# Unconstrained Dysfluency Modeling for Dysfluent Speech Transcription and
Detection

###### Abstract

Dysfluent speech modeling requires time-accurate and silence-aware
transcription at both the word-level and phonetic-level. However, current
research in dysfluency modeling primarily focuses on either transcription or
detection, and the performance of each aspect remains limited. In this work,
we present an unconstrained dysfluency modeling (UDM) approach that addresses
both transcription and detection in an automatic and hierarchical manner. UDM
eliminates the need for extensive manual annotation by providing a
comprehensive solution. Furthermore, we introduce a simulated dysfluent
dataset called VCTK++ to enhance the capabilities of UDM in phonetic
transcription. Our experimental results demonstrate the effectiveness and
robustness of our proposed methods in both transcription and detection tasks.

Index Terms: dysfluent speech, transcription, detection

##  1 Introduction

In the field of speech analysis, a clear and universally agreed-upon
definition of dysfluent speech is yet to be established. Dysfluencies are
commonly associated with speech disorders such as stuttering, aphasia [1], and
dyslexia [2], characterized by disruptions in the normal flow of speech [3].
However, dysfluency also appears in normal conversational speech or
spontaneous speech [4], where individuals may experience hesitations or
interruptions while speaking. In this context, we refer to dysfluent speech as
any form of speech characterized by abnormal patterns such as repetition,
prolongation, and irregular pauses, as discussed in [5]. Within the domain of
dysfluent speech modeling, there is also a lack of a unified paradigm.
Research efforts in this area can be roughly categorized as transcription and
detection.

Current state-of-the-art word transcription models [6, 7, 8, 9, 10] and
phonetic transcription models [11] often struggle to accurately transcribe
dysfluent speech. As a result, human annotations are still required and
transcribers commonly resort to a two-step process. They first obtain
automatic transcriptions and then manually annotate the missing portions, a
laborious and time-consuming task that is comparable to manual annotation from
scratch. WhisperX[12] recently extends Whisper [6] by generating timestamps
for individual words. However, it still delivers limited performance for
dysfluent speech. Moreover, time-accurate phonetic transcriptions might be a
better representation to capture various dysfluency types. Another requirement
is that phonetic transcriptions should be sensitive to silences as it might
indicate a block or poor breath-speech coordination. [5] recently proposed a
time-accurate and silence-aware neural forced aligner, where a weighted
finite-state transducer (WFST) is introduced for modeling dysfluency patterns
such as repetition. However, this approach assumes that there is minimal
deviation between the reference and ”real” transcribed text. In real-life
dysfluent speech, such as the example shown in Figure 1, this assumption may
not hold true.

Research on dysfluency detection has traditionally been conducted
independently of dysfluency transcription and has recently been dominated by
end-to-end methods. These approaches typically focus on either utterance-level
detection [13, 14, 15, 16], or frame-level detection [17, 18]. However, these
studies primarily address data-driven classification problems and do not
explicitly incorporate dysfluency transcription into their detection methods.
A unified framework that integrates dysfluency transcription and detection is
essential to develop an efficient and robust dysfluency modeling system.

In this study, we propose an unconstrained dysfluency modeling (UDM) approach
that integrates dysfluent speech transcription and pattern detection in an
automatic manner with no human effort. Since real dysfluent speech is
unconstrained and word transcription is unknown (as shown in the ”Human
Transcription” in Figure 1, which is largely different from reference text),
we develop hierarchical transcription methods. Firstly, we introduce an
unconstrained forced aligner with a dynamic alignment search module to
generate text-independent alignments. Secondly, we propose a Text Refresher
that leverages the alignment input to refine the state-of-the-art Whisper [6]
transcription. For the detection component, we employ 2D-alignment to
automatically detect various phonetic and word-level dysfluency patterns,
including repetition, missing, replacement, insertion, deletion, and irregular
pauses. To further enhance performance, we curate a dysfluent dataset called
VCTK++ to boost the capacity of our unconstrained forced aligner (UFA).
Experimental results demonstrate the effectiveness of our proposed framework
in both dysfluent speech transcription and dysfluency pattern detection.

Fig. 1: Unconstrained Dysfluency Modeling (Transcription and Detection) for
aphasia speech instance. Here is an example of aphasia speech. The reference
text is ”You wish to know all about my grandfather,” while the human
transcription or ground truth differs significantly from the reference.
Whisper recognizes it as perfect speech, while UFA is able to capture most of
the dysfluency patterns. An audio sample of this can be found here111Audio
sample for Fig.1:https://shorturl.at/rTW26.

##  2 Unconstrained Dysfluency Transcription

###  2.1 Unconstrained Forced Aligner (UFA)

Fig. 2: UFA Module

Unconstrained forced aligner predicts alignment with weak text supervision. As
shown in Fig. 2, a speech segment is passed into WavLM [19] encoder which
generates latent representations. A conformer module [20] is followed to
predict both alignment and boundary information. The alignment and boundary
targets used in UFA are derived from the Montreal Forced Aligner (MFA) [21].
This integration with MFA allows UFA to operate with weak text supervision.
During the inference stage, there is no need for text input, making the
alignment process unconstrained. The conformer module comprises of four
conformer [20] encoder layers. The hidden size, number of attention heads,
filter size, and dropout for each conformer layer are [1024, 4, 5, 0.1],
[1024, 8, 3, 0.1], [1024, 8, 3, 0.1], [1024, 4, 3, 0.1] respectively. Two
linear layers are simply applied as phoneme classifier and boundary predictor.
For the phoneme classifier, UFA optimizes the softmax cross-entropy objective,
while logistic regression is utilized for boundary prediction. Specifically,
it predicts floating numbers between 0 (non-boundary) and 1 (boundary).

####  2.1.1 Dynamic Alignment Search

For the inference of dysfluent speech, such as aphasia speech, obtaining an
accurate text transcription is often not achievable (as shown in the ”Human
Transcription” in Figure 1). Consequently, alignment should be decoded without
text supervision. We propose a boundary-aware dynamic alignment search
algorithm, which is the extension of Viterbi algorithm. Let us denote the
phoneme logits as logits $\in\mathbb{R}^{B,T,D}$, the boundary predictions as
boundaries $\in\mathbb{R}^{B,T}$, and the bi-gram phoneme language model as
transition_probs $\in\mathbb{R}^{D,D}$, where $(B,T,D)$ represents the batch
size, time steps, and phoneme dictionary size, respectively. The search
algorithm is outlined in Algorithm 1.

It is worth noting that the transitions between consecutive phonemes near the
boundary should be assigned lower importance to mitigate the risk of phoneme
omissions. For instance, consider the correct alignment as SIL SIL SIL Y Y Y.
In some cases, when the predicted probability for ”Y” is low, there is a
possibility that the prediction of ”Y” might be overlooked due to the higher
self-transition probability of SIL. Consequently, the final prediction could
erroneously become SIL SIL SIL SIL SIL SIL. The bi-gram phoneme language model
is derived by applying maximum likelihood estimation to the VCTK [22] forced
alignment obtained from MFA [21].

Algorithm 1 Boundary-Aware Dynamic Alignment Search

1:procedure Decode(logits, boundaries, transitional_probs)

2: $B,T,D\leftarrow$ shape of logits

3: Initialize trellis and backpointers

4: for $t$ in range($1$, $T$) do

5: for $d$ in range($D$) do

6: trellis[:, $t$, $d$], backpointers[:, $t$, $d$] $\leftarrow$
max_argmax(trellis[:, $t-1$, :] $+$ $(1-\textit{boundaries})$[:, $t$] $\times$
transition_probs[$d$, :])

7: end for

8: end for

9: Derive best_path from trellis and backpointers

10: return best_path

11:end procedure

###  2.2 ASR Transcription and Text-Refresher

State-of-the-art ASR models [6, 8, 23] are commonly trained using a robust
language model constraint, ensuring a high level of accuracy in transcribing
dysfluent or disordered speech, thereby generating nearly perfect
transcriptions. However, to perform word-level dysfluency analysis, it is
necessary to introduce imperfections. In this study, we propose Text Refresher
to achieve this objective.

First, we obtain a perfect transcription using Whisper-large [6]. We then
obtain its corresponding phoneme transcription using CMU dictionary [24].
Subsequently, in Text Refresher, we perform Dynamic Time Warping (DTW) between
the phoneme transcription of the Whisper output and the output of the
Unsupervised Forced Aligner (UFA). Our primary focus is on identifying
insertions and deletions. If a word (represented as a phoneme sequence) in the
Whisper output does not align with the correct word (phoneme sequence) in the
UFA output, we remove that word. For example, in the case illustrated in
Figure 1, the word ”to” is deleted. On the other hand, if a word (phoneme
sequence) in the UFA output does not align with any word (phoneme sequence) in
the Whisper output, we insert that word. Our observations indicate that in
Aphasia speech, most word-level imperfections that Whisper cannot transcribe
are primarily from deletions or insertions.

###  2.3 Unconstrained Transcription Evaluation

####  2.3.1 Phonetic Transcription(Alignment) Evaluation

In order to evaluate how accurately the speech is transcribed at the frame
level, we report Micro F1 Score and Macro F1 Score of phoneme transcription.
Note that our F1 scores evaluate how many phonemes are correctly predicted.
This is different from [25] which evaluates how many time steps are correctly
predicted as phonetic boundaries. In order to evaluate the phoneme
segmentation performance within our methods, we propose the duration-aware
phoneme error rate (dPER). dPER extends Phoneme Error Rate (PER) by weighing
each operation (substitution, insertion, deletion) with its duration. Denote
$\hat{S},\hat{I},\hat{D},\hat{C}$ as the weighted value of substitutions,
insertions, deletions, and correct samples. Denote $p_{i}$ and $p_{j}$ as the
current two phonemes we are comparing in the reference sequence and prediction
sequence respectively. Denote $d(p_{i})$ and $d(p_{j})$ as their durations
(number of repetitions). Whatever the error type is detected, we propose the
following updating rule: $\hat{S}\rightarrow\hat{S}+d(p_{i})+d(p_{j})$,
$\hat{I}\rightarrow\hat{I}+d(p_{j})$, $\hat{D}\rightarrow\hat{D}+d(p_{i})$,
$\hat{C}\rightarrow\hat{C}+|d(p_{i})-d(p_{j})|$. The ultimate formula is:

$\text{dPER}=\frac{\hat{S}+\hat{D}+\hat{I}}{\hat{S}+\hat{D}+\hat{C}}$

(1)

####  2.3.2 Imperfect Word Transcription Evaluation

In contrast to conventional ASR tasks, evaluating the performance in word-
level dysfluency analysis requires the utilization of imperfect word targets.
In this study, we employ manual word annotation of Aphasia speech as the
target reference and report the imperfect Word Error Rate (iWER).

