Limited regularity of solutions to fractional heat and Schrödinger equations
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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.
|Tidsskrift||Discrete and Continuous Dynamical Systems- Series A|
|Status||Udgivet - 2019|