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

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.