Title: Principal Eigenvalue of Mixed Problem for the Fractional Laplacian: Moving the Boundary Conditions,
Abstract: We analyze the behavior of the eigenvalues of the following non local mixed
problem $\left\{ \begin{array}{rcll} (-\Delta)^{s} u &=& \lambda_1(D) \ u
&\inn\Omega,\\ u&=&0&\inn D,\\ \mathcal{N}_{s}u&=&0&\inn N. \end{array}\right $
Our goal is to construct different sequences of problems by modifying the
configuration of the sets $D$ and $N$, and to provide sufficient and necessary
conditions on the size and the location of these sets in order to obtain
sequences of eigenvalues that in the limit recover the eigenvalues of the
Dirichlet or Neumann problem. We will see that the non locality plays a crucial
role here, since the sets $D$ and $N$ can have infinite measure, a phenomenon
that does not appear in the local case (see for example \cite{D,D2,CP}).