Abstract
The k-Hessian operator σ k is the k-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the k-Hessian equation σ k (D 2 u)=f with Dirichlet boundary condition u=0 is variational; indeed, this problem can be studied by means of the k-Hessian energy −∫uσ k (D 2 u). 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 language | English (US) |
|---|---|
| Pages (from-to) | 2895-2916 |
| Number of pages | 22 |
| Journal | Journal of Functional Analysis |
| Volume | 275 |
| Issue number | 11 |
| DOIs | |
| State | Published - Dec 1 2018 |
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All Science Journal Classification (ASJC) codes
- Analysis
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A Dirichlet's principle for the k-Hessian. / Case, Jeffrey Steven; Wang, Yi.
In: Journal of Functional Analysis, Vol. 275, No. 11, 01.12.2018, p. 2895-2916.Research output: Contribution to journal › Article
TY - JOUR
T1 - A Dirichlet's principle for the k-Hessian
AU - Case, Jeffrey Steven
AU - Wang, Yi
PY - 2018/12/1
Y1 - 2018/12/1
N2 - The k-Hessian operator σ k is the k-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the k-Hessian equation σ k (D 2 u)=f with Dirichlet boundary condition u=0 is variational; indeed, this problem can be studied by means of the k-Hessian energy −∫uσ k (D 2 u). 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.
AB - The k-Hessian operator σ k is the k-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the k-Hessian equation σ k (D 2 u)=f with Dirichlet boundary condition u=0 is variational; indeed, this problem can be studied by means of the k-Hessian energy −∫uσ k (D 2 u). 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.
UR - http://www.scopus.com/inward/record.url?scp=85053164621&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85053164621&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2018.08.024
DO - 10.1016/j.jfa.2018.08.024
M3 - Article
AN - SCOPUS:85053164621
VL - 275
SP - 2895
EP - 2916
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
SN - 0022-1236
IS - 11
ER -