Positivity, Duality and Fully nonlinear elliptic PDEs

Giovedi' 14 Aprile 2011 - Marco Cirant


Giovedi' 14 aprile 2011 alle ore 12:15 in aula 2BC30 Marco Cirant (Universita' di Padova) terra` un seminario dal titolo "Positivity, Duality and Fully nonlinear elliptic PDEs".

In a recent work (CPAM, 2009), F. R. Harvey and H. B. Lawson studied the Dirichlet problem for fully nonlinear (degenerate) elliptic equations of the form F(D2 u)=0 on a bounded domain using the new language of Dirichlet Duality. In their approach the equation is replaced by a suitable subset of the symmetric n x n matrices, satisfying a positivity condition (following an idea of N. V. Krylov); associated to that set is a dual set which gives a dual Dirichlet problem. This pairing leads naturally to a notion of weak solution (that is equivalent to the standard notion of viscosity solution) and clarifies some aspects of the problem, including comparison and appropriate conditions on the boundary of the domain; moreover, this level of generality enables to treat different branches of the homogeneous Monge-Ampere equation det (D2 u) = 0. Although a consistent part of the theory is a clever reformulation of viscosity theory, comparison principle is proved using a result of Z. Slodkowski on the largest eigenvalue of a convex function (instead of R. Jensen's maximum principle). In this talk we present the key ingredients of the theory and a generalization to equations of the form F(x, D2 u)=0. We replace the equation by a map that takes values into the powerset of the symmetric matrices; in order to solve the Dirichlet problem in this more general framework we need to introduce further assumptions on the regularity of the map, which guarantee a crucial property of translation of subsolutions. Last, we apply the theory to the inhomogeneous Monge-Ampere equation det (D2 u) = f(x).

Rif. int. M. Bardi, P. Mannucci, C. Marchi

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