Some geometric estimates for fractional Poincaré inequalities

This thesis is devoted to study the following two problems: giving geometric lower bounds for the first eigenvalue of the fractional Dirichlet-Laplacian of order $s$; determining the sharp constant for the fractional Hardy inequality. The two problems are tightly connected: indeed, for an open set having finite inradius, every lower bound on the sharp Hardy constant immediately translates into a lower bound for the first eigenvalue. \par For the first problem, we prove a geometric lower bound in terms of the {\it inradius} of the set, in the case of open planar sets having nontrivial topology. This is valid for $1/2<s<1$ only, and we show that this condition is sharp. Moreover, by constructing suitable counter-examples, we prove that our lower bound is optimal, in many respects. The result is obtained through some non-trivial adaptions to the fractional case of techniques employed by Hayman and Taylor, to handle the case of the usual Laplacian operator. In adapting these techniques, we will develop some technical tools, which are interesting in themselves, in the context of fractional Sobolev spaces. \par For the second problem, we determine the sharp constant in the fractional Hardy inequality for open convex sets in every dimension. This is done by constructing suitable local weak supersolutions with geometric content, to the relevant Euler-Lagrange equation. The latter is a weighted eigenvalue--type equation, containing a negative power of the distance from the boundary. For $1/2\le s<1$, such supersolution is given by a suitable power of the distance function. The case $0<s<1/2$ is much more difficult, since such a construction fails. In this case, we employ a directional decomposition method, which permits to reduce the problem to dimension $1$. This technique is due to Loss and Sloane, in the fractional setting. In particular, we can compute the sharp constant in the whole range $0<s<1$. This completes a result which was left open in the literature.