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Limited regularity of solutions to fractional heat and Schrödinger equations

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  • 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.

    Mathematics Subject Classification: Primary: 35K05, 35K35; Secondary: 35S11, 47G30, 60G52.

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