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- W2911961287 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 $Omegasubset {mathbb R}^n$ 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-Ampere 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." @default.
- W2911961287 created "2019-02-21" @default.
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- W2911961287 date "2019-01-21" @default.
- W2911961287 modified "2023-09-27" @default.
- W2911961287 title "A generalization of pde's from a Krylov point of view" @default.
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