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

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

2008-06-15

arxiv.org/abs/0806.2473v4

J. Differential Equations 246 (2009) 2958-2987.

Mathematics

Analysis of PDEs

32 pages

Scientific paper

10.1016/j.jde.2008.10.026

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.

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