On viscosity solutions to the Dirichlet problem for elliptic branches of inhomogeneous fully nonlinear equations

Abstract

For scalar fully nonlinear partial differential equations F (x, D 2 u(x)) = 0 with x ∈ Ω R N , we present a general theory for obtaining comparison principles and well posedness for the associated Dirichlet problem, where F (x, ·) need not be monotone on all of S(N ), the space of symmetric N ×N matrices. We treat admissible viscosity solutions u of elliptic branches of the equation in the sense of Krylov [20] and extend the program initiated by Harvey and Lawson [11] in the homogeneous case when F does not depend on x. In particular, for the set valued map Θ defining the elliptic branch by way of the differential inclusion D 2 u(x) ∈ ∂Θ(x), we identify a uniform continuity property which ensures the validity of the comparison principle and the applicability of Perron's method for the differential inclusion on suitably convex domains, where the needed boundary convexity is characterized by Θ. Structural conditions on F are then derived which ensure the existence of an elliptic map Θ with the needed regularity. Concrete applications are given in which standard structural conditions on F may fail and without the request of convexity conditions in the equation. Examples include perturbed Monge-Ampère equations and equations prescribing eigenvalues of the Hessian.

Principal eigenvalues for k-Hessian operators by maximum principle methods

“…Proof. The existence and uniqueness for Σ k -admissible viscosity solutions follows from the main results in [7]. See Theorem 1.2 as applied in section 5 of that paper.…”

“…As suggested in the title, we will make use of various comparison and maximum principles for Σ k -admissible viscosity subsolutions and supersolution in the sense of Definition 2.5 and the subsequent remarks and examples. While they will be special cases of the results in [12], [7] and [8], for the convenience of the reader we will give the precise statements and some indication of the proofs. In all that follows Ω will be an open bounded domain in R N .…”

“…The admissible supersolution definition takes its inspiration from Krylov [26] and was developed in [7] for equations of the form F (x, D 2 u) = 0. In the convex Monge-Ampère case k = N of (2.31), an analogous definition was given by Ishii-Lions [18].…”

The Inhomogeneous Dirichlet problem for natural operators on manifolds

“…The final ingredient is the following. The pure second order case of the following result follows from the work of Cirant and Payne [7]. In fact their work is much more general; they consider operators of the form f (x, D 2 u).…”

“…-Of course an interesting case of the work here is when (F, f ) = (F, f ) is itself constant coefficient in euclidian space. This case (pure second-order) is contained in the work of Cirant and Payne [7], where other quite nice theorems are proved.…”