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The k-Hessian Equation

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Geometric Analysis and PDEs

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1977))

Abstract

Abstract The k-Hessian is the k-trace, or the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix. When k ≥ 2, the k-Hessian equation is a fully nonlinear partial differential equations. It is elliptic when restricted to k-admissible functions. In this paper we establish the existence and regularity of k-admissible solutions to the Dirichlet problem of the k-Hessian equation. By a gradient flow method we prove a Sobolev type inequality for k-admissible functions vanishing on the boundary, and study the corresponding variational problems. We also extend the definition of k-admissibility to non-smooth functions and prove a weak continuity of the k-Hessian operator. The weak continuity enables us to deduce a Wolff potential estimate. As an application we prove the Hölder continuity of weak solutions to the k-Hessian equation. These results are mainly from the papers [CNS2, W2, CW1, TW2, Ld] in the references of the paper.

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Authors and Affiliations

  1. Mathematical Sciences Institute, Australian National University, 0200, Canberra, ACT, Australia

    Xu-Jia Wang

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  1. Xu-Jia Wang
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Correspondence to Xu-Jia Wang .

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Editors and Affiliations

  1. Dept. Mathematics, Princeton University, Washington Rd., Princeton, 08544-1000, U.S.A.

    Sun-Yung Alice Chang

  2. Department of Mathematics, SISSA, Via Beirut 2-4, Trieste, 34014, Italy

    Antonio Ambrosetti

  3. Studi Avanzati (SISSA), Scuola Internazionale Superiore di, via Beirut 2-4, Trieste, 34014, Italy

    Andrea Malchiodi

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Wang, XJ. (2009). The k-Hessian Equation. In: Chang, SY., Ambrosetti, A., Malchiodi, A. (eds) Geometric Analysis and PDEs. Lecture Notes in Mathematics(), vol 1977. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01674-5_5

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