Hopf’s lemma and radial symmetry for the Logarithmic Laplacian problem

In this paper, we prove Hopf’s lemma for the Logarithmic Laplacian. At first, we introduce the strong minimum principle. Then Hopf’s lemma for the Logarithmic Laplacian in the ball is proved. On this basis, Hopf’s lemma of the Logarithmic Laplacian is extended to any open set with the property of th...

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Bibliographic Details
Published inFractional calculus & applied analysis Vol. 27; no. 4; pp. 1906 - 1916
Main Authors Zhang, Lihong, Nie, Xiaofeng
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.08.2024
Subjects
Abstract Harmonic Analysis
Analysis
Functional Analysis
Integral Transforms
Mathematics
Mathematics and Statistics
Operational Calculus
Original Paper
35B06
Hopf’s lemma
Logarithmic Laplacian
Strong minimum principle
Radial symmetry
35J05
35B09
35S05
35B50 (primary)
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Summary:In this paper, we prove Hopf’s lemma for the Logarithmic Laplacian. At first, we introduce the strong minimum principle. Then Hopf’s lemma for the Logarithmic Laplacian in the ball is proved. On this basis, Hopf’s lemma of the Logarithmic Laplacian is extended to any open set with the property of the interior ball. Finally, an example is given to explain Hopf’s lemma can be applied to the study of the symmetry of the positive solution of the nonlinear Logarithmic Laplacian problem by the moving plane method.
ISSN:1311-0454
1314-2224
DOI:10.1007/s13540-024-00285-1