A generalization of pde's from a Krylov point of view

Abstract

We introduce and investigate the notion of a `generalized equation' of the form $f(D^2 u)=0$, based on the notions of subequations and Dirichlet duality. Precisely, a subset ${\mathbb H}\subset {\rm Sym}^2({\mathbb R}^n)$ is a generalized equation if it is an intersection ${\mathbb H} = {\mathbb E}\cap (-\widetilde{\mathbb G})$ where ${\mathbb E}$ and ${\mathbb G}$ are subequations and $\widetilde{\mathbb G}$ is the subequation dual to ${\mathbb G}$. We utilize a viscosity definition of `solution' to ${\mathbb H}$. The mirror of ${\mathbb H}$ is defined by ${\mathbb H}^* \equiv {\mathbb G}\cap (-\widetilde {\mathbb E})$. One of the main results here concerns the Dirichlet problem on arbitrary bounded domains $Ω\subset {\mathbb R}^n$ for solutions to ${\mathbb H}$ with prescribed boundary function $φ\in C(\partial Ω)$. We prove that: (A) Uniqueness holds $\iff$ ${\mathbb H}$ has no interior, and (B) Existence holds $\iff$ ${\mathbb H}^*$ has no interior. For (B) the appropriate boundary convexity of $\partial Ω$ must be assumed. Many examples of generalized equations are discussed, including the constrained Laplacian, the twisted Monge-Ampère equation, and the $C^{1,1}$-equation. The closed sets ${\mathbb H}$ which can be written as generalized equations are intrinsically characterized. For such an ${\mathbb H}$ the set of subequation pairs with ${\mathbb H} = {\mathbb E}\cap (-\widetilde{\mathbb G})$ is partially ordered, and there is a canonical least element, contained in all others. Harmonics for the canonical equation are harmonic for all others giving ${\mathbb H}$. A general form of the main theorem, which holds on any manifold, is also established.