Limited regularity of solutions to fractional heat and Schrödinger equations
Publikation: Bidrag til tidsskrift › Tidsskriftartikel › fagfællebedømt
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 (Equation presented), is known to be solvable in relatively low-order Sobolev or Holder spaces. We now show that in contrast with differential operator cases, the regularity of u in x at δΩ 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, supp u ⊂ Ω, with V(x) ϵ C ∞ . 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, δΩ) a singularity.
Originalsprog | Engelsk |
---|---|
Tidsskrift | Discrete and Continuous Dynamical Systems- Series A |
Vol/bind | 39 |
Udgave nummer | 6 |
Sider (fra-til) | 3609-3634 |
Antal sider | 26 |
ISSN | 1078-0947 |
DOI | |
Status | Udgivet - 2019 |
Links
- https://arxiv.org/pdf/1806.10021.pdf
Accepteret manuskript
ID: 236554799