_{\varepsilon}(x_{0})\subseteq B_{\rho}$ such that $\|b\|_{L^{n}(B_{\varepsilon}(x_{0}))}$ is sufficiently small and $\min_{\overline{B_{\varepsilon}(x_{0})}}u$ is not too large, both quantities measured in terms of the universal constants. After having done this, the result in part 1 concludes the argument.

The condition that $\|b\|_{L^{n}(B_{\varepsilon}(x_{0}))}$ is small can be
arranged from a geometric argument. The maximum principle allows to find an
annular region $B_{r+\varepsilon}\setminus B_{r-\varepsilon}\subseteq
B_{1}\setminus B_{1/2}$ such that $\|b\|_{L^{n}(B_{r+\varepsilon}\setminus
B_{r-\varepsilon})}$ is sufficiently small and $\min_{\partial B_{r}}u\leq 2$.
Finally $B_{\varepsilon}(x_{0})$ and $B_{r+\varepsilon}\setminus
B_{r-\varepsilon}$ can be connected by some thin cylinder $T$ where
$\|b\|_{L^{n}(T)}$ is also small. This geometric arrangement allows to
transmit the minimum around $\partial B_{r}$ to
$\overline{B_{\varepsilon}(x_{0})}$ using the localization lemma with small
drift.

Details can be consulted in [Moo15, Proposition 6.1].

Measure Estimates for Parabolic Equations

Figure 24. The test functions are planes of the form $x\mapsto p\cdot x-h$
that at time $t=-1$ are below $u$. We bring them form the past until they
touch the graph of $u$ at some time $t_{0}$ and some point $x_{0}$.

1. (ABP Maximum Principle) The test functions and contact set are illustrated in the Figure 24. Let $T\colon A\to{\mathbb{R}}^{n}\times{\mathbb{R}}$ such that $(p_{0},h_{0})=T(x_{0},t_{0})$ if and only if $u>\varphi_{p_{0},h_{0}}$ in $B_{1}\times(-1,t_{0})$, with equality at $(x_{0},t_{0})$. This means that $T=(Du,x\cdot Du-u)$.

For $m=\max_{\overline{B_{1}}}u_{-}$ we get that

$\\{(p,h)\in{\mathbb{R}}^{n}\times{\mathbb{R}}\,:\,h\in(0,m/2),p\in
B_{h/2}\\}\subseteq T(A).$

The Jacobian of $T$ is computed by

$\det\begin{pmatrix}D^{2}u&D(\partial_{t}u)\\\ x^{T}D^{2}u&x\cdot
D(\partial_{t}u)-\partial_{t}u\end{pmatrix}=-\partial_{t}u\det(D^{2}u).$

For $(x_{0},t_{0})\in A$ and $e\in\operatorname{eig}(D^{2}u(x_{0},t_{0}))$

$0\leq\lambda e\leq\partial_{t}u(x_{0},t_{0})+f(x_{0},t_{0})\leq
f(x_{0},t_{0}).$

Therefore $0\leq\det(D_{x,t}T)\leq Cf^{n+1}$ over the contact set.

Details can be consulted in [Tso85].

2. (Measure Estimate) Let $\varphi_{y_{0},s_{0}}(x,t):=(t-s_{0})-2|x-y_{0}|^{2}$ for $(y_{0},s_{0})\in{\mathbb{R}}^{n}\times{\mathbb{R}}$. Notice that $\varphi_{0,-1}(0,0)>0$ and $\varphi_{0,-1}<0$ in $\partial_{\text{par}}(B_{1}\times(-1,0])$. For some $\rho\in(0,1)$ sufficiently small and $(y_{0},s_{0})\in B:=B_{\rho}\times[-1,-1+\rho)$, we still have that $\varphi_{y_{0},s_{0}}<0$ in $\partial_{\text{par}}(B_{1}\times(-1,0])$ and $\theta:=\min_{(y_{0},s_{0})\in B}\varphi_{y_{0},s_{0}}(0,0)>0$.

