Geometric Analysis and PDEs
Cetraro, June 11-76, 2007


Fully Nonlinear Elliptic Equations

Prof. Xu-Jia Wang, Australian National University, Canberra, Australia


Abstract


We study a class of fully nonlinear partial differential equations, namely the k-Hessian equations for k >= 2. The k-Hessian is the k-trace, or the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix. When k = 1 and k = n, it is respectively the Laplace and the Monge-Ampere equations.

The k-Hessian equation is elliptic when restricted to k-admissible functions, that is when the eigenvalues of the Hessian matrix lie in an appropriate convex cone. A function is k-admissible if it is subharmonic when k = 1 or convex when k = n.

Regularity for the Dirichlet problem of the Monge-Ampere equation and the k-Hessian equation has been established in [1,2,3]. The k-Hessian equation is also variational, which enables one to study the equation by variational and gradient flow methods. This is respectively carried out for the Monge-Ampere equation in [4] and the k-Hessian equation in [5,6]. Main theorems includes Sobolev type inequalities and existence of nontrivial solutions in the superlinear cases.

In another trend of research [7,8], various properties in the Newton potential theory were extended to the k-Hessian equation. In particular the Wolff potential estimate, built upon the Hessian measure and its weak continuity [7], was established in [8]. This estimate gives a necessary and sufficient condition for the Hölder continuity of admissible functions.

All these results indicate that the k-Hessian equation, including the Monge-Ampere equation, enjoys many similar properties as the Laplace equation. Furthermore, the in volved techniques are useful in the study of the so-called k-Yamabe problem.

Pre-requisite: Part I (Chapters 2-9) of the book [1] by Gilbarg and Trudinger.


References


[1] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Second edition, Springer-Verlag, 1983.

[2] L. Caffarelli, L. Nirenberg, J. Spruck, Dirichlet problem for nonlinear second order elliptic equations III, Functions of the eigenvalues of the Hessian, Acta Math., 155(1985), 261-301.

[3] N.S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math., 175(1995), 151-164.

[4] K.S. Chou (Tso), On a real Monge-Ampére functional, Invent. Math., 101(1990), 425-448.

[5] X.-J. Wang, A class of fully nonlinear elliptic equations and related functionals, Indiana Univ. Math. J., 43(1994), 25-54.

[6] K.S. Chou and X.-J. Wang, Variational theory for Hessian equations, Comm. Pure Appl. Math., 54(2001), 1029-1064.

[7] N.S. Trudinger and X.-J. Wang, Hessian measures II, Ann. Math., 150(1999), 579-604.

[8] D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J., 111(2002), 1-49.