##  3 Unconstrained Dysfluency Detection

We develop rule-based methods for detecting time-accurate phonetic-level
dysfluencies, including Phonetic Errors (Missing, Deletion, Replacement),
Repetition, and Irregular Pause. Additionally, our methods also cover word-
level dysfluencies, including Missing, Insertion, Replacement, and Repetition.

###  3.1 Phonetic-Level Dysfluency Detection

Phonetic-level dysfluency detection is empirically performed by comparing the
predicted forced alignment with the reference ground truth text. The initial
and crucial step in this process involves accurately aligning the forced
alignment with the reference text. This alignment necessitates non-
monotonicity whenever a phonetic dysfluency, such as repetition or insertion,
occurs. Traditional aligners, such as Dynamic Time Warping (DTW) and MFA [21],
cannot handle non-monotonic cases and are inadequate for this purpose.

To address this limitation, we extract the phoneme center embeddings from the
phoneme classifier in the Unconstrained Forced Aligner (UFA) (Fig. 2). By
obtaining the phoneme embedding sequences for both the reference text and the
forced alignment, we compute the dot product between these sequences. As a
result, we generate a 2D similarity matrix that serves as the alignment
representation. In the forced alignment, each phoneme may align with multiple
occurrences of the same phoneme in the reference text, particularly when the
reference text contains repeated phonemes. For instance, in the phrase ”Please
call Stella” represented as ”P L IY Z K AO L S T EH L AH,” each occurrence of
”L” in the forced alignment aligns with all three ”L” phonemes in the
reference text. To ensure that only one phoneme in the reference aligns with
the current phoneme in the forced alignment, we employ Viterbi search on the
2D similarity matrix. This process yields the final 2D alignment, denoted as
alignment-2d, which is primarily monotonic. As illustrated in Figure 1 and
Figure 3, the alignment-2d is visualized through green plots, highlighting the
relationship between the forced alignment and the reference text.

In addition to the alignment-2d, we also require a ground truth alignment-2d,
which represents the expected alignment between the forced alignment of nearly
perfect speech and the reference text. This ground truth alignment is strictly
monotonic. To obtain it, we apply Dynamic Time Warping (DTW) between the
forced alignment and the reference text, resulting in the alignment
represented by the red plots in Figure 3. We denote this as alignment-2D-DTW.

Finally, the detection of phonetic dysfluency becomes straightforward with the
availability of the alignment-2D and alignment-2D-DTW. As illustrated in
Figure 3, in the case of normal speech, these two alignments align perfectly
with each other. However, if there is a significant decrease in the
alignment-2D-DTW while lacking any intersection in the corresponding row, it
indicates a missing phoneme, as depicted in Fig 3 (b). If a row in
alignment-2D-DTW encounters multiple columns in alignment-2D, and there are
repeated phonemes present, it indicates a repetition. This is depicted in
Figure 3 (d). Conversely, if a row in alignment-2D-DTW already aligns with
alignment-2D and simultaneously aligns with the surrounding column in
alignment-2D, it signifies an insertion. This is illustrated in Figure 3 (c).
If a row in alignment-2D-DTW does not overlap with any horizontal regions in
alignment-2D, but only overlaps with a single vertical block in alignment-2D,
it is recognized as a replacement. This is depicted in Figure 3 (e). Lastly,
any pauses occurring within a complete sentence are identified as irregular
pauses, as shown in Figure 3 (f). It should be noted that within this rule-
based detection framework, the precise timing of all five types of
dysfluencies can be accurately identified with a resolution of 20ms.

Fig. 3: Phonetic-Level Dysfluency Detection. Audio samples can be found here.
333Audio sample for Fig.3:https://rb.gy/fmiv7

.

###  3.2 Word-level Dysfluency Detection

We address missing, insertion, replacement, and repetition as part of our
word-level dysfluency detection. To detect word-level dysfluency, we follow a
similar methodology as phonetic-level dysfluency detection, which involves
obtaining alignment-2D and alignment-2D-DTW. However, in the case of word-
level dysfluency detection, we do not utilize word embeddings. Instead, we
employ perfect matching between the words in the reference and predicted
texts, without the need for embedding dot product calculations. Duration,
including silence, is not taken into account in this particular analysis since
it is already incorporated in the phonetic component.

###  3.3 Dysfluency Evaluation

We conduct dysfluency evaluation on segments of Aphasia speech. In each
Aphasia speech segment, manual annotations are made for all types of
dysfluencies, including their accurate timing. For the evaluation of phonetic-
level dysfluency, we report the F1 score (Micro and Macro) for dysfluency type
identification. Additionally, we measure the accuracy of dysfluency detection
in terms of time alignment. To achieve this, we calculate the Intersection
over Union (IoU) between our predicted time boundaries and the ground truth
boundaries. If the IoU is greater than 0.5, the dysfluency is identified as
detected. We also report the F1 score for this matching evaluation, referred
to as the matching score. For the evaluation of word-level dysfluency, we
simply report the F1 score (Micro and Macro) without considering the timing
aspects.

##  4 Experiments

###  4.1 Datasets and Pre-processing

##### VCTK [22]

It is a multi-speaker accented corpus containing 44 hours of fluent speech.
Following [26, 27, 28], we randomly select 90$\%$ of speakers as training set
and the remaining as dev set. VCTK is used to train Unconstrained Forced
Aligner.

##### VCTK++

For each waveform in VCTK and its forced alignment (from MFA [21]), we applied
simulations regarding the following stutter types. (i) Repetitions: Phonemes
are randomly sampled within the waveform, appended by a variable-length sample
of silence, and inserted into the original sound file. The silence sample is
set to vary between 200ms and 500ms in multiples of 20 to match the framerate
of the phoneme alignments. (ii) Prolongations: Phonemes are randomly selected,
excluding phonemes that cannot be reasonably prolonged, such as hard
consonants or silence tokens. The sound sample containing the phoneme is then
stretched by a random factor anywhere from 5x to 10x using Waveform Similarity
Overlap-Add (WSOLA)[29]. The original phoneme is then replaced by the
stretched variant in the waveform. (iii) Blocks: Phonemes are selected from a
list of commonly blocked sounds, such as consonants or combinations of hard
phonemes. With each simulation, we maintain the phoneme alignments such that
the phoneme timestamps line up with the individual stutters, generating new
alignments that act as ground truth for inference. See supplemental material
for details. Here is an example of our augmented data.
https://shorturl.at/aeGKR

Method | WavLM Size | Training Data |  Micro F1 ($\%$, $\uparrow$) |  Macro F1 ($\%$, $\uparrow$) |  dPER ($\%$, $\downarrow$) |  Micro F1 ($\%$, $\uparrow$) |  Macro F1 ($\%$, $\uparrow$) |  dPER ($\%$, $\downarrow$)
---|---|---|---|---|---|---|---|---
| Buckeye Test Set | VCTK++ Test Set
WavLM-CTC-VAD | Large | None | 50.1 | 47.3 | 86.9 | 48.8 | 45.7 | 88.0
WavLM-CTC-MFA | Large | None | 49.8 | 28.7 | 53.9 | 47.6 | 26.0 | 54.2
UFA | Base | VCTK | 68.9 | 55.6 | 53.3 | 78.8 | 59.5 | 53.4
UFA | Base | VCTK+Buckeye | 65.9 | 51.6 | 63.6 | 75.2 | 56.0 | 60.0
UFA | Large | VCTK+Buckeye | 70.3 | 55.0 | 46.2 | 80.7 | 66.4 | 45.8
UFA | Large | VCTK | 71.3 | 60.0 | 46.0 | 81.7 | 72.0 | 44.0
– Boundary-aware | Large | VCTK | 68.9 | 52.0 | 49.9 | 78.4 | 62.9 | 47.8
UFA | Large |  VCTK++ | 73.5 | 64.0 | 41.0 | 93.6 | 90.8 | 38.0
– Boundary-aware | Large |  VCTK++ | 71.0 | 63.7 | 44.3 | 91.1 | 90.0 | 42.1

Table 1: Phonetic transcription evaluation on two dysfluent corpora: Buckeye
and VCTK++ test set.

Method |  iWER ($\%$, $\downarrow$)
---|---
Whisper-Large [6] |  11.3
+Text Refresher |  9.7
+VCTK++ |  9.2

Table 2: Word Transcription Evaluation on Aphasia Speech

##### Buckeye [4]

It contains over 40 hours of recordings from 40 native speakers of American
English. The corpus contains quite a few portions of dysfluent speech with
time-accurate annotation. We follow [25] to make the train/val/test split. The
Buckeye corpus is utilized for training the Unconstrained Forced Aligner and
for Phonetic Transcription Evaluation.

##### Aphasia Speech

From our clinical collaborators, our dysfluent data comprises ten participants
diagnosed with Primary Progressive Aphasia (PPA), which exhibits various
manifestations of dysfluencies. The dataset consists of audio recordings
capturing interactions between patients and speech-language pathologists
(SLPs). Our primary focus lies in the audio input of patients reading the
Grandfather passage, resulting in approximately 20 minutes of speech data. The
Aphasia speech dataset is employed for the evaluation of Imperfect Word
Transcription and Dysfluency Detection.

##### Phonetic Dictionary

We remove stress-aware phoneme labels (e.g. AE0, AE1$\rightarrow$AE). The
phoneme dictionary adopted in this paper contains 39 monophones from CMU
phoneme dictionary [24] along with one additional silence label. For Buckeye
corpus, we manually translate the out-of-dictionary phonemes into CMU
monophones. Here is the translation paradigm: AEN$\rightarrow$AE N,
EYN$\rightarrow$EY N, IYN$\rightarrow$IY N, TQ$\rightarrow$T,
IHN$\rightarrow$IH N, OWN$\rightarrow$OW N, NX$\rightarrow$N,
EHN$\rightarrow$EH N, DX$\rightarrow$T, EN$\rightarrow$AH N,
OYN$\rightarrow$OY N, EM$\rightarrow$EH M, ENG$\rightarrow$EH NG,
EL$\rightarrow$EH L, AAN$\rightarrow$AA N, AHN$\rightarrow$AH N,
AWN$\rightarrow$AW N.

