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

TL;DR

This paper investigates principal eigenvalues and extends the anti-maximum principle for fully nonlinear elliptic equations, establishing solution existence under new eigenvalue sign conditions.

Contribution

It proves the existence of solutions when both principal eigenvalues are negative with a positive second eigenvalue, generalizing the anti-maximum principle for such operators.

Findings

Solutions exist when both principal eigenvalues are negative and the second is positive.

Generalization of the anti-maximum principle to homogeneous, fully nonlinear operators.

Identification of conditions under which Dirichlet problems have solutions.

Abstract

We study the fully nonlinear elliptic equation $F(D^2u,Du,u,x) = f$ in a smooth bounded domain $\Omega$, under the assumption 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 Cl\'{e}ment and Peletier to homogeneous, fully nonlinear operators.