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arXiv · 2412.02762

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

Gyula Csató·Albert Mas
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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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Gyula Csató, Albert Mas. 2024-12-03. Examples of optimal Hölder regularity in semilinear equations involving the fractional Laplacian. https://arxiv.org/abs/2412.02762

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