Examples of optimal Hölder regularity in semilinear equations involving the fractional Laplacian

2024/12/03 by Gyula Csató, Csató, Gyula, Albert Mas Blesa +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2412.02762

Abstract

We discuss the Hölder regularity of solutions to the semilinear equation involving the fractional Laplacian (-Δ)^s u=f(u) in one dimension. We put in evidence a new regularity phenomenon which is a combined effect of the nonlocality and the semilinearity of the equation, since it does not happen neither for local semilinear equations, nor for nonlocal linear equations. Namely, for nonlinearities f in C^β and when 2s+β<1, the solution is not always C^2s+β-ε for all ε>0. Instead, in general the solution u is at most C^2s/(1-β).

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