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 $Omegasubset {mathbb R}^n$ 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.
تحميل البحث