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

We introduce and investigate the notion of a “generalized equation”, which extends nonlinear elliptic equations, and which is based on the notions of subequations and Dirichlet duality. Precisely, a subset H⊂Sym²(Rⁿ) is a generalized equation if it is an intersection H=E∩(−G˜) where E and G are subequations and G˜ is the subequation dual to G. We utilize a viscosity definition of “solution” to H. The mirror of H is defined by H^⁎≡G∩(−E˜). One of the main results (Theorem 2.6) concerns the Dirichlet problem on arbitrary bounded domains Ω⊂Rⁿ for solutions to H with prescribed boundary function φ∈C(∂Ω). We prove that: (A) Uniqueness holds ⇔ H has no interior, and (B) Existence holds ⇔ H^⁎ has no interior. For (B) the appropriate boundary convexity of ∂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 H⊂Sym²(Rⁿ) which can be written as generalized equations are intrinsically characterized. For such an H the set of subequation pairs (E,G) with H=E∩(−G˜) is partially ordered (see (1.10)). If (E,G)≺(E^′,G^′), then any solution for the first is also a solution for the second. Furthermore, in this ordered set there is a canonical least element, contained in all others. A general form of the main theorem, which holds on any manifold, is also established.