Umberto Biccari

Local Elliptic Regularity for the Dirichlet Fractional Laplacian

Abstract We prove the W loc 2 ⁢ s , p

Addendum: Local Elliptic Regularity for the Dirichlet Fractional Laplacian

{W_{{\mathrm{loc}}}^{2s,p}}

Local regularity for fractional heat equations

local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian on an arbitrary bounded open set of ℝ N

Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function

{\mathbb{R}^{N}}

Null controllability of linear and semilinear nonlocal heat equations with integral kernel.

. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different methods. First working directly in the variational formulation of the elliptic problem and then employing the heat kernel representation of solutions.

Null-controllability properties of the wave equation with a second order memory term

Abstract In [1], for 1 < p < ∞ {1<p<\infty} , we proved the W loc 2 ⁢ s , p {W^{2s,p}_{\mathrm{loc}}} local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian ( - Δ ) s {(-\Delta)^{s}} on an arbitrary bounded open set of ℝ N {\mathbb{R}^{N}} . Here we make a more precise and rigorous statement. In fact, for 1 < p < 2 {1<p<2} and s ≠ 1 2 {s\neq\frac{1}{2}} , local regularity does not hold in the Sobolev space W loc 2 ⁢ s , p {W^{2s,p}_{\mathrm{loc}}} , but rather in the larger Besov space ( B p , 2 2 ⁢ s ) loc {(B^{2s}_{p,2})_{\mathrm{loc}}} .