Optimal elliptic regularity: A comparison between local and nonlocal equations

Sunra J. N. Mosconi

doi:10.3934/dcdss.2018030

Article Contents

2018, Volume 11, Issue 3: 547-559. Doi: 10.3934/dcdss.2018030

This issue Previous Article Multiple solutions of fractional Kirchhoff equations involving a critical nonlinearity Next Article Bifurcation results for problems with fractional Trudinger-Moser nonlinearity

Optimal elliptic regularity: A comparison between local and nonlocal equations

Sunra J. N. Mosconi

Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6 I-95125 Catania, Italy

Received: May 2017

Revised: August 2017

Published: June 2018

^*The author was partially supported by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (INdAM).

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Abstract

Given $L≥1$ , we discuss the problem of determining the highest $α=α(L)$ such that any solution to a homogeneous elliptic equation in divergence form with ellipticity ratio bounded by $L$ is in $C^α_{\rm loc}$ . This problem can be formulated both in the classical and non-local framework. In the classical case it is known that $α(L)≳ {\rm exp}(-CL^β)$ , for some $C, β≥q$ depending on the dimension $N≥q$ . We show that in the non-local case, $α(L)≳ L^{-1-δ}$ for all $δ>0$ .

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Mathematics Subject Classification: 6E35, 35B40, 49K22.

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