Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity

We analyze the asymptotic behavior of eigenvalues and eigenfunctions of an elliptic operator with mixed boundary conditions on cylindrical domains when the length of the cylinder goes to infinity. We identify the correct limiting problem and show in particular, that in general the limiting behavior is very different from the one for the Dirichlet boundary conditions.Comment: Asymptotic Analysis, 201

Fractional boundary Hardy inequality for the critical cases

We establish fractional boundary Hardy-type inequality for the critical cases by introducing a logarithmically singular weight term for various domains in $\mathbb{R}^{d}$, $d \geq 1$. We show that this particular weight function is optimal, as the inequality becomes false when using weight functions more singular than this one. Additionally, we extend our results to include a weighted fractional boundary Hardy-type inequality for the critical case, employing the same type of weight function.Comment: 29 page

Magnetic fractional Poincar\'e inequality in punctured domains

We study Poincar\'e-Wirtinger type inequalities in the framework of magnetic fractional Sobolev spaces. In the local case, Lieb-Seiringer-Yngvason [E. Lieb, R. Seiringer, and J. Yngvason, Poincar\'e inequalities in punctured domains, Ann. of Math., 2003] showed that, if a bounded domain $\Omega$ is the union of two disjoint sets $\Gamma$ and $\Lambda$, then the $L^p$-norm of a function calculated on $\Omega$ is dominated by the sum of magnetic seminorms of the function, calculated on $\Gamma$ and $\Lambda$ separately. We show that the straightforward generalisation of their result to nonlocal setup does not hold true in general. We provide an alternative formulation of the problem for the nonlocal case. As an auxiliary result, we also show that the set of eigenvalues of the magnetic fractional Laplacian is discrete