Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators

Abstract

We study uniformly elliptic fully nonlinear equations of the type F (D 2 u, Du, u, x) = f (x). We show that convex positively 1-homogeneous operators possess two principal eigenvalues and eigenfunctions, and study these objects; we obtain existence and uniqueness results for nonproper operators whose principal eigenvalues (in some cases, only one of them) are positive; finally, we obtain an existence result for nonproper Isaac's equations.

Existence and regularity results for fully nonlinear equations with singularities

“…This is the main goal of this paper. We are going to show that the recently developed theory of HamiltonJacobi-Bellman and Isaacs equations [8], [22], [5], [9], [29] and [1] permits us to prove the same existence results as in the semi-linear setting, when weak (viscosity) solutions to (1.1) are considered. The regularity result requires a new approach, since none of the methods used in the previous papers on singular problems applies in the fully nonlinear setting.…”

“…For detailed description of the theory and the numerous applications of HJB operators, we refer to the book [15] and to the surveys [24], [33] and [3]. In particular, it is shown in [26] and [29] that, under (S) the operator F has two real "principal half-eigenvalues" λ + 1 (F ) ≤ λ − 1 (F ) that correspond to a positive and a negative eigenfunction and…”

Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls

“…Recently there has been much interest in eigenvalue problems for fully nonlinear PDE since the work of P.-L. Lions [15]. See [3,13,4,18,1,17] for these developments. See also [2,8,12] for some earlier related works.…”

Existence, uniqueness and qualitative properties of positive solutions of quasilinear elliptic equations

“…An important feature of the notion of generalized principal eigenvalue is that if Ω is a smooth and bounded domain, λ(K V , Ω) coincides with the principal eigenvalue λ 1,V (Ω), while if Ω is unbounded λ(K V , Ω) is well defined and can be expressed by a variational formula. For related definitions of generalized principal eigenvalues, the reader is referred to [7,8,9,11] for linear operators, [4,28] for fully nonlinear operators and [13] for singular fully nonlinear operators. To our knowledge, no investigation of generalized principal eigenvalue for quasilinear operators has been previously obtained.…”

“…Note that since C 1 c (Ω) is dense in W 1,p 0 (Ω) with respect to W 1,p norm, the infimum in (1.4) can be taken over C 1 c (Ω). When Ω is an arbitrary (possibly unbounded) domain, following Berestycki et al [7,8,9,11,28], we define This type of eigenvalue was first introduced in a celebrated work of BerestyckiNirenberg-Varadhan [8] for second order operators in bounded (not necessarily smooth) domains, and then was developed to second order operators in unbounded domains [7,9,11]. An important feature of the notion of generalized principal eigenvalue is that if Ω is a smooth and bounded domain, λ(K V , Ω) coincides with the principal eigenvalue λ 1,V (Ω), while if Ω is unbounded λ(K V , Ω) is well defined and can be expressed by a variational formula.…”