First eigenvalue and maximum principle for fully nonlinear singular operators

By Isabeau Birindelli, Françoise Demengel. Advances in Differential Equations, 2004

104 citations recorded by the source corpus.

In this paper we introduce the notion of first eigenvalue for fully nonlinear operators which are nonvariational but homogeneous. It is unnecessary to emphasize the importance of knowing the spectrum of a linear operator. When the operator is a uniformly elliptic operator of second order Lu = tr(A(x)D2u) associated with a Dirichlet problem in a bounded domain Ω the spectrum is a point spectrum bounded from below and the first eigenvalue λ is paramount. It is well known that λ is positive and it satisfies: • There exists a positive function φ satisfying { Lφ + λφ = 0 in Ω φ = 0 on ∂Ω.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e66283106925904339f4c939daf444e334107ada

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Citations in the corpus

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].”
Harnack inequality for singular fully nonlinear operators and some existence results (2010, methodology): “In order to obtain a solution for this equation we need an extra assumption so we can use a comparison principle proved by Birindelli and Demengel in [ 2 ], which is stated in Theorem 6.1.”
Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations (2008, result): “Similar results have been achieved by Birindelli and Demengel [4, 5], Ishii and Yoshimura [21], and much earlier by Lions [24], using stochastic methods, in the case F is an infimum or supremum of a collection of linear operators.”
Nonexistence of Positive Supersolutions of Elliptic Equations via the Maximum Principle (2010, unknown): “We believe that modifications of our arguments will yield analogous results for inequalities involving elliptic operators with lower order terms as well as other types of operators, for instance mean-curvature-type operators, nondivergence form extensions of the p-Laplacian studied by Birindelli and”