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.
Categories: math.AP math.DG