Viscosity Solutions Of Partial Differential Equations On Riemannian Manifolds And Stochastic Partial Differential Equations

This paper is a survey on viscosity solutions of partial differential equations. We focus on the existence, uniqueness, regularity of Hamilton-Jacobi equations and second order partial differential equations on Riemannian manifold, fully nonlinear stochastic partial dif-ferential equations, nonlinear stochastic partial differential equations.Firstly we introduce existence, uniqueness and regularity of Hamilton-Jacobi equations on Riemannian manifold.We will restrict ourselves mainly to one of the most interesting examples of first-order Hamilton-Jacobi equations, namely equations of the formWe can have the following maximum principle.Theorem 1 Let f,g:M→R be bounded uniformly continuous functions, and F:T*M→R be intrinsically uniformly coutinuous. If u is a bounded viscosity subsolution of u+F(du)=f, and v is a bounded viscosity supersolution of v+F(dv)=g, then v-u≥inf(g-f).Takingf=g=0, we can prove uniquness of the equations:Theorem 2 Let M be a complete Riemannian manifold which is uniformly locally convex and has a strictly positive injectivity radius. Let F:T*M→R be a intrinsically uniformly continuous function. Aussume also that there is a constant A>0, so that-A≤F(x,0x)≤A, (?)x∈M. Then, there exists a unique bounded viscosity solution of the equation u+F(du)=0.Then, Daniel Azagra, Juan Ferrera, Fernando Lopez-Mesas prove the maxmiun princi-ple for evolution Hamilton-Jacobi equation Theorem 3 Let M be a complete Riemannian manifold with strictly positive convexity and injectivity radii, let u0, v0:M→R be two bounded, uniformly continuous functions, and let F:T*M→R be an intrinsically uniformly continuous function. Assume that u is a bounded viscosity subsolution of And that v is a bounded viscosity supersolution of Then (?) (u-v)≤(?)(u0-v0).In the case when u0=v0, we can immediately have the following uniqueness result:Corollary 4 Let M be a complete Riemannian manifold with strictly positive