A Dirichlet's principle for the \(k\)-Hessian
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 \(\mathcal{Q}_k: (C^2(X)\cap C^2(M)\cap C^1(\overline{X}))^{k+1}\rightarrow \mathbb{R}\) such that the functional \(\mathcal{E}_k\) defined by \[ \mathcal{E}_k(u)=\mathcal{Q}_k(u,\dots,u) \] satisfies \((1)\). For \(k=1\), the obtained result gives back the well-known trace inequality for the harmonic function \(u_f\) in \(X\) such that \(u_{\mid \partial X}=f\). For \(k=2\), an explicit expression for \(\mathcal{E}_k\) is given.
scientific article; zbMATH DE number 6948916
scientific article