A Dirichlet's principle for the \(k\)-Hessian
type: Item
instance of: scholarly article
title: A Dirichlet's principle for the \(k\)-Hessian
zbMATH Open document ID: 1405.35054
author: Jeffrey S. Case
author: Yi Wang
published in: Journal of Functional Analysis
publication date: 2018-10-08T00:00:00Z
review text: The authors establish the Dirichlet variational principle for the solutions of the Dirichlet boundary value problem for the \(k\)-Hessian equation \(\sigma_k(D^2 u)=0\) with a non-zero boundary data \(f\). More precisely, let \(k\in \mathbb{N}\), let \(X\subset \mathbb{R}^n\) be a bounded \((k-1)\)-convex domain, and let \(f\in C^\infty(\partial X)\). Given a \(n\times n\)-matrix \(A\), consider the \(k\)-th elementary symmetric function \[ \sigma_k (A):=\sum\lambda_{i_1}\cdot \dots\cdot \lambda_{i_k} \] of the eigenvalues \(\lambda_1,\dots,\lambda_n\) of \(A\), where the summation is over all integers \(1\leq i_1,\dots,i_k\leq n\) such that \(i_1<\dots<i_k\). Consider also the elliptic \(k\)-cone \[ \Gamma_k^+:=\{u\in C^\infty(\overline{X}): \sigma_j(D^2 u)>0, \;1\leq j\leq k\} \] and the set \[ \mathcal{C}_{f,k}:=\{u\in C^\infty(\overline{X}): u_{\mid \partial X}=f \;\;\text{and} \;\;u\in \Gamma_k^+\}, \] where \(u_{\mid \partial X}\) is the trace of \(u\) on \(\partial X\), \(D^2 u\) is the Hessian of \(u\), and \(\sigma_j(D^2 u)\) is the \(j\)-Hessian of \(u\). Finally, let \(\overline{\mathcal{C}_{f,k}}\) be the closures in the space \(C^{1,1}(\overline{X})\) of the set \(\mathcal{C}_{f,k}\). It is known that the problem \[ \begin{cases} \sigma_k(D^2u)=0, &\text{in }X, \\ u=f, &\text{on } \partial X. \end{cases} \] has a unique solution \(u_f\in \overline{\mathcal{C}_{f,k}}\). The authors establish the Dirichlet variational principle for \(u_f\) by proving that there is an energy functional \[ \mathcal{E}_k:C^2(X)\cap C^2(M)\cap C^1(\overline{X})\rightarrow \mathbb{R}, \] given by the \(k\)-Hessian energy plus a suitable boundary functional, such that \[ \mathcal{E}_k(u)\geq \mathcal{E}_k(u_f),\tag{1} \] for all \(u\in \overline{\mathcal{C}_{f,k}}\) with \(\mathcal{E}_k(u)= \mathcal{E}_k(u_f)\), if and only if \(u=u_f\). The authors prove the existence of such a functional by showing that there exists a symmetric multilinear form \(\mathcal{Q}_k: (C^2(X)\cap C^…
reviewed by: Giovanni Anello
Mathematics Subject Classification ID: 35J66
Mathematics Subject Classification ID: 35A23
zbMATH DE Number: 6948916
zbMATH Keywords: Hessian equation
zbMATH Keywords: fully nonlinear PDE
zbMATH Keywords: Dirichlet principle
zbMATH Keywords: trace inequality
MaRDI profile type: Publication
full work available at URL: j.jfa.2018.08.024
OpenAlex ID: W2890919253
cites work: The Dirichlet problem for nonlinear second order elliptic equations. III: Functions of the eigenvalues of the Hessian
cites work: An Extension Problem Related to the Fractional Laplacian
cites work: Conformal deformation on manifolds with boundary
cites work: Q3262807
cites work: The Dirichlet Problem for Degenerate Hessian Equations
cites work: Q4342727
cites work: Q5758550
cites work: \(C^{1,1}\) solution of the Dirichlet problem for degenerate \(k\)-Hessian equations
cites work: Q4300995
cites work: The k-Hessian Equation
DOI: 10.1016/J.JFA.2018.08.024
description: scientific article; zbMATH DE number 6948916