| Abstract: |
| In this talk we will analyze the non local problem
\begin{equation*}\label{prob}
\left\{
\begin{array}{rcll}
(-\Delta)^{s}u&=&\lambda_1 u &\mbox{ in }\Omega,\\
u&=&0 &\mbox{ in }D,\\
\mathcal{N}_{s}u&=&0 &\mbox{ in }N,
\end{array}
\right.
\end{equation*}
where $s\in (0,1)$, $\lambda_1$ is the first eigenvalue associated to the problem, $N$ and $D$ are two open sets of $\mathbb{R}^N\setminus \Omega$ of positive measure satisfying
$$
D\cap N=\emptyset, \ \ \ \ \left | \mathbb R^N\setminus (\Omega\cup D \cup N)\right |=0,
$$
and $\mathcal{N}_{s}u$ denotes the non local Neumann boundary condition of $u$. A natural question here is how the configuration of the sets $D$ and $N$ determines the behavior of $u$. Is it similar to the solution of the Dirichlet problem when $N$ is small? does it behave like the Neumann eigenfunction when $N$ is large?
When $s=1$ and the boundary conditions are prescribed on $\partial\Omega$ it is known that $u$ behaves in very different ways depending on the size and the location of $D$ and $N$.
As we will see, the fact that in the fractional case the boundary is the whole $\mathbb{R}^N\setminus \Omega$ completely changes the possible configurations of the sets (one can even have both sets of unbounded measure). The purpose of this talk will be to translate the notion of ``small boundary set and to analyze wether $D$ and $N$ can be prescribed to recover the classical results.
\medskip
This is a joint work with T. Leonori, I. Peral, A. Primo and F. Soria. |
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