Limited regularity of solutions to fractional heat and Schrödinger 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 $\Omega \subset {\mathbb{R}}^n$: $r^{+}Pu(x,t)+{\partial }_{t}u(x,t)=f(x,t)$ 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 $\Omega$, $v(x)=0$ 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.

Accepted in Discrete Continuous Dynamical Systems, the adapted final version shown here, 26 pages

References in corpus (1)

• The fractional Schrödinger equation with general nonnegative potentials. The weighted space approach

Cited by in corpus (1)

• Exact Green's formula for the fractional Laplacian and perturbations