Limited regularity of solutions to fractional heat and Schrödinger equations

Abstract

When $P$ is the fractional Laplacian $(-Δ)^a$, $0<a<1$, or a pseudodifferential generalization thereof, the Dirichlet problem for the associated heat equation over a smooth set $Ω\subset{\Bbb R}^n$: $r^+Pu(x,t)+\partial_tu(x,t)=f(x,t)$ on $Ω\times \,]0,T[\,$, $u(x,t)=0$ for $x\notinΩ$, $u(x,0)=0$, is known to be solvable in relatively low-order Sobolev or Hölder spaces. We now show that in contrast with differential operator cases, the regularity of $u$ in $x$ at $\partialΩ$ 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 Schrödinger Dirichlet problem $r^+Pv(x)+Vv(x)=g(x)$ on $Ω$, $v(x)=0$ for $x\notin Ω$, 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 $dist(x,\partialΩ)^a$ singularity.