A Dirichlet's principle for the k-Hessian

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Abstract

The k-Hessian operator σk is the k-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the k-Hessian equation σk(D2u)=f with Dirichlet boundary condition u=0 is variational; indeed, this problem can be studied by means of the k-Hessian energy −∫uσk(D2u). We construct a natural boundary functional which, when added to the k-Hessian energy, yields as its critical points solutions of k-Hessian equations with general non-vanishing boundary data. As a consequence, we establish a Dirichlet's principle for k-admissible functions with prescribed Dirichlet boundary data.

Original languageEnglish (US)
Pages (from-to)2895-2916
Number of pages22
JournalJournal of Functional Analysis
Volume275
Issue number11
DOIs
StatePublished - Dec 1 2018

All Science Journal Classification (ASJC) codes

  • Analysis

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