Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE

Abstract

Two results are proved in the paper. The first is a uniqueness theorem for viscosity solutions of Dirichlet boundary value problems for Bellman-Isaacs equations with just measurable lower order terms. The second is a proof that there always exist maximal and minimal viscosity solutions of Dirichlet boundary value problems for fully nonlinear, uniformly elliptic PDE that are measurable in the x-variable.2000 Mathematics Subject Classification. 35J60, 35J65, 35J25, 49L25.

Nonuniqueness for the Dirichlet Problem for Fully Nonlinear Elliptic Operators and the Ambrosetti–Prodi Phenomenon

“…This is satisfied for instance when H is Hölder continuous in x with a sufficiently large Hölder constant or when H is convex in M and H(M, 0, 0, x) is uniformly continuous. Many other conditions which ensure uniqueness for proper equations can be found in [19], [31], [14], [32]. For instance, F can be a Hamilton-Jacobi-Bellman (HJB) operator, that is, a supremum of linear second order operators with bounded coefficients and continuous second order coefficients -see [37] for examples and discussions.…”

Existence of solutions to a fully nonlinear free transmission problem

“…and the functions v n = u − ϕ n are L p -viscosity supersolutions of (16). It is well known -see for instance [15, Theorem III.1-( 1)], together with Section V.1 there -that ( 16) satisfies comparison principle; so we have u n ≤ v n on Ω.…”

“…A consequential aspect of (1) concerns the dependence of the operator on the solutions. We observe the equation does not satisfy the usual structure conditions (e.g., [6, Condition (SC)]) under which existence of L p -viscosity solutions is well known [8,16,30]. As a result, the existence of L p -viscosity solutions to (1) requires a different strategy, which is based on the approach employed in [14].…”

Viscosity solutions of the Monge–Ampère equation with the right hand side in $L^p$

“…solution with W 2,p regularity guarantees uniqueness of L p -viscosity solutions. The paper [14] gives a remarkable contribution to the uniqueness issue. The authors prove C 1,α regularity of the viscosity solutions under a certain structure condition.…”