# Boundary regularity, Pohozaev identities, and nonexistence results

## TL;DR

This paper surveys recent advances in boundary regularity, Pohozaev identities, and nonexistence results for Dirichlet problems involving fractional Laplacian and related integro-differential operators.

## Contribution

It provides a detailed survey, simplified proofs, and new insights into boundary regularity and Pohozaev identities for fractional operators, highlighting their applications.

## Key findings

- Boundary regularity results for fractional Laplacian
- Pohozaev identities for integro-differential operators
- Nonexistence and unique continuation derived from identities

## Abstract

In this expository paper we survey some recent results on Dirichlet problems of the form $Lu=f(x,u)$ in $\Omega$, $u\equiv0$ in $\mathbb R^n\backslash\Omega$. We first discuss in detail the boundary regularity of solutions, stating the main known results of Grubb and of the author and Serra. We also give a simplified proof of one of such results, focusing on the main ideas and on the blow-up techniques that we developed in \cite{RS-K,RS-stable}. After this, we present the Pohozaev identities established in \cite{RS-Poh,RSV,Grubb-Poh} and give a sketch of their proofs, which use strongly the fine boundary regularity results discussed previously. Finally, we show how these Pohozaev identities can be used to deduce nonexistence of solutions or unique continuation properties. The operators $L$ under consideration are integro-differential operator of order $2s$, $s\in(0,1)$, the model case being the fractional Laplacian $L=(-\Delta)^s$.