Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations

By S. Armstrong. 2008

72 citations recorded by the source corpus.

We study the fully nonlinear elliptic equation (0.1)F(D2u,Du,u,x)=f in a smooth bounded domain Ω, under the assumption that the nonlinearity F is uniformly elliptic and positively homogeneous. Recently, it has been shown that such operators have two principal “half” eigenvalues, and that the corresponding Dirichlet problem possesses solutions, if both of the principal eigenvalues are positive. In this paper, we prove the existence of solutions of the Dirichlet problem if both principal eigenvalues are negative, provided the “second” eigenvalue is positive, and generalize the anti-maximum principle of Clement and Peletier [P. Clement, L.A. Peletier, An anti-maximum principle for second-order elliptic operators, J. Differential Equations 34 (2) (1979) 218–229] to homogeneous, fully nonlinear operators.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/666d0544cf6f95077dc964aa906c825bcd6887b4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

In Control and Systems Engineering

Citations in the corpus

Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators (2009, background): “The concept of principal eigenvalue for boundary value problems of elliptic operators has been extended, in the last decades, to quasi-nonlinear and fullynonlinear equations (somehow ”abusing” the name of eigenvalue) see [1, 2, 28, 31, 24, 6, 25] etc.”
A family of degenerate elliptic operators: Maximum principle and its consequences (2016, unknown): “In the case of uniformly elliptic fully nonlinear operators, this has been proved to be the case in different context (see [3, 6, 9, 20, 24, 27]).”
Fundamental solutions of homogeneous fully nonlinear elliptic equations (2009, background): “Indeed, several of the ideas we employ in our proof of Theorem 3 are related to the principal eigenvalue theory for fully nonlinear operators developed in [20, 3, 24, 1].”