##### Audio Segmentation

For VCTK, we train on the entire utterance without segmentation. For Buckeye
data, we follow [25] to segment the long utterance by the ground truth
transcription. We make sure that the beginning and ending silence length would
be no longer than 3s, resulting in the length of all segments ranging from 2s
to 17s. Different from [25], we keep all silence labels but still remove the
untranscriptable labels such as ’LAUGH’, ’IVER’, etc. For patient speech, we
apply the online Silero VAD [30] with a default threshold of 0.5 to make the
segments. We keep all of the silences and this results in the length of all
segments ranging from 2s to 15s. All audio samples have a sampling rate of 16K
Hz.

###  4.2 Phonetic Transcription Experiments

We train our unconstrained forced aligner (UFA) using three types of data:
VCTK only, VCTK+Buckeye, and VCTK++. Additionally, we conduct an ablation
study to examine the impact of the boundary-aware constraint in the dynamic
search algorithm. This is achieved by removing the constraint from the search
algorithm. Across all experiments, we utilize the same configuration settings,
employing the Adam optimizer with an initial learning rate of 1e-3, which is
decayed by 0.9 at each step. Each model converges after approximately 10
epochs, as determined by achieving a 90$\%$ phoneme classification accuracy on
the development set.

Furthermore, we investigate two alternative forced aligners for comparison
purposes: WavLM-CTC-VAD and WavLM-CTC-MFA. In WavLM-CTC-VAD, we combine the
CTC phoneme alignment [31] obtained from WavLM-CTC [11] with Voice Activity
Detection (VAD) segmentation. By assigning blank tokens and incorporating
silence segments identified using online Silero VAD [30], we obtain a silence-
aware transcription. The VAD threshold is set to the default value of 0.5, and
the minimum and maximum speech durations are defined as 250ms and infinity,
respectively. In WavLM-CTC-MFA, we employ the Montreal Forced Aligner (MFA)
[21] to derive silence-aware phoneme alignment. We utilize WavLM-CTC [11] to
generate the initial phoneme transcription, and we leverage a pre-trained
English ARPA acoustic model. A pronunciation dictionary maps phonemes (as
word-level items) to phonemes (as phonemic pronunciation breakdowns). The
default beam size of 10 is applied for MFA. In the phoneme-to-phoneme
dictionary, the parameters for each phoneme mapping include a pronunciation
probability of 0.99, a silence probability of 0.05, and final silence and non-
silence correction terms of 1.0. For both methods, no additional training data
is needed. Phonetic transcription results are shown in Table. 1.

###  4.3 Imperfect Word Transcription Experiments

We utilize Whisper-large [6] as our ASR transcriber. We begin by presenting
the results obtained directly from Whisper-large. Subsequently, we employ Text
Refresher to refine the Whisper transcription and report the updated results.
By default, Text Refresher incorporates the UFA-WavLM-Large-VCTK alignment.
Additionally, for ablation purposes, we consider the UFA-WavLM-Large-VCTK++
alignment as input, which demonstrated superior performance as indicated in
Table 1. The comprehensive results are presented in Table 2.

###  4.4 Dysfluency Detection

Methods |  F1 ($\%$, $\uparrow$) |  MS ($\%$, $\uparrow$) |  Human F1 ($\%$, $\uparrow$) |  Human MS ($\%$, $\uparrow$)
---|---|---|---|---
UFA-VCTK | 62.4 | 55.2 | 90.4 | 85.6
UFA-VCTK++ | 64.5 | 60.2 | 90.6 | 86.0

Table 3: Phonetic Dysfluency evaluation on Aphasia speech.

The preliminary experiments presented in Table 1 indicate that both WavLM-CTC-
VAD and WavLM-CTC-MFA do not exhibit significant improvements in phonetic
transcription performance. Furthermore, the joint training of the VCTK and
Buckeye corpora does not enhance the overall performance. Hence, we restrict
our evaluation to two variants of the Unconstrained Forced Aligner (UFA): UFA-
WavLM-Large-VCTK and UFA-WavLM-Large-VCTK++. To assess the efficacy of our
rule-based detection algorithm, we also perform manual detection using the
predicted alignment from UFA and human-created targets. The results are
presented in Table 3 and Table 4. MS is Matching Score, as stated in Sec. 3.3.

###  4.5 Results and Discussion

Methods |  F1 ($\%$, $\uparrow$) |  Human F1 ($\%$, $\uparrow$)
---|---|---
Whisper-Large [6] | 64.0 | 86.4
+Text Refresher(VCTK) | 66.8 | 88.0
+Text Refresher(VCTK++) | 68.4 | 89.1

Table 4: Word Dysfluency evaluation on Aphasia speech.

####  4.5.1 Transcription Analysis

We begin by examining the phonetic transcription results, as presented in
Table 1. Both WavLM-CTC-VAD and WavLM-CTC-MFA demonstrate commendable zero-
shot silence-aware phonetic transcription capabilities. However, their
performance remains limited and is even inferior to the UFA trained with the
WavLM base model. Interestingly, incorporating the Buckeye data during
training does not yield any performance improvement. We hypothesize that the
presence of noise in the Buckeye corpus, being a dysfluent dataset itself,
hinders performance. Additionally, including the LibriSpeech dataset in VCTK
training does not lead to performance enhancement. This suggests that UFA has
already reached a certain limit in terms of data scalability. Consequently,
the subsequent ablations and dysfluency detection experiments are conducted
solely using UFA-WavLM-Large-VCTK. During our ablation study, we consistently
observed performance improvements by incorporating boundary prediction
information in the dynamic alignment search, as described in Section 2.1.1.
Moreover, our experiments on VCTK++ consistently demonstrated enhanced
performance compared to the original VCTK dataset, highlighting the robustness
introduced by VCTK++. In terms of word transcription results, as shown in
Table 2, we found that Whisper-Large exhibited the lowest performance due to
its overpowering language modeling. However, with the introduction of Text
Refresher and the incorporation of VCTK++, we observed an improvement in the
imperfect Word Error Rate (iWER), further boosting the overall performance.

####  4.5.2 Dysfluency Analysis

Since there is no previous work on hierarchical (word/phoneme) and fine-
grained (time-accurate) dysfluency detection models like ours, we conducted
ablation experiments to compare our proposed rule-based detection methods
against ourselves. The results, as shown in Table 3 and Table 4, indicate
impressive performance in terms of F1 scores and matching scores (MS),
demonstrating the ability of our methods to accurately capture most
dysfluencies. However, it is important to note that our methods still fall
short of human detection performance, highlighting their inherent limitations.

##  5 Conclusion and Limitations

We propose an unconstrained dysfluency modeling approach that combines
transcription and detection, which has been proven effective in both tasks.
However, there are several limitations that should be addressed in future
research. First, our detection experiments primarily focus on aphasia speech,
which limits the generalizability. Future work should explore diverse and
open-domain dysfluent datasets, which may lack manual annotations. Second, our
approach relies on phoneme-level forced alignment as the key representation
for detection. However, it is worth investigating alternative speech units,
such as self-supervised discrete units [32] and articulatory units [33, 34,
35, 36], to improve alignment modeling. Third, while our detection method
performs well on ”easy” speech samples, it struggles with more challenging and
abnormal speech patterns. Even in the current setting, the detection scores
are still below human performance. Lastly, it is worth exploring the
application of LLM-guided speech models [37, 38] to advance dysfluency
modeling in a prompt manner, which remains an open problem. Hopefully our
proposed approach can serve as a promising paradigm for the field of
dysfluency modeling.

##  6 Acknowledgement

Thanks for support from UC Noyce Initiative, Society of Hellman Fellows,
NIH/NIDCD and the Schwab Innovation fund.

## References

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Nittka’s invariance criterion and
Hilbert space valued parabolic equations in $L_{p}$

W. Arendt1, A.F.M. ter Elst2 and M. Sauter1

Abstract

  * Nittka gave an efficient criterion on a form defined on $L_{2}(\Omega)$ which implies that the associated semigroup is $L_{p}$-invariant for some given $p\in(1,\infty)$. We extend this criterion to the Hilbert space valued $L_{2}(\Omega,H)$. As an application we consider elliptic systems of pure second order. Our main result shows that the induced semigroup is $L_{p}$-contractive for all $p\in[p_{-},p_{+}]$ for some $1<p_{-}<2<p_{+}<\infty$.

October 2023.

MSC (2020): 35B45, 35K15 46B20.

Keywords: Semigroup, $L_{p}$-contraction, sesquilinear forms, strictly convex
Banach space.

Home institutions:
1. Institute of Applied Analysis 2. Department of Mathematics Ulm University University of Auckland Helmholtzstr. 18 Private bag 92019 89081 Ulm Auckland 1142 Germany New Zealand

##  1 Introduction

Let $(S_{t})_{t>0}$ be a $C_{0}$-semigroup on $L_{2}(\Omega)$, with
$\Omega\subset\mathds{R}^{d}$, which is associated with a closed sesquilinear
form. Ouhabaz [Ouh1] gave a convenient criterion on the form characterising
when the semigroup is $L_{\infty}$-contractive. It is more complicated to
describe $L_{p}$-contractivity if $1<p<\infty$. The reason is the fact that
there is no explicit formula which describes the orthogonal projection from
$L_{2}(\Omega)$ to the closed convex set $\\{u\in L_{2}(\Omega):\|u\|_{p}\leq
1\\}$ if $1<p<\infty$ and $p\neq 2$.

Nonetheless, Nittka [Nit] succeeded to overcome the difficulty by a structural
analysis and developed an efficient criterion for the $L_{p}$-contractivity of
a $C_{0}$-semigroup on $L_{2}(\Omega)$ which is associated with a form.

The purpose of this paper is twofold. Our first aim is to present Nittka’s
result. We do this within the more general setting of the vector-valued space
$L_{2}(\Omega,H)$, where $H$ is a Hilbert space. The result is the following.

######  Theorem 1.1.

Let $(\Omega,{\cal B},\mu)$ be a $\sigma$-finite measure space. Let $H$ be a
Hilbert space. Fix $p\in(1,\infty)$. Define

$C=\\{u\in L_{2}(\Omega,H)\cap L_{p}(\Omega,H):\|u\|_{L_{p}(\Omega,H)}\leq
1\\}.$

Let $P$ be the orthogonal projection of $L_{2}(\Omega,H)$ onto $C$. Let ${\cal
V}$ be a Hilbert space which is continuously and densely embedded in
$L_{2}(\Omega,H)$. Let $\gothic{a}\colon{\cal V}\times{\cal V}\to\mathds{C}$
be a continuous elliptic sesquilinear form, let $A$ be the operator in
$L_{2}(\Omega,H)$ associated with $\gothic{a}$ and let $S$ be the semigroup
generated by $-A$. Then the following are equivalent.

