- 'Serena Dipierro${}^{(1)}$'
- 'Ovidiu Savin${}^{(2)}$'
- 'Enrico Valdinoci${}^{(1)}$'
title: |
    Boundary properties of fractional objects:\
    flexibility of linear equations\
    and rigidity of minimal graphs
---

\(1) – Department of Mathematics and Statistics\
University of Western Australia\
35 Stirling Highway, WA6009 Crawley (Australia)\

\(2) – Department of Mathematics\
Columbia University\
2990 Broadway, NY 10027 New York (USA)

E-mail addresses: [serena.dipierro@uwa.edu.au]{},\
[savin@math.columbia.edu]{},\
[enrico.valdinoci@uwa.edu.au]{}

Introduction
============

Boundary behavior of fractional objects
---------------------------------------

This article investigates the geometric properties at the boundary of solutions of fractional problems. Two similar, but structurally significantly different, situations are taken into account. On the one hand, we will consider the solution of the [*linear fractional equation*]{} $$\begin{cases}
(-\Delta)^\sigma u=0 & {\mbox{ in }} B_1\cap \{x_n>0\},\\
u=0 & {\mbox{ in }} \{x_n<0\},\end{cases}$$ where $\sigma\in(0,1)$, and, for $x'=(x_1,\dots,x_{n-1})$ with $|x'|<1$, we consider the “fractional boundary derivative” $$\label{56:62734899fi02}
\displaystyle\lim_{x_n\searrow0} \frac{u(x',x_n)}{x_n^\sigma}
.$$ Interestingly, the function in  plays an important role in understanding fractional equations, see [@MR3168912]. In particular, while classical elliptic equations are smooth up to the boundary, the solutions of fractional equations with prescribed exterior datum are in general not better than Hölder continuous with exponent $\sigma$, and therefore the function in  is the crucial ingredient to detect the growth 