If $u$ is non-negative and $u(0,0)\leq\theta$, then for every
$(y_{0},s_{0})\in B$, the graph of $\varphi_{y_{0},s_{0}}$ must eventually
touch the graph of $u$ from below at some point in $B_{1}\times(-1,0]$.
Consider in this way the contact set

$\displaystyle A:=\\{(x_{0},t_{0})\in B_{1}\times(-1,0]\,:\,\text{For some
$(y_{0},s_{0})\in B$,}$ $u>\varphi_{y_{0},s_{0}}$ in $B_{1}\times(-1,t_{0})$,
$\displaystyle\text{and
$u(x_{0},t_{0})=\varphi_{y_{0},s_{0}}(x_{0},t_{0})$}\\},$

and the surjective map $T\colon A\to B$ such that
$(y_{0},s_{0})=T(x_{0},t_{0})$ if and only if $u>\varphi_{y_{0},s_{0}}$ in
$B_{1}\times(-1,t_{0})$, with equality at $(x_{0},t_{0})$. This means that

$T=\left(x+\frac{1}{4}Du,t-u-\frac{1}{8}|Du|^{2}\right),$

and its Jacobian is

$\det(D_{x,t}T)=\left(1-\partial_{t}u\right)\det\left(I+\frac{1}{4}D^{2}u\right).$

Using the uniform ellipticity we get that

$0\leq\det(D_{x,t}T)\leq C(1+f^{n+1}).$

3. (Localization) The argument is an adaptation of the Lemma 3.10. It now requires to design a test function $\varphi=\varphi(x,t)\colon B_{1+\rho/2}\times(-1-\rho/2,0]\to{\mathbb{R}}$ such that

$\begin{cases}\varphi<0\text{ in }B_{1+\rho/2}\times(-1-\rho/2,0]\setminus
B_{1-\rho/2}\times(-1,0],\\\ \varphi>0\text{ in
}\overline{B_{1/2+\rho/2}}\times[-1/2-\rho/2,0],\\\
\partial_{t}\varphi-\mathcal{P}^{-}_{\lambda,\Lambda}(D^{2}\varphi)<0\text{ in
}B_{1+\rho/2}\times(-1-\rho/2,0]\setminus
B_{\rho/2}\times(-1-\rho/2,-1+\rho/2].\end{cases}$

Notice that these characteristics resemble the backwards fundamental solution
centered at $(0,-1)$ (Figure 25). This problem can be reduced to an ODE
assuming that $\varphi$ is a truncation of a function of the form
$(t+1)^{-\alpha}\psi(|x|^{2}/(t+1))$.

See [IS12, Lemma 4.16] for further details.

Figure 25. The function $\varphi$ is negative in (a parabolic neighborhood of)
$\partial_{\text{par}}(B_{1}\times(-1,0])$ (red region), positive in (a
parabolic neighborhood of) $B_{1/2}\times(-1/2,0]$ (yellow region), and
satisfies
$\partial_{t}\varphi-\mathcal{P}^{-}_{\lambda,\Lambda}(D^{2}\varphi)<0$ in (a
parabolic neighborhood of) $B_{1}\times(-1,0]\setminus
B_{\rho}\times(-1,-1+\rho]$ (complement of the green region). We leave some
room so that for every $(y_{0},s_{0})\in B_{\rho/2}\times[0,\rho/2)$, the
translations $\varphi_{y_{0},s_{0}}(x,t):=\varphi(x-y_{0},t-s_{0})$ keep
similar properties.

4. (Localized Measure Estimate) Follows by combining the previous parts.