  * (i)

$\|S_{t}u\|_{L_{p}(\Omega,H)}\leq\|u\|_{L_{p}(\Omega,H)}$ for all $u\in
L_{2}(\Omega,H)\cap L_{p}(\Omega,H)$ and $t>0$.

  * (ii)

$P{\cal V}\subset{\cal V}$ and $\mathop{\rm
Re}\gothic{a}(u,\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u)\geq 0$ for all $u\in{\cal
V}$ with $\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u\in{\cal V}$.

Our proof in Section 2 is slightly different from Nittka’s since we exploit
the strict convexity of the space $L_{p}(\Omega,H)$ for all $1<p<\infty$,
which we prove in Appendix B.

Our second aim is to apply the criterion to purely second-order Hilbert space
valued elliptic operators with Neumann boundary conditions. They generate a
contractive $C_{0}$-semigroup $(S_{t})_{t>0}$ on $L_{2}(\Omega,H)$. Of
particular interest are systems, that is $H=\mathds{C}^{d}$. In Section 3 we
show that there is an interval $[p_{-},p_{+}]$, with $1<p_{-}<2<p_{+}<\infty$,
such that the semigroup $S$ extends to a contractive $C_{0}$-semigroup on
$L_{p}(\Omega,H)$ for all $p\in[p_{-},p_{+}]$. To prove the needed estimates,
we use a chain rule formula, which is quite delicate and will be proved in
Appendix A. In the scalar case our results may be compared with Cialdea–Maz’ya
[CM], who introduced an algebraic version of $L_{p}$-dissipativity and
presented an algebraic characterisation for scalar-valued elliptic operators.
This algebraic characterisation was refined by Carbonaro–Dragičević [CD] and
they used the result of Nittka to describe contractive $C_{0}$-semigroups on
$L_{p}(\Omega)$ via the notion that they called $p$-ellipticity.

##  2 Nittka’s criterion for $L_{p}$-contractivity

Let $(\Omega,{\cal B},\mu)$ be a $\sigma$-finite measure space. Let $H$ be a
Hilbert space. For all $p\in[1,\infty)$ we write $L_{p}=L_{p}(\Omega,H)$. If
$u\in L_{p}$, then we write $\|u\|_{p}=\|u\|_{L_{p}(\Omega,H)}$ and
$\|u\|_{H}\colon\Omega\to\mathds{R}$ is the function from $\Omega$ into
$\mathds{R}$ such that

$\|u\|_{H}(x)=\|u(x)\|_{H}$

for all $x\in\Omega$. Further we write ${\cal H}=L_{2}=L_{2}(\Omega,H)$.
Throughout this paper we fix $p\in(1,\infty)$. Define

$C=\\{u\in{\cal H}\cap L_{p}:\|u\|_{p}\leq 1\\}.$

Clearly $C$ is convex and it follows from Fatou’s lemma that $C$ is closed in
${\cal H}$. Let $P\colon{\cal H}\to C$ be the orthogonal projection. For all
$u\in{\cal H}$ define

$N(u)=\\{h\in{\cal H}:\mathop{\rm Re}(h,v-u)_{\cal H}\leq 0\mbox{ for all
}v\in C\\}.$

Then $N(u)$ is a closed cone in ${\cal H}$ with $0\in N(u)$. We state an easy
property regarding $C$ and $N(u)$.

######  Lemma 2.1.

For all $f\in{\cal H}$ there exist unique $u\in C$ and $h\in N(u)$ such that
$f=u+h$. Actually, $u=Pf$.

###### Proof.

Let $u=Pf$ and $h=f-Pf$. Then $u\in C$ and $\mathop{\rm Re}(h,v-u)_{\cal
H}=\mathop{\rm Re}(f-Pf,\penalty-10v-Pf)_{\cal H}\leq 0$ for all $v\in C$.
This proves existence. Let also $\tilde{u}\in C$ and $\tilde{h}\in
N(\tilde{u})$ be such that $f=\tilde{u}+\tilde{h}$. Then $\mathop{\rm
Re}(h,\tilde{u}-u)_{\cal H}\leq 0$ and similarly $\mathop{\rm
Re}(\tilde{h},u-\tilde{u})_{\cal H}\leq 0$. Hence $\mathop{\rm
Re}(h-\tilde{h},\tilde{u}-u)_{\cal H}\leq 0$. Since $h-\tilde{h}=\tilde{u}-u$
this implies that $\mathop{\rm Re}\|h-\tilde{h}\|_{\cal H}^{2}\leq 0$ and the
statement follows. ∎

If $u\colon\Omega\to H$ is a function, then define $\mathop{\rm
sgn}u\colon\Omega\to H$ by

$(\mathop{\rm
sgn}u)(x)=\left\\{\begin{array}[]{ll}\frac{1}{\|u(x)\|_{H}}\,u(x)&\mbox{if
}u(x)\neq 0,\\\\[5.0pt] 0&\mbox{if }u(x)=0.\end{array}\right.$

Note that $\mathop{\rm sgn}u=\lim_{\varepsilon\downarrow 0}u_{\varepsilon}$
pointwise, where the function $u_{\varepsilon}\colon\Omega\to H$ is defined by
$u_{\varepsilon}(x)=\frac{1}{\sqrt{\|u(x)\|_{H}^{2}+\varepsilon}}\,u(x)$.
Hence $\mathop{\rm sgn}u$ is (Bochner) measurable if $u$ is (Bochner)
measurable. Then also $\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u$ is (Bochner)
measurable whenever $u$ is (Bochner) measurable.

For a description of $N(u)$ we use that $L_{p}(\Omega,H)$ is strictly convex.
See Appendix B for a proof. Recall that a Banach space $E$ is called strictly
convex if for all $\xi,\eta\in E$ with $\|\xi\|_{E}=1=\|\eta\|_{E}$ and
$\xi\neq\eta$ it follows that $\|\xi+\eta\|_{E}<2$.

######  Lemma 2.2.

Let $E$ be a Banach space. Assume that $E^{*}$ strictly convex and let $x\in
E$. Then there exists a unique $f\in E^{*}$ such that $\mathop{\rm
Re}f(x)=\|x\|_{E}^{2}=\|f\|_{E^{*}}^{2}$. This unique $f$ satisfies
$f(x)=\|x\|_{E}^{2}$.

###### Proof.

The existence of an $f\in E^{*}$ such that
$f(x)=\|x\|_{E}^{2}=\|f\|_{E^{*}}^{2}$ is a well-known consequence of the
Hahn–Banach theorem. For the uniqueness, we may assume without loss of
generality that $\|x\|_{E}=1$. Suppose also $g\in E^{*}$ with $\mathop{\rm
Re}g(x)=\|x\|_{E}^{2}=\|g\|_{E^{*}}^{2}$ and $g\neq f$. Define
$h=\frac{1}{2}\,(f+g)$. Then $\|h\|_{E^{*}}<1$ by the strict convexity of
$E^{*}$. But then

$1={\textstyle\frac{1}{2}}\,\mathop{\rm Re}(f(x)+g(x))=\mathop{\rm
Re}h(x)\leq\|h\|_{E^{*}}\,\|x\|_{E}<1.$

This is a contradiction. ∎

Now we are able to give a characterisation for $N(u)$.

######  Proposition 2.3.

Let $u\in C$. Then the following are equivalent.

  * (i)

$N(u)\neq\\{0\\}$.

  * (ii)

$\|u\|_{p}=1$ and $\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u\in{\cal H}$.

If these conditions are valid, then $N(u)=\\{t\,\|u\|_{H}^{p-1}\,\mathop{\rm
sgn}u:t\in[0,\infty)\\}$.

###### Proof.

‘(ii)$\Rightarrow$(i)’. Let $v\in C$. Then the Cauchy–Schwarz inequality and
the Hölder inequality give

$\displaystyle\mathop{\rm Re}(\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u,v)_{\cal H}$
$\displaystyle\leq$ $\displaystyle\int_{\Omega}\|u\|_{H}^{p-1}\,\|v\|_{H}$
$\displaystyle\leq$
$\displaystyle\Big{(}\int_{\Omega}\|u\|_{H}^{(p-1)p^{\prime}}\Big{)}^{1/p^{\prime}}\,\|v\|_{p}$
$\displaystyle\leq$ $\displaystyle\|u\|_{p}^{p/p^{\prime}}=1=\mathop{\rm
Re}(\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u,u)_{\cal H}.$

So $\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u\in N(u)$ and
$\\{t\,\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u:t\in[0,\infty)\\}\subset N(u)$. In
particular, $N(u)\neq\\{0\\}$.

‘(i)$\Rightarrow$(ii)’. We first show that $\|u\|_{p}=1$. Suppose that
$\|u\|_{p}<1$. Let $h\in N(u)$. Let $w\in L_{2}\cap L_{p}$ with $\|w\|_{p}\leq
1-\|u\|_{p}$. Then $u+w\in C$. Since $h\in N(u)$ one deduces that $\mathop{\rm
Re}(h,w)_{\cal H}\leq 0$. Because $L_{2}\cap L_{p}$ is dense in ${\cal H}$ it
follows that $h=0$. Hence (ii) implies that $\|u\|_{p}=1$.

