TY - JOUR
T1 - Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations
AU - Armstrong, Scott N.
N1 - Copyright:
Copyright 2009 Elsevier B.V., All rights reserved.
PY - 2009/4/1
Y1 - 2009/4/1
N2 - We study the fully nonlinear elliptic equation(0.1)F (D2 u, D u, 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 Clément and Peletier [P. Clément, 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.
AB - We study the fully nonlinear elliptic equation(0.1)F (D2 u, D u, 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 Clément and Peletier [P. Clément, 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.
KW - Anti-maximum principle
KW - Dirichlet problem
KW - Fully nonlinear elliptic equation
KW - Principal eigenvalue
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U2 - 10.1016/j.jde.2008.10.026
DO - 10.1016/j.jde.2008.10.026
M3 - Article
AN - SCOPUS:60349114337
VL - 246
SP - 2958
EP - 2987
JO - Journal of Differential Equations
JF - Journal of Differential Equations
SN - 0022-0396
IS - 7
ER -