Measure Estimates for the Hessian I

1. We can prove it in two steps as in Lemma 3.8 followed by Lemma 3.10.

Let $x_{0}\in A^{1}_{\\{0\\}}\cap\overline{B_{1/2}}$, and consider for
$y_{0}\in B_{1/4}(x_{0})$

$\varphi_{0}^{1}+(\varphi_{y_{0}}^{1}+1/32)=\varphi_{y_{0}/2}^{2}+c_{y_{0}}.$

For $y_{0}\in B_{1/4}(x_{0})$ we have that $\varphi_{y_{0}}^{1}(x_{0})+1/32>0$
and $\varphi_{y_{0}}^{1}+1/32<0$ in ${\mathbb{R}}^{n}\setminus B_{1/2}(x_{0})$
(Figure 26). Therefore

$\emptyset\neq(u-\varphi_{y_{0}/2}^{2})\subseteq B_{1}.$

From this one can show that $|A^{2}_{{\mathbb{R}}^{n}}\cap B_{1}|\geq\eta$.

The localization lemma can be shown modifying the test function from Lemma
3.10. instead of truncating it with a constant we can cap it with a paraboloid
that we fit in a $C^{1,1}$ fashion.

See [CC95, Lemma 7.6] for an alternative proof using the ABP maximum
principle.

Dyadic Covering Lemma for Parabolic Equations

1. (Staking Lemma) We can reduce the problem to the case of zero spatial dimension by using Fubini’s Theorem. Consider then $\mathcal{I}$ to be a collection of disjoint open intervals in $(-1,0)$. For every $I=(t-r,t)$ and $m\geq 1$, let $I^{m}:=(t,t+mr)$. Then we have to show that

$\left|\bigcup_{I\in\mathcal{I}}I^{m}\right|\geq\frac{m}{m+1}\left|\bigcup_{I\in\mathcal{I}}(I^{m}\cup
I)\right|.$

Let $T$ be a mapping between intervals such that for $I=(a,b)$,
$T(I)=(a-(b-a)/m,b)$. In particular, $T(I^{m})=I^{m}\cup I$. Extend $T$ to
unions of disjoint open intervals $E=\bigcup I$ such that $T(E)=\bigcup T(I)$.
We still have that the length of $T(E)$ is less than or equal to $(m+1)/m$
times the length of $E$.

On the other hand, any open set of ${\mathbb{R}}$ can we decomposed in a
unique way as a numerable union of disjoint open intervals. If $E=\bigcup I$
is a union of non-necessarily disjoint intervals, we get by an inductive
argument that $T(E)\supseteq\bigcup T(I)$.

We finally recover the desired estimate by applying the previous observations
to the stack $\bigcup_{I\in\mathcal{I}}I^{m}$

$\frac{m+1}{m}\left|\bigcup_{I\in\mathcal{I}}I^{m}\right|\geq\left|T\left(\bigcup_{I\in\mathcal{I}}I^{m}\right)\right|\geq\left|\bigcup_{I\in\mathcal{I}}T(I^{m})\right|=\left|\bigcup_{I\in\mathcal{I}}I^{m}\cup
I\right|.$

Figure 26. Assume without loss of generality that $\varphi_{0}^{1}$ touches
$u$ from below at $x_{0}\in B_{1/2}$. For $y_{0}\in B_{1/4}(x_{0})$, the
paraboloids $\varphi_{y_{0}}^{1}+1/32$ are given such that $u$ must cross the
sum $\psi_{y_{0}}:=\varphi_{0}^{1}+(\varphi_{y_{0}}^{1}+1/32)$. Indeed
$\psi_{y_{0}}(x_{0})>\varphi_{0}^{1}(x_{0})=u(x_{0})$ and
$\psi_{y_{0}}<\varphi_{0}^{1}\leq u$ in ${\mathbb{R}}^{n}\setminus
B_{1/2}(x_{0})$.

Measure Estimates for the Hessian II

1. The set $\\{m((\mathcal{P}^{-}_{\lambda,\Lambda}(D^{2}u))_{+}^{n})\geq\delta^{n}\\}$ is considered because we need to have that the $L^{n}$-norm of the right-hand side remains small after a rescaling of the form $u_{r}(x)=r^{-2}u(rx+x_{0})$. After this observation, the idea is to use the result from Problem 13.