Next let $h\in N(u)$ with $h\neq 0$. If $v\in L_{2}(\Omega,H)\cap
L_{p}(\Omega,H)$, then $\mathop{\rm Re}(h,v)_{\cal
H}\leq\|v\|_{p}\,\mathop{\rm Re}(h,u)_{\cal H}$. So $h\in
L_{p^{\prime}}(\Omega,H)$ and $\|h\|_{p^{\prime}}\leq\mathop{\rm
Re}(h,u)_{\cal H}$. If $\mathop{\rm Re}(h,u)_{\cal H}=0$, then
$\|h\|_{p^{\prime}}=0$ and $h=0$, which is a contradiction. So $\mathop{\rm
Re}(h,u)_{\cal H}\neq 0$. Multiplying $h$ with a strictly positive constant we
may assume that $\mathop{\rm Re}(h,u)_{\cal H}=1$. Then

$\|h\|_{p^{\prime}}\leq 1=\mathop{\rm Re}(h,u)_{\cal
H}\leq\|h\|_{p^{\prime}}\,\|u\|_{p}\leq\|h\|_{p^{\prime}}.$

So $\|h\|_{p^{\prime}}=1=\mathop{\rm Re}(h,u)_{\cal H}=\mathop{\rm Re}\langle
h,u\rangle_{L_{p^{\prime}}\times L_{p}}$. We proved that

$\mathop{\rm Re}\langle h,u\rangle_{L_{p^{\prime}}\times
L_{p}}=\|h\|_{p^{\prime}}^{2}=\|u\|_{p}^{2}.$

On the other hand,

$\int_{\Omega}\Big{\|}\|u\|_{H}^{p-1}\,\mathop{\rm
sgn}u\Big{\|}_{H}^{p^{\prime}}=\int_{\Omega}\|u\|_{H}^{(p-1)p^{\prime}}=\int_{\Omega}\|u\|_{H}^{p}=1,$

so $\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u\in L_{p^{\prime}}$ and furthermore

$\mathop{\rm Re}\langle\|u\|_{H}^{p-1}\,\mathop{\rm
sgn}u,u\rangle_{L_{p^{\prime}}\times
L_{p}}=\int_{\Omega}\|u\|_{H}^{p}=1=\Big{\|}\|u\|_{H}^{p-1}\,\mathop{\rm
sgn}u\Big{\|}_{p^{\prime}}^{2}=\|u\|_{p}^{2}.$

By Lemma 2.2 we obtain that $\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u=h\in{\cal H}$.
Moreover, $N(u)\subset\\{t\,\|u\|_{H}^{p-1}\,\mathop{\rm
sgn}u:t\in[0,\infty)\\}$. ∎

Let ${\cal V}$ be a Hilbert space that is continuously and densely embedded in
${\cal H}=L_{2}(\Omega,H)$. Let $\gothic{a}\colon{\cal V}\times{\cal
V}\to\mathds{C}$ be a sesquilinear form such that $\gothic{a}$ is continuous,
that is there is an $M>0$ such that

$|\gothic{a}(u,v)|\leq M\,\|u\|_{\cal V}\,\|v\|_{\cal V}$

for all $u,v\in{\cal V}$, and $\gothic{a}$ is elliptic, that is there are
$\mu>0$ and $\omega\in\mathds{R}$ such that

$\mathop{\rm Re}\gothic{a}(u,u)+\omega\,||u\|_{\cal H}^{2}\geq\mu\,\|u\|_{\cal
V}^{2}$ (1)

for all $u\in{\cal V}$. Then there is a unique operator $A$ in ${\cal H}$
whose graph is

$\mathop{\rm graph}(A)=\\{(u,f):u\in{\cal V},\;f\in{\cal H}\mbox{ and
}\gothic{a}(u,v)=(f,v)_{\cal H}\mbox{ for all }v\in{\cal V}\\}.$

We call $A$ the operator associated with the form $\gothic{a}$. Then $-A$
generates a holomorphic $C_{0}$-semigroup $(S_{t})_{t>0}$ in ${\cal H}$
satisfying $\|S_{t}\|_{{\cal H}\to{\cal H}}\leq e^{\omega t}$ for all $t>0$,
where $\omega$ is as in (1).

###### Proof of Theorem 1.1..

‘(i)$\Rightarrow$(ii)’. It follows from [Ouh2] Theorem 2.2 1)$\Rightarrow$2)
that $P{\cal V}\subset{\cal V}$ and $\mathop{\rm Re}\gothic{a}(Pf,f-Pf)\geq 0$
for all $f\in{\cal V}$. Since $\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u\in
L_{2}(\Omega,H)$ by assumption, one deduces that

$\int_{\Omega}\|u\|_{H}^{p}=\int_{\Omega}(\|u\|_{H}^{p-1}\,\mathop{\rm
sgn}u,\|u\|_{H}\,\mathop{\rm sgn}u)_{H}<\infty.$

So $u\in L_{p}$. Without loss of generality we may assume that $\|u\|_{p}=1$.
Then $u\in C$ and $\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u\in{\cal H}$, so
$\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u\in N(u)$ by Proposition 2.3. Set
$f=u+\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u\in{\cal V}\subset{\cal H}$. Then the
uniqueness of Lemma 2.1 gives $u=Pf$. Consequently $\mathop{\rm
Re}\gothic{a}(u,\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u)=\mathop{\rm
Re}\gothic{a}(Pf,f-Pf)\geq 0$.

‘(ii)$\Rightarrow$(i)’. Let $v\in{\cal V}$. We shall show that $\mathop{\rm
Re}\gothic{a}(Pv,v-Pv)\geq 0$. If $v\in C$, then this is trivial, so we may
assume that $v\not\in C$. Set $u=Pv$ and $h=v-u$. Then $h\in N(u)$ by Lemma
2.1. Also $h\neq 0$, so Proposition 2.3 implies that $\|u\|_{p}=1$ and
$\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u\in{\cal H}$. Moreover, there exists a
$t\in[0,\infty)$ such that $h=t\,\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u$. Then
$t\neq 0$. Since $u=Pv\in P{\cal V}\subset{\cal V}$ by assumption, one deduces
that $\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u=\frac{1}{t}(v-u)\in{\cal V}$. Hence
$\mathop{\rm Re}\gothic{a}(Pv,v-Pv)=t\mathop{\rm
Re}\gothic{a}(u,\|u\|_{H}^{p-1}\,\mathop{\rm sgn}u)\geq 0$. Now it follows
from [Ouh2] Theorem 2.2 2)$\Rightarrow$1) that $S_{t}C\subset C$ for all
$t>0$. This obviously implies Statement (i). ∎

##  3 Application to Hilbert space valued parabolic problems

Let $\Omega\subset\mathds{R}^{d}$ be an open set. Let $H$ be a separable
Hilbert space. For all $k,l\in\\{1,\ldots,d\\}$ let
$c_{kl}\colon\Omega\to{\cal L}(H)$ be a bounded function such that
$x\mapsto(c_{kl}(x)\,\xi,\eta)_{H}$ is measurable from $\Omega$ into
$\mathds{C}$ for all $\xi,\eta\in H$. Let $\mu>0$. We assume that

$\mathop{\rm
Re}\sum_{k,l=1}^{d}(c_{kl}(x)\,\xi_{l},\xi_{k})_{H}\geq\mu\sum_{k=1}^{d}\|\xi_{k}\|_{H}^{2}$

for all $x\in\Omega$ and $\xi_{1},\ldots,\xi_{d}\in H$. Further let $M>0$ be
such that

$\sum_{k=1}^{d}\Big{\|}\sum_{l=1}^{d}c_{kl}(x)\,\xi_{l}\Big{\|}_{H}^{2}\leq
M^{2}\,\sum_{k=1}^{d}\|\xi_{k}\|_{H}^{2}$

for all $x\in\Omega$ and $\xi_{1},\ldots,\xi_{d}\in H$. For simplicity we
consider Neumann boundary conditions. Define ${\cal V}=H^{1}(\Omega,H)$ and
$\gothic{a}\colon{\cal V}\times{\cal V}\to\mathds{C}$ by

$\gothic{a}(u,v)=\sum_{k,l=1}^{d}\int_{\Omega}(c_{kl}(x)\,(\partial_{l}u)(x),(\partial_{k}v)(x))_{H}\,dx.$

Then $\gothic{a}$ is a continuous elliptic sesquilinear form. Let $A$ be the
operator associated with $\gothic{a}$ and let $S$ be the semigroup generated
by $-A$.

######  Theorem 3.1.

Let $p\in(1,\infty)$ and suppose that

$\frac{\mu}{M}\geq
2\Big{|}\frac{p-2}{p}\Big{|}+\Big{|}\frac{p-2}{p}\Big{|}^{2}.$

Then $S$ extends consistently to a contraction semigroup in $L_{p}(\Omega,H)$.

Note that the condition in Theorem 3.1 is invariant by taking the dual
exponent, that is, if $p\in(1,\infty)$ satisfies the condition, then so does
$q$, where $\frac{1}{p}+\frac{1}{q}=1$. If one is satisfied with some small
interval, $1<p_{-}<2<p_{+}<\infty$ such that $S$ is $L_{p}$-contractive for
all $p\in[p_{-},p^{+}]$, then a significantly easier and less technical proof
than the following can be given.

###### Proof.

Using duality, without loss of generality we may assume that $p>2$. We argue
as in [CM], [ELSV] and [Ege]. Let $u\in H^{1}(\Omega,H)$. For all
$n\in\mathds{N}$ define

$v_{n}=(\|u\|_{H}^{\frac{p-2}{2}}\wedge n)\,u\quad,\quad
w_{n}=(\|u\|_{H}^{p-2}\wedge
n^{2})\,u\quad\mbox{and}\quad\chi_{n}=\mathds{1}_{[\|u\|_{H}^{p-2}<n^{2}]}.$

If follows from Proposition A.1 (cf. [AK] Theorems 3.3 and 4.2) that
$v_{n},w_{n}\in H^{1}(\Omega,H)$ with

$\displaystyle\nabla v_{n}$ $\displaystyle=$ $\displaystyle
n\,(\mathds{1}-\chi_{n})\,\nabla
u+\chi_{n}\,\|u\|_{H}^{\frac{p-2}{2}}\Big{(}\nabla
u+{\textstyle\frac{p-2}{2}}\,(\nabla\|u\|_{H})\,\mathop{\rm
sgn}u\Big{)},\mbox{ and}$ $\displaystyle\nabla w_{n}$ $\displaystyle=$
$\displaystyle n^{2}\,(\mathds{1}-\chi_{n})\,\nabla
u+\chi_{n}\,\|u\|_{H}^{p-2}\Big{(}\nabla
u+(p-2)\,(\nabla\|u\|_{H})\,\mathop{\rm sgn}u\Big{)}.$ (2)

