Optimal boundary regularity and a Hopf-type lemma for Dirichlet problems involving the logarithmic Laplacian

Víctor Hernández-Santamaría; Luis Fernando López Ríos; Alberto Saldaña

doi:10.3934/dcds.2024084

Article Contents

2025, Volume 45, Issue 1: 1-36. Doi: 10.3934/dcds.2024084

This issue Previous Article Next Article Regularity and Liouville theorem on an integral equation of Allen-Cahn type

Optimal boundary regularity and a Hopf-type lemma for Dirichlet problems involving the logarithmic Laplacian

1.
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, 04510 Coyoacán, Ciudad de México, Mexico
2.
Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Circuito Escolar s/n, Ciudad Universitaria, C.P. 04510, Ciudad de México, Mexico

^*Corresponding author: Alberto Saldaña
^*Corresponding author: Alberto Saldaña

Received: February 2024

Revised: June 2024

Early access: June 2024

Published: January 2025

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Abstract

We study the optimal boundary regularity of solutions to Dirichlet problems involving the logarithmic Laplacian. Our proofs are based on the construction of suitable barriers via the Kelvin transform and direct computations. As applications of our results, we show a Hopf-type lemma for nonnegative weak solutions and the uniqueness of solutions to some nonlinear problems.

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Mathematics Subject Classification: Primary: 35S15, 35B65; Secondary: 35A02.

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Figure 1. The shape of the initial barriers for $ N = 1 $ and $ N = 2 $

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Figure 2. The Kelvin transform can be used to obtain a new barrier. In the picture, the dotted line represents the boundary of the unitary circle. The red line segment (representing a subset of a hyperplane) is sent with the Kelvin transform to the red half-circle (representing a halfsphere)

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Figure 3. The function $ \ell $

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Figure 4. Domains of integration

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