2. Assume without loss of generality that $\|u\|_{L^{\infty}(B_{3\sqrt{n}})}\leq 1$ and $\|(\mathcal{P}^{-}_{\lambda,\Lambda}(D^{2}u))_{+}\|_{L^{n}(B_{3\sqrt{n}})}\leq\delta$.

The previous part can be used to show the following diminish of the
distribution

$\displaystyle|Q_{1}\setminus A^{M^{k+1}}_{{\mathbb{R}}^{n}}|$
$\displaystyle\leq(1-\eta)|(Q_{1}\setminus
A^{M^{k+1}}_{{\mathbb{R}}^{n}})\cup\\{m((\mathcal{P}^{-}_{\lambda,\Lambda}(D^{2}u))_{+}^{n})\geq(\delta
M^{k})^{n}\\}|$
$\displaystyle\leq(1-\eta)^{k+1}+\sum_{i=0}^{k}(1-\eta)^{k-i}|\\{m((\mathcal{P}^{-}_{\lambda,\Lambda}(D^{2}u))_{+}^{n})\geq(\delta
M^{i})^{n}\\}|$

Finally use the Hardy–Littlewood maximal inequality to bound the distribution
of the maximal function in terms of $\delta$.

Details can be consulted in [CC95, Lemma 7.7 and Lemma 7.8].

Fractional Laplacian

2. (Weak Harnack) Let $\beta\in C^{\infty}_{c}(B_{5/8})$ radial bump function taking values between $0$ and $1$, and such that $\beta=1$ in $B_{1/2}$. Let $\theta\in(0,1]$ such that the graph of $\theta\beta$ touches the graph of $u$ from below at $x_{0}\in B_{5/8}$, and let

$v=u(x_{0})\beta+\mathbbm{1}_{B_{1}\setminus B_{3/4}}u.$

(Figure 27).

Figure 27. The function $v$ is defined piece-wise as a small bump that touches
$u$ from below in $B_{5/8}$, and it is equal to $u$ outside $B_{3/4}$. Then
the comparison principle between $u$ and $v$ leads to the desired measure
estimate.

Then the desired estimate can be completed from the following computation

$1\geq\Delta^{\sigma/2}u(x_{0})\geq\Delta^{\sigma/2}v(x_{0})=u(x_{0})\Delta^{\sigma/2}\beta(x_{0})+C\int_{B_{1}\setminus
B_{3/4}}\frac{u(y)}{|y-x_{0}|^{n+\sigma}}dy.$

3. (Hölder Estimate) The challenge in the implementation of the diminish of oscillation is the non-local aspect of the equation. Even if we assume without loss of generality that $u$ takes values between $0$ and $1$ over $B_{1}$, the weak Harnack estimate we showed requires $u$ to be positive in the whole of ${\mathbb{R}}^{n}$.

First, we can truncate the tail of the solution by considering instead
$v=\mathbbm{1}_{B_{1}}u$, this introduces an additional term to the right-
hand, bounded by a multiple of
$\pm\int_{{\mathbb{R}}^{n}}|u|/(1+|x|^{n+\sigma})dx$.

We assume in an inductive way and without loss of generality that $u$
satisfies for some small $\alpha,\delta\in(0,1)$

$\begin{cases}|u|\leq(1+\delta)|x|^{\alpha}\text{ in
}{\mathbb{R}}^{n}\setminus B_{1},\\\ |u|\leq 1\text{ in }B_{1},\\\
\|(-\Delta)^{\sigma}u\|_{L^{\infty}(B_{1})}\leq\delta.\end{cases}$

Then we consider whether $u$ is positive or negative at least half of the
measure in $B_{1}\setminus B_{3/4}$. If we have that $|\\{u\geq
0\\}\cap(B_{1}\setminus B_{3/4})|\geq|B_{1}\setminus B_{3/4}|/2$, then we
translate and truncate the function as $v=(u+1)_{+}$. For $\alpha$
sufficiently small we can assure that the right-hand side is also small. Then,
by the weak Harnack estimate there is $\theta\in(0,1)$ such that
$u+1=v\geq\theta>0$ in $B_{1/2}$.

