# American Institute of Mathematical Sciences

June  2019, 39(6): 3609-3634. doi: 10.3934/dcds.2019148

## Limited regularity of solutions to fractional heat and Schrödinger equations

 Department of Mathematical Sciences, Copenhagen University, Universitetsparken 5, DK-2100 Copenhagen, Denmark

Received  October 2018 Revised  December 2018 Published  February 2019

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 Hölder 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 Schrödinger Dirichlet problem $r^+Pv(x)+Vv(x) = g(x)$ on $\Omega$, $\text{supp } v\subset\overline\Omega$, 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 $\text{dist}(x, \partial\Omega )^a$ singularity.

Citation: Gerd Grubb. Limited regularity of solutions to fractional heat and Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3609-3634. doi: 10.3934/dcds.2019148
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