A note on the point-wise behaviour of bounded solutions for a non-standard elliptic operator

Laura Baldelli; Simone Ciani; Igor Skrypnik; Vincenzo Vespri

doi:10.3934/dcdss.2022143

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2024, Volume 17, Issue 5&6: 1718-1732. Doi: 10.3934/dcdss.2022143

This issue Previous Article Reconstruction of a convolution kernel in an integrodifferential problem with a fractional time derivative Next Article Mean value formulas for classical solutions to some degenerate elliptic equations in Carnot groups

A note on the point-wise behaviour of bounded solutions for a non-standard elliptic operator

1.
Institute of Mathematics, Polish Academy of Sciences, Poland
2.
Technical University of Darmstadt, Department of Mathematics, Germany
3.
Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Ukraine
4.
Università degli Studi di Firenze, Dipartimento di Matematica e Informatica "Ulisse Dini", Italy

^*Corresponding author: Simone Ciani
^*Corresponding author: Simone Ciani

To celebrate Jerry Goldstein's 80th genethliac

Received: April 2022

Revised: June 2022

Early access: July 2022

Published: May 2024

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Abstract

In this brief note we discuss local Hölder continuity for solutions to anisotropic elliptic equations of the type

$ \begin{equation*} \label{prototype} \sum\limits_{i = 1}^s \partial_{ii} u+ \sum\limits_{i = s+1}^N \partial_i \bigg(A_i(x, u, \nabla u) \bigg) = 0, \quad x \in \Omega \subset \subset \mathbb R^N \quad \text{ for } \quad 1\leq s \leq N-1, \end{equation*} $

where each operator $ A_i $ behaves directionally as the singular $ p $-Laplacian, $ 1< p < 2 $ and the supercritical condition $ p+(N-s)(p-2)>0 $ holds true. We show that the Harnack inequality can be proved without the continuity of solutions and that in turn this implies Hölder continuity of solutions.

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Mathematics Subject Classification: Primary: 35J75, 35K92; Secondary: 35B65.

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