On oblique derivative problems for fully nonlinear second-order elliptic PDE's on domains with corners
Paul G. Dupuis, Hitoshi Ishii · Hokkaido Mathematical Journal · 1991
For x\in U and a real function f on U, D^{+}f(x) and D^{-}f(x) denote the superdifferential and the subdifferential of f at x , respectively, that is, D^{+}f(x)=\{p\in R^{N} : f(x+h)\leq f(x)+\langle p, h\rangle+o (|h|) for x+h\in U and as harrow 0 }, and On oblique derivative problems for fully nonlinear second-Order elliptic PDE's on domains with comers 137 D^{-}f(x)=\{p\in R^{N} : f(x+h)\geq f(x)+\langle p, h\rangle+o(|h|)for x+h\in U and as harrow 0 }.For x\in U the superdifferential D^{2,+}f(x) and the subdifferential D^{2,-}f(x) of order 2 at x\in U are defined by D^{2,+}f(x)=\{(p, A)\in R^{N}\cross S^{N} : f(x+h) \leq f(x)+\langle p, h\rangle+\frac{1}{2}\langle Ah, h\rangle +o(|h|^{2}) for x+h\in U and as harrow 0 } and D^{2,-}f(x)=\{(p, A)\in R^{N}\cross S^{N} : f(x+h) \geq f(x)+\langle p, h\rangle+\frac{1}{2}\langle Ah, h\rangle +o(|h|^{2}) for x+h\in U and as harrow 0 }, respectively.Let U be an open subset of R^{N} C^{1.+}(U) denotes the set of all real functions f on U such that f\in C^{0.1}(U) and D^{+}f(x) eq\emptyset for all x\in U. C^{2,+}(U) denotes the set of all real functions f\in C^{0,1}(U) having the property: for each compact K\subset U there is a constant C such that if x\in K, then (p, CI)\in D^{2,+}f(x) for some p\in R^{N}-Note that C^{2,+}(U)\subset C^{1.+}(U)