On the Hölder regularity for the fractional Schrödinger equation and its improvement for radial data

2016/04/01 by Lemm, Marius
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1604.00415

Abstract

We consider the linear, time-independent fractional Schrödinger equation (-Δ)^s ψ+Vψ=f. We are interested in the local Hölder exponents of distributional solutions ψ, assuming local L^p integrability of the functions V and f. By standard arguments, we obtain the formula 2s-N/p for the local Hölder exponent of ψ where we take some extra care regarding endpoint cases. For our main result, we assume that V and f (but not necessarily ψ) are radial functions, a situation which is commonplace in applications. We find that the regularity theory "becomes one-dimensional" in the sense that the Hölder exponent improves from 2s-N/p to 2s-1/p away from the origin. Similar results hold for ∇ψ as well.

Related

• Hölder regularity for the fractional p-Laplacian, revisited
• Regularity for the fractional p-Laplace equation
• Regularity of Solution of the Schrödinger Equation on Symmetric Space
• Higher Hölder regularity for the fractional p-Laplace equation in the subquadratic case
• Higher Hölder regularity for fractional (p,q)-Laplace equations