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 $\Omega\subset {\mathbb R}^n$ for solutions to ${{\mathbb H}}$ with prescribed boundary function $\varphi \in C(\partial \Omega)$. 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 $\partial \Omega$ must be assumed. Many examples of generalized equations are discussed, including the constrained Laplacian, the twisted Monge-Amp\`ere 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.