Note that $\chi_{n}\,\|v_{n}\|_{H}=\chi_{n}\,\|u\|_{H}^{p/2}$. Hence

$\displaystyle(\mathds{1}-\chi_{n})\,\nabla u$ $\displaystyle=$
$\displaystyle\frac{1}{n}\,(\mathds{1}-\chi_{n})\,\nabla v_{n},$
$\displaystyle\chi_{n}\,\|v_{n}\|_{H}^{\frac{p-2}{p}}\,\nabla u$
$\displaystyle=$ $\displaystyle\chi_{n}\,\Big{(}\nabla
v_{n}-{\textstyle\frac{p-2}{p}}\,(\nabla\|v_{n}\|_{H})\,\mathop{\rm
sgn}v_{n}\Big{)},\mbox{ and}$ $\displaystyle\nabla w_{n}$ $\displaystyle=$
$\displaystyle n\,(\mathds{1}-\chi_{n})\,\nabla
v_{n}+\chi_{n}\,\,\|v_{n}\|_{H}^{\frac{p-2}{p}}\Big{(}\nabla
v_{n}+{\textstyle\frac{p-2}{p}}\,(\nabla\|v_{n}\|_{H})\,\mathop{\rm
sgn}v_{n}\Big{)}.$

Therefore for almost all $x\in\Omega$ one obtains

$\displaystyle\sum_{k,l=1}^{d}\mathop{\rm
Re}(c_{kl}\,\partial_{l}u,\partial_{k}w_{n})_{H}$ $\displaystyle=$
$\displaystyle(\mathds{1}-\chi_{n})\sum_{k,l=1}^{d}\mathop{\rm
Re}(c_{kl}\,\partial_{l}u,n\,\partial_{k}v_{n})_{H}$ $\displaystyle\hskip
28.45274pt{}+\chi_{n}\sum_{k,l=1}^{d}\mathop{\rm
Re}(c_{kl}\,\|v_{n}\|_{H}^{\frac{p-2}{p}}\,\partial_{l}u,\Big{(}\partial_{k}v_{n}+{\textstyle\frac{p-2}{p}}\,(\partial_{k}\|v_{n}\|_{H})\,\mathop{\rm
sgn}v_{n}\Big{)})_{H}$ $\displaystyle=$
$\displaystyle(\mathds{1}-\chi_{n})\sum_{k,l=1}^{d}\mathop{\rm
Re}(c_{kl}\,\partial_{l}v_{n},\partial_{k}v_{n})_{H}$ $\displaystyle\hskip
2.84526pt{}+\chi_{n}\sum_{k,l=1}^{d}\mathop{\rm
Re}(c_{kl}\,\Big{(}\partial_{l}v_{n}-{\textstyle\frac{p-2}{p}}\,(\partial_{l}\|v_{n}\|_{H})\,\mathop{\rm
sgn}v_{n}\Big{)},\Big{(}\partial_{k}v_{n}+{\textstyle\frac{p-2}{p}}\,(\partial_{k}\|v_{n}\|_{H})\,\mathop{\rm
sgn}v_{n}\Big{)})_{H}$ $\displaystyle=$
$\displaystyle\sum_{k,l=1}^{d}\mathop{\rm
Re}(c_{kl}\,\partial_{l}v_{n},\partial_{k}v_{n})_{H}+{\textstyle\frac{p-2}{p}}\,\chi_{n}\,\mathop{\rm
Re}((c_{kl}-c_{lk}^{*})\,\partial_{l}v_{n},(\partial_{k}\|v_{n}\|_{H})\,\mathop{\rm
sgn}v_{n})_{H}$ $\displaystyle\hskip
28.45274pt{}-\Big{(}{\textstyle\frac{p-2}{p}}\Big{)}^{2}\,\mathop{\rm
Re}(c_{kl}\,(\partial_{l}\|v_{n}\|_{H})\,\mathop{\rm
sgn}v_{n},(\partial_{k}\|v_{n}\|_{H})\,\mathop{\rm sgn}v_{n})_{H}$
$\displaystyle\geq$
$\displaystyle\Big{(}\mu-2M\,{\textstyle\frac{p-2}{p}}-M\,\Big{(}{\textstyle\frac{p-2}{p}}\Big{)}^{2}\Big{)}\sum_{k=1}^{d}\|\partial_{k}v_{n}\|_{H}^{2},$

where we used that
$\sum_{k=1}^{d}|\partial_{k}\|v_{n}\|_{H}|^{2}\leq\sum_{k=1}^{d}\|\partial_{k}v_{n}\|_{H}^{2}$
and $\|\mathop{\rm sgn}v_{n}\|_{H}\leq 1$ by Lemma A.5. By the assumption on
$p$ we obtain that

$\sum_{k,l=1}^{d}\mathop{\rm
Re}(c_{kl}\,\partial_{l}u,\partial_{k}w_{n})_{H}\geq 0$

almost everywhere. This is for all $n\in\mathds{N}$.

Next take the limit $n\to\infty$ and use (2). Then

$\sum_{k,l=1}^{d}\mathop{\rm
Re}(c_{kl}\,\partial_{l}u,\|u\|_{H}^{p-2}\,(\partial_{k}u+(p-2)\,(\partial_{k}\|u\|_{H})\,\mathop{\rm
sgn}u))_{H}\geq 0$ (3)

almost everywhere.

We assume from now on that in addition $\|u\|_{H}^{p-2}\,u\in{\cal V}$. Let
$k\in\\{1,\ldots,d\\}$. We shall show that
$\partial_{k}(\|u\|_{H}^{p-2}\,u)=f_{k}$ almost everywhere, where

$f_{k}=\|u\|_{H}^{p-2}\,\Big{(}\partial_{k}u+(p-2)\,(\partial_{k}\|u\|_{H})\,\mathop{\rm
sgn}u\Big{)}.$

Write $r=2\,\frac{p-1}{p-2}\in(2,\infty)$ and let $q\in(1,2)$ be such that
$\frac{1}{2}+\frac{1}{r}=\frac{1}{q}$. Since $\|u\|_{H}^{p-1}\in
L_{2}(\Omega)$ one deduces that
$\int_{\Omega}(\|u\|_{H}^{p-2})^{r}=\int_{\Omega}(\|u\|_{H}^{p-1})^{2}<\infty$.
So $\|u\|_{H}^{p-2}\in L_{r}(\Omega)$. If $n\in\mathds{N}$, then
$n^{2}\,(\mathds{1}-\chi_{n})\,\|\partial_{k}u\|_{H}\leq(\mathds{1}-\chi_{n})\,\|u\|_{H}^{p-2}\,\|\partial_{k}u\|_{H}$
and hence (2) gives

$\|\partial_{k}w_{n}\|_{H}\leq\|u\|_{H}^{p-2}\Big{(}\|\partial_{k}u\|_{H}+(p-2)\,\Big{|}\partial_{k}\|u\|_{H}\Big{|}\Big{)}$
(4)

Note that the right hand side of (4) does not depend on $n$ and is an element
of $L_{q}(\Omega)$. Also $\lim\partial_{k}w_{n}=f_{k}$ almost everywhere.
Hence $\lim\partial_{k}w_{n}=f_{k}$ in $L_{q}(\Omega,H)$. It is easy to see
that $\lim w_{n}=\|u\|_{H}^{p-2}\,u$ in $L_{2}(\Omega,H)$ since
$\|u\|_{H}^{p-2}\,u\in L_{2}(\Omega,H)$. Let $\varphi\in
C_{c}^{\infty}(\Omega,H)$. Then

$\displaystyle\int_{\Omega}(\partial_{k}(\|u\|_{H}^{p-2}\,u),\varphi)_{H}$
$\displaystyle=$
$\displaystyle-\int_{\Omega}(\|u\|_{H}^{p-2}\,u,\partial_{k}\varphi)_{H}$
$\displaystyle=$
$\displaystyle-\lim_{n\to\infty}\int_{\Omega}(w_{n},\partial_{k}\varphi)_{H}=\lim_{n\to\infty}\int_{\Omega}(\partial_{k}w_{n},\varphi)_{H}=\int_{\Omega}(f_{k},\varphi)_{H}.$

So $\partial_{k}(\|u\|_{H}^{p-2}\,u)=f_{k}$ almost everywhere.

Finally (3) implies that

$\sum_{k,l=1}^{d}\mathop{\rm
Re}(c_{kl}\,\partial_{l}u,\partial_{k}(\|u\|_{H}^{p-2}\,u))_{H}\geq 0$

almost everywhere. Integrating over $\Omega$ gives $\mathop{\rm
Re}\gothic{a}(u,\|u\|_{H}^{p-2}\,u)\geq 0$. Now apply Theorem 1.1.

By the way, with some more work one can show that
$\lim_{n\to\infty}\partial_{k}w_{n}=f_{k}$ in $L_{2}(\Omega,H)$. ∎

We comment on related results concerning the extension of $S$ to a not
necessarily contractive $C_{0}$-semigroup on $L_{p}$. Hofmann, Mayboroda and
McIntosh [HMM] showed for $H=\mathds{C}$, $\Omega=\mathds{R}^{d}$ and $d\geq
3$ that the semigroup $S$ can be extended to a $C_{0}$-semigroup on
$L_{p}(\Omega)$ if $p\in[\frac{2d}{d+2},\frac{2d}{d-2}]$. Conversely, for each
$p\in(1,\frac{2d}{d+2})\cup(\frac{2d}{d-2},\infty)$ they construct an elliptic
operator such that the associated semigroup $(S_{t})_{t>0}$ cannot be extended
consistently to a bounded semigroup $L_{p}(\mathds{R}^{d})$. The extension
results are based on off-diagonal Davies–Gaffney estimates; cf. also [Dav1]
Theorem 25 and [Aus] Section 3.1.

Davies had already pointed out in the introduction of [Dav2] that his proof of
[Dav1] Theorem 25 extends to the vector-valued case. Moreover, by [Dav2]
Theorem 10 for each $p\in(1,\frac{2d}{d+2})\cup(\frac{2d}{d-2},\infty)$ there
exists an elliptic system with $H=\mathds{C}^{d}$, $\Omega=\mathds{R}^{d}$,
$d\geq 3$ and with real symmetric coefficients such that the operator $S_{t}$
does not continuously extend to $L_{p}$ for any $t>0$.

We shall give a corresponding extension result for our setting, which we
obtain readily from standard estimates and tracing Auscher’s proof of [Aus]
Proposition 3.2.

######  Theorem 3.2.

Suppose $\Omega=\mathds{R}^{d}$ or $\Omega\subset\mathds{R}^{d}$ is open and
Lipschitz. Then the semigroup $S$ extends to a $C_{0}$-semigroup with growth
bound $0$ on $L_{p}(\Omega,H)$ for all $p\in(\frac{2d}{d+2},\frac{2d}{d-2})$
if $d\geq 3$ and for all $p\in(1,\infty)$ if $d\in\\{1,2\\}$.

