Limited regularity of solutions to fractional heat and Schr\"odinger equations

TL;DR

This paper investigates the limited regularity of solutions to fractional heat and Schrödinger equations at the boundary, showing that solutions cannot generally be smoothed beyond certain estimates unless specific boundary conditions vanish.

Contribution

It demonstrates the inherent regularity limitations for solutions to fractional PDEs at boundaries, contrasting with classical differential operators, and characterizes the Dirichlet domains in terms of boundary singularities.

Findings

Solutions lack higher regularity at the boundary unless Neumann conditions vanish.

Regularity is constrained by boundary singularities of the form dist(x,∂Ω)^a.

Provides a detailed description of Dirichlet domains for fractional operators.

Abstract

When $P$ is the fractional Laplacian $(-\Delta )^a$, $0<a<1$, or a pseudodifferential generalization thereof, the Dirichlet problem for the associated heat equation over a smooth set $\Omega \subset{\Bbb R}^n$: $r^+Pu(x,t)+\partial_tu(x,t)=f(x,t)$ on $\Omega \times \,]0,T[\,$, $u(x,t)=0$ for $x\notin\Omega $, $u(x,0)=0$, is known to be solvable in relatively low-order Sobolev or H\"older spaces. We now show that in contrast with differential operator cases, the regularity of $u$ in $x$ at $\partial\Omega $ when $f$ is very smooth cannot in general be improved beyond a certain estimate. An improvement requires the vanishing of a Neumann boundary value. --- There is a similar result for the Schr\"odinger Dirichlet problem $r^+Pv(x)+Vv(x)=g(x)$ on $\Omega $, $v(x)=0$ for $x\notin \Omega $, with $V(x)\in C^\infty $. The proofs involve a precise description, of interest in itself, of the Dirichlet domains in terms of regular functions and functions with a $dist(x,\partial\Omega)^a$ singularity.