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Boundary regularity, Pohozaev identities, and nonexistence results. (arXiv:1705.05525v1 [math.AP])
来源于:arXiv
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
$L=(-\Delta)^s$. 查看全文>>