###### Proof.

We outline the arguments for $d\geq 3$. Let $\omega>0$. Since $\gothic{a}$ is
elliptic for this choice of $\omega$, by [Tan] Lemma 3.6.2 (3.60) there exists
a $c>0$ such that $\|e^{-\omega t}\,S_{t}u\|_{\cal V}\leq
c\,t^{-1/2}\,\|u\|_{2}$ for all $u\in L_{2}(\Omega,H)$ and $t>0$. Combining
this with the Sobolev embedding ${\cal V}\hookrightarrow L_{2d/(d-2)}$, we
obtain that there exists a $C>0$ such that

$\|e^{-\omega t}\,S_{t}u\|_{2d/(d-2)}\leq C\,t^{-1/2}\,\|u\|_{2}$

for all $u\in L_{2}(\Omega,H)$ and $t>0$. Then by duality

$\|e^{-\omega t}\,S_{t}^{*}u\|_{2}\leq C\,t^{-1/2}\,\|u\|_{2d/(d+2)}$

for all $u\in L_{2}\cap L_{2d/(d+2)}$ and $t>0$.

Next, it follows from inspection of the proof of [Aus] Proposition 3.2 that
the parts (2) and (3) of [Aus] Proposition 3.2 extend to the vector-valued
case and general open sets $\Omega$, and are applicable to the
$C_{0}$-semigroup $(T_{t})_{t>0}$ given by $T_{t}=e^{-\omega t}\,S_{t}^{*}$.
For the extension of part (2) one needs $L_{2}$–$L_{2}$ off-diagonal estimates
that can be obtained, for example, as in [AE] Theorem 4.2. Moreover, the
vector-valued version of the Riesz–Thorin theorem follows from [GLY] Lemma
2.6. Let $q\in(\frac{2d}{d+2},2)$. By the extension of [Aus] Proposition
3.2(2) we obtain that $T$ satisfies $L_{q}$–$L_{2}$ off-diagonal estimates,
which implies by the extension of [Aus] Proposition 3.2(3) that $T$ is
uniformly bounded in $L_{q}$. Dualizing again, we obtain the statement for all
$p\in(2,\frac{2d}{d-2})$. By considering the adjoint form, applying the result
for $p>2$ and taking the dual we obtain the statement for
$p\in(\frac{2d}{d+2},2)$. ∎

######  Remark 3.3.

We comment on the admissible ranges for $p$ in Theorems 3.1 and 3.2.
Remarkably, it is possible that the range given in Theorem 3.1 for contractive
extensions is larger than the one given in Theorem 3.2 for extensions with
growth bound $0$. For example, this occurs if $\frac{\mu}{M}\geq\frac{1}{2}$,
say, and $d$ is sufficiently large.

##  Appendix A The derivative of a truncation

Let $\Omega\subset\mathds{R}^{d}$ be an open set. Let $H$ be a Hilbert space.
The principal aim in this section is to prove the following chain rule.

######  Proposition A.1.

Let $\alpha>0$ and $M>0$. Let $u\in H^{1}(\Omega,H)$. Define
$v=(\|u\|_{H}^{\alpha}\wedge M)\,u$. Then $v\in H^{1}(\Omega,H)$ and

$\partial_{k}v=\alpha\,\mathds{1}_{[\|u\|_{H}^{\alpha}<M]}\,\|u\|_{H}^{\alpha}\,\mathop{\rm
Re}(\mathop{\rm sgn}u,\partial_{k}u)\,\mathop{\rm sgn}u+(\|u\|_{H}\wedge
M)\,\partial_{k}u$

for all $k\in\\{1,\ldots,d\\}$.

The proof involves some work. We use the following approximation by smooth
functions.

######  Lemma A.2.

The space $C^{\infty}(\Omega,H)\cap H^{1}(\Omega,H)$ is dense in
$H^{1}(\Omega,H)$.

###### Proof.

This follows as in the scalar case in [AF] Theorem 3.17. ∎

For an approximation argument the next lemma is useful.

######  Lemma A.3.

For all $n\in\mathds{N}$ let $u_{n}\in H^{1}(\Omega,H)$. Let
$u,g_{1},\ldots,g_{d}\in L_{2}(\Omega,H)$. Suppose that $\lim u_{n}=u$ in
$L_{2}(\Omega,H)$ and $\lim\partial_{k}u_{n}=g_{k}$ in $L_{2}(\Omega,H)$ for
all $k\in\\{1,\ldots,d\\}$. Then $u\in H^{1}(\Omega,H)$ and
$\partial_{k}u=g_{k}$ for all $k\in\\{1,\ldots,d\\}$.

###### Proof.

Let $\varphi\in C_{c}^{\infty}(\Omega)$. Let $k\in\\{1,\ldots,d\\}$. Then
$-\int_{\Omega}u_{n}\,\partial_{k}\varphi=\int_{\Omega}(\partial_{k}u_{n})\,\varphi$
for all $n\in\mathds{N}$. Then the lemma follows by taking the limit
$n\to\infty$. ∎

For the proof of Proposition A.1 we shall approximate the function $t\mapsto
t^{\alpha}\wedge M$ with smooth functions. The next technical lemma gives
sufficient conditions in order to apply a chain rule. Note that we do not
require that $f^{\prime}$ is bounded.

######  Lemma A.4.

Let $f\in C^{1}(0,\infty)$. Suppose that $f$ is bounded, $\lim_{t\downarrow
0}f(t)=0$, $\lim_{t\downarrow 0}t\,f^{\prime}(t)=0$ and
$\sup_{t\in(0,\infty)}t\,|f^{\prime}(t)|<\infty$. Let $u\in H^{1}(\Omega,H)$.
Define $v=f(\|u\|_{H})\,u$. Then $v\in H^{1}(\Omega,H)$ and

$\partial_{k}v=\left\\{\begin{array}[]{ll}\|u\|_{H}\,f^{\prime}(\|u\|_{H})\,\mathop{\rm
Re}(\mathop{\rm sgn}u,\partial_{k}u)_{H}\,\mathop{\rm
sgn}u+f(\|u\|_{H})\,\partial_{k}u&\mbox{on }[u\neq 0],\\\\[5.0pt] 0&\mbox{on
}[u=0].\end{array}\right.$ (5)

for all $k\in\\{1,\ldots,d\\}$.

###### Proof.

Let $\varepsilon>0$. Define
$v_{\varepsilon}=f(\sqrt{\|u\|_{H}^{2}+\varepsilon})\,u$. If $u\in
C^{1}(\Omega,H)$, then $v_{\varepsilon}\in C^{1}(\Omega,H)$ and

$\partial_{k}v_{\varepsilon}=\sqrt{\|u\|_{H}^{2}+\varepsilon}\,f^{\prime}(\sqrt{\|u\|_{H}^{2}+\varepsilon})\,\frac{\mathop{\rm
Re}(u,\partial_{k}u)_{H}}{\sqrt{\|u\|_{H}^{2}+\varepsilon}}\,\frac{1}{\sqrt{\|u\|_{H}^{2}+\varepsilon}}\,u+f(\sqrt{\|u\|_{H}^{2}+\varepsilon})\,\partial_{k}u$
(6)

for all $k\in\\{1,\ldots,d\\}$. Then by Lemmas A.2 and A.3 this extends to all
$u\in H^{1}(\Omega,H)$ and (6) is valid. Finally choose
$\varepsilon=\frac{1}{n}$, take the limit $n\to\infty$ and use again Lemma
A.3. ∎

Now we are able to prove Proposition A.1.

###### Proof of Proposition A.1..

For all $n\in\mathds{N}$ define $f_{n},f\colon(0,\infty)\to\mathds{R}$ by

$\displaystyle f(t)$ $\displaystyle=$ $\displaystyle t^{\alpha}\wedge M,$
$\displaystyle f_{n}(t)$ $\displaystyle=$
$\displaystyle{\textstyle\frac{1}{2}}\,(t^{\alpha}+\sqrt{M^{2}+n^{-1}}-\sqrt{|t^{\alpha}-M|^{2}+n^{-1}}).$

Then $\lim f_{n}(t)=f(t)$ for all $t\in(0,\infty)$. Also $\lim_{t\downarrow
0}f_{n}(t)=0$ for all $n\in\mathds{N}$. Let $n\in\mathds{N}$. Then $f_{n}\in
C^{1}(0,\infty)$ and

$f_{n}^{\prime}(t)={\textstyle\frac{1}{2}}\,\alpha\,t^{\alpha-1}\Big{(}1-\frac{t^{\alpha}-M}{\sqrt{(t^{\alpha}-M)^{2}+n^{-1}}}\Big{)}$

for all $t\in(0,\infty)$. In particular, $f_{n}$ is increasing. Moreover,
$\lim_{t\downarrow 0}t\,f_{n}^{\prime}(t)=0$. In addition,
$\lim_{n\to\infty}f_{n}^{\prime}(t)=\alpha\,t^{\alpha-1}$ if $t^{\alpha}\leq
M$ and $\lim_{n\to\infty}f_{n}^{\prime}(t)=0$ if $t^{\alpha}>M$.

Let $n\in\mathds{N}$ and $t\in(0,\infty)$. If $t^{\alpha}\leq M$, then

$0\leq
t\,f_{n}^{\prime}(t)={\textstyle\frac{1}{2}}\,\alpha\,t^{\alpha}\Big{(}1+\frac{M-t^{\alpha}}{\sqrt{(t^{\alpha}-M)^{2}+n^{-1}}}\Big{)}\leq\alpha\,M.$

Alternatively, if $t^{\alpha}>M$, then

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
t\,f_{n}^{\prime}(t)={\textstyle\frac{1}{2}}\,\alpha\,\frac{t^{\alpha}-M+M}{\sqrt{(t^{\alpha}-M)^{2}+n^{-1}}}\Big{(}\sqrt{(t^{\alpha}-M)^{2}+n^{-1}}-\sqrt{(t^{\alpha}-M)^{2}}\Big{)}$
$\displaystyle\leq$
$\displaystyle{\textstyle\frac{1}{2}}\,\alpha\Big{(}1+\frac{M}{\sqrt{n^{-1}}}\Big{)}\sqrt{n^{-1}}\leq{\textstyle\frac{1}{2}}\,\alpha\,(1+M).$

So

$\sup_{n\in\mathds{N}}\sup_{t\in(0,\infty)}t\,|f^{\prime}(t)|\leq\alpha\,(M+1).$
(7)

If $n\in\mathds{N}$ and $t\in(0,\infty)$, then

$0\leq
f_{n}(t)\leq{\textstyle\frac{1}{2}}\,(t^{\alpha}+M+1-\sqrt{|t^{\alpha}-M|^{2}+n^{-1}})\leq{\textstyle\frac{1}{2}}\,(t^{\alpha}+M+1-|t^{\alpha}-M|)={\textstyle\frac{1}{2}}+f(t)\leq{\textstyle\frac{1}{2}}+M.$

So $f_{n}$ is bounded and even

$\sup_{n\in\mathds{N}}\sup_{t\in(0,\infty)}|f_{n}(t)|\leq{\textstyle\frac{1}{2}}+M.$
(8)

Hence all conditions of Lemma A.4 are satisfied for all the $f_{n}$.