We need to focus on the relation between the constants $\alpha$, $\delta$, and
$\theta$, such as the inductive step holds. Assume that we improved the lower
bound as in the previous paragraph. Then we consider
$v(x)=(u(x/2)-\theta/2)/(1-\theta/2)$ such that $|v|\leq 1$ in $B_{1}$.

To keep $|v|\leq(1+\delta)|x|^{\alpha}$ in $B_{1}\setminus B_{2}$ we ask that
$(2-\theta-2^{1-\alpha})2^{\alpha+1}\geq\theta(2-\theta)$. This can be
achieved by choosing $\theta$ sufficiently small in terms of $\alpha$.

Details can be consulted in [Sil06].

Interior Gradient Estimates

1. Follows as in the proof of Corollary 2.10.

Local Maximum Principle - Uniformly Elliptic

1. Follows as in the solution of Problem 6.

Harnack Inequality - Uniformly Elliptic

1. Follows by combining the weak Harnack inequality and the local maximum principle. Details can be consulted in [CC95, Section 4.2]

Interior Hessian Estimates for Convex Operators

1. $v:=-v_{2}$ satisfies $\mathcal{P}^{+}_{\lambda,\Lambda}(D^{2}v)\geq 0$, then by the local maximum principle we just need to provide a bound for $\|v\|_{L^{\varepsilon}(B_{1})}$. This follows by the measure estimate for the Hessian (Problem 15) with a slightly larger exponent.

2. We have to notice that $v_{A}$ is the difference of a convex combination and $u$ and $u$ itself,

$v_{A}(x)=\lim_{\varepsilon\to
0}\frac{C}{\varepsilon^{2}}\fint_{B_{\varepsilon}}(u(x+Ay)-u(x))dy.$

3. Assume by contradiction that for some projection $P_{0}$ it happens that $v_{P_{0}}$ gets smaller than $-(1-\theta)$ for some $\theta\in(0,1)$ to be fixed sufficiently small.

Given that $v_{P_{0}}\geq-1$ satisfies
$\mathcal{P}_{\lambda,\Lambda}^{-}(D^{2}v_{P_{0}})\leq 0$, we obtain by the
weak Harnack inequality that $|\\{v_{P_{0}}>-1+\theta^{1/2}\\}\cap
B_{1/4}|\leq C\theta^{\varepsilon/2}$.

Over the complementary set $E:=\\{v_{P_{0}}\leq-1+\theta^{1/2}\\}\cap B_{1/4}$
set we obtain that

$0\leq\underbrace{\max_{\begin{subarray}{c}P\in{\mathbb{R}}^{n\times
n}_{\text{sym}}\\\
P^{2}=P\end{subarray}}v_{P}}_{=:v_{+}}-v_{I-P_{0}}=v_{P_{0}}-\underbrace{\min_{\begin{subarray}{c}P\in{\mathbb{R}}^{n\times
n}_{\text{sym}}\\\ P^{2}=P\end{subarray}}v_{P}}_{=:-v_{-}}\leq\theta^{1/2}$

By uniform ellipticity and $F(0)=0$,

$0\leq\mathcal{P}^{+}_{\lambda,\Lambda}(D^{2}u)=\Lambda v_{+}-\lambda v_{-}.$

Then for $\theta$ sufficiently small we get that
$v_{I-P_{0}}\geq\lambda/(2\Lambda)$ in $E$.

However, $w:=\lambda/(2\Lambda)-v_{I-P_{0}}$ satisfies
$\mathcal{P}^{+}_{\lambda,\Lambda}(D^{2}w)\geq 0$ and then by the local
maximum principle we obtain the following contradiction (for $\theta$ chosen
sufficiently small)

$\lambda/(2\Lambda)=w(0)\leq C\|w_{+}\|_{L^{1}(B_{1/4})}\leq
C\theta^{\varepsilon/2}.$

Details can be consulted in [CS10].

Imbert-Silvestre’s Approach

1. Details can be consulted in [IS16, Proposition 3.3].

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