Let $u\in H^{1}(\Omega,H)$. For all $n\in\mathds{N}$ define
$v_{n}=f_{n}(u)\,u$. Then $v_{n}\in H^{1}(\Omega,H)$ with derivatives given by
(5) and $f$ replaced by $f_{n}$. The Lebesgue dominated convergence theorem
and the uniform bounds (8) and (7) imply that $\lim v_{n}=v$ and
$\lim\partial_{k}v_{n}=\partial_{k}v$ in $L_{2}(\Omega,H)$ for all
$k\in\\{1,\ldots,d\\}$. Then the proposition follows from Lemma A.3. ∎

Almost the same arguments show that the norm of an $H^{1}(\Omega,H)$-function
is in the Sobolev space.

######  Lemma A.5.

Let $u\in H^{1}(\Omega,H)$. Then $\|u\|_{H}\in H^{1}(\Omega)$ and
$\partial_{k}\|u\|_{H}=\mathop{\rm Re}(\mathop{\rm sgn}u,\partial_{k}u)$ for
all $k\in\\{1,\ldots,d\\}$.

###### Proof.

Let $\varepsilon>0$. For all $u\in H^{1}(\Omega,H)$ define
$u_{\varepsilon}\colon\Omega\to H$ by
$u_{\varepsilon}=\sqrt{\|u\|_{H}^{2}+\varepsilon}$. Let $\varphi\in
C_{c}^{\infty}(\Omega)$ and $k\in\\{1,\ldots,d\\}$. If $u\in
C^{\infty}(\Omega,H)\cap H^{1}(\Omega,H)$, then $u_{\varepsilon}\in
C^{\infty}(\Omega,H)$ with classical partial derivative
$\partial_{k}u_{\varepsilon}=\frac{\mathop{\rm
Re}(u,\partial_{k}u)_{H}}{u_{\varepsilon}}$. Hence

$-\int_{\Omega}u_{\varepsilon}\,\partial_{k}\varphi=\int_{\Omega}\frac{\mathop{\rm
Re}(u,\partial_{k}u)_{H}}{u_{\varepsilon}}\,\varphi.$ (9)

Using approximation and Lemma A.2, it follows that (9) is valid for all $u\in
H^{1}(\Omega,H)$. Finally choose $\varepsilon=\frac{1}{n}$ and take the limit
$n\to\infty$. ∎

##  Appendix B Strict convexity of $L_{p}(\Omega,H)$

As before let $(\Omega,{\cal B},\mu)$ be a $\sigma$-finite measure space and
$H$ a Hilbert space. In order to make this paper more self-contained, we give
a direct proof of the following theorem. At the end of this section we give
information on more general results.

######  Theorem B.1.

Let $p\in(1,\infty)$ and $u,v\in L_{p}(\Omega,H)$ with
$\|u\|_{p}=\|v\|_{p}=1$. If $\|u+v\|_{p}=2$, then $u=v$.

For the proof of Theorem B.1 we use three lemmas.

######  Lemma B.2.

Let $\xi,\eta\in H$ and suppose that
$\|\xi+\eta\|_{H}=\|\xi\|_{H}+\|\eta\|_{H}$. If $\eta\neq 0$, then there is a
$\lambda\in[0,\infty)$ such that $\xi=\lambda\,\eta$.

###### Proof.

The equality implies that $\mathop{\rm
Re}(\xi,\eta)_{H}=\|\xi\|_{H}\,\|\eta\|_{H}$. This gives equality in the
Cauchy–Schwarz inequality. Hence there is a $\lambda\in\mathds{C}$ such that
$\xi=\lambda\,\eta$. Using again the equality, one deduces that
$|1+\lambda|=|\lambda|+1$ and therefore $\lambda\in[0,\infty)$. ∎

Let $p,q\in(1,\infty)$ and suppose that $\frac{1}{p}+\frac{1}{q}=1$.

######  Lemma B.3.

Let $a,b\in[0,\infty)$. Then $a\,b\leq\frac{1}{p}\,a^{p}+\frac{1}{q}\,b^{q}$
and the equality holds if and only if $a^{p}=b^{q}$.

###### Proof.

This follows from the concavity of the logarithm. ∎

######  Lemma B.4.

Let $f\in L_{p}(\Omega)$ and $g\in L_{q}(\Omega)$ with $f\neq 0$ and $g\neq
0$. Suppose that
$\int_{\Omega}|f|\,|g|=\|f\|_{L_{p}(\Omega)}\,\|g\|_{L_{q}(\Omega)}$. Then
there exists a $\lambda>0$ such that $|f|^{p}=\lambda\,|g|^{q}$ almost
everywhere.

###### Proof.

We may assume that $\|f\|_{L_{p}(\Omega)}=1=\|g\|_{L_{q}(\Omega)}$. Then Lemma
B.3 gives

$1=\int_{\Omega}|f|\,|g|\leq\int_{\Omega}{\textstyle\frac{1}{p}}\,|f|^{p}+{\textstyle\frac{1}{q}}\,|g|^{q}={\textstyle\frac{1}{p}}\,\|f\|_{L_{p}(\Omega)}+{\textstyle\frac{1}{q}}\,\|g\|_{L_{q}(\Omega)}=1.$

Hence $|f|\,|g|={\textstyle\frac{1}{p}}\,|f|^{p}+\frac{1}{q}\,|g|^{q}$ almost
everywhere and the lemma follows from Lemma B.3. ∎

###### Proof of Theorem B.1..

Using the triangle inequality on $H$, twice the Hölder inequality on
$L_{p}(\Omega)$ and the assumption $\|u+v\|_{p}=\|u\|_{p}+\|v\|_{p}$ one
obtains

$\displaystyle\|u+v\|_{p}^{p}$ $\displaystyle=$
$\displaystyle\int_{\Omega}\|u+v\|_{H}\,\|u+v\|_{H}^{p-1}$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}\|u\|_{H}\,\|u+v\|_{H}^{p-1}+\int_{\Omega}\|v\|_{H}\,\|u+v\|_{H}^{p-1}$
$\displaystyle\leq$
$\displaystyle\Big{(}\int_{\Omega}\|u\|_{H}^{p}\Big{)}^{1/p}\Big{(}\int_{\Omega}\|u+v\|_{H}^{(p-1)q}\Big{)}^{1/q}+\Big{(}\int_{\Omega}\|v\|_{H}^{p}\Big{)}^{1/p}\Big{(}\int_{\Omega}\|u+v\|_{H}^{(p-1)q}\Big{)}^{1/q}$
$\displaystyle=$
$\displaystyle(\|u\|_{p}+\|v\|_{p})\,\|u+v\|_{p}^{p/q}=\|u+v\|_{p}^{\frac{p}{q}+1}=\|u+v\|_{p}^{p}.$

Hence all three inequalities are equalities. The first gives that there is a
null-set $N_{1}\subset\Omega$ such that
$\|u+v\|_{H}(x)=\|u\|_{H}(x)+\|v\|_{H}(x)$ for all $x\in\Omega\setminus N_{1}$
such that $\|u+v\|_{H}(x)\neq 0$. Recall that $\|u\|_{p}=1$, so $\|u\|_{H}\neq
0\in L_{p}(\Omega)$. Similarly $\|u+v\|_{H}\neq 0\in L_{p}(\Omega)$ and
therefore $\|u+v\|_{H}^{p-1}\neq 0\in L_{q}(\Omega)$. Hence the equality in
the first Hölder inequality together with Lemma B.4 gives that there are
$\alpha>0$ and a null-set $N_{2}\subset\Omega$ such that
$\|u\|_{H}^{p}(x)=\alpha\,\|u+v\|_{H}^{(p-1)q}(x)$ for all
$x\in\Omega\setminus N_{2}$. Similarly there are $\beta>0$ and a null-set
$N_{3}\subset\Omega$ such that
$\|v\|_{H}^{p}(x)=\beta\,\|u+v\|_{H}^{(p-1)q}(x)$ for all $x\in\Omega\setminus
N_{3}$. Hence $\|u\|_{H}^{p}=\gamma\,\|v\|_{H}^{p}$ on
$\Omega\setminus(N_{2}\cup N_{3})$, where $\gamma=\frac{\alpha}{\beta}$. Since
$\|u\|_{p}=\|v\|_{p}=1$, one deduces that $\gamma=1$.

Now let $x\in\Omega\setminus(N_{1}\cup N_{2}\cup N_{3})$. If
$\|u+v\|_{H}(x)=0$, then $\|u\|_{H}^{p}(x)=\alpha\,\|u+v\|_{H}^{(p-1)q}(x)=0$
and $u(x)=0$. Similarly $v(x)=0$ and therefore $u(x)=v(x)$. Alternatively, if
$\|u+v\|_{H}(x)\neq 0$, then $v(x)\neq 0$ since $x\not\in N_{3}$. Moreover,
$\|u(x)+v(x)\|_{H}=\|u(x)\|_{H}+\|v(x)\|_{H}$ and Lemma B.2 implies that there
is a $\lambda\in[0,\infty)$ such that $u(x)=\lambda\,v(x)$. But
$\|u(x)\|_{H}^{p}=\|v(x)\|_{H}^{p}$ and consequently $u(x)=v(x)$. Therefore
$u=v$ almost everywhere. ∎

With a small modification one can prove that $L_{p}(\Omega,E)$ is strictly
convex if $E$ is strictly convex and $p\in(1,\infty)$. In fact, a stronger
result than Theorem B.1 is known. The space $L_{p}(\Omega,E)$ is uniformly
convex if $E$ is uniformly convex and $p\in(1,\infty)$. See [DHM] and the
references therein.

### Acknowledgement.

The second-named author is most grateful for the hospitality extended to him
during a fruitful stay at Ulm University. He wishes to thank Ulm University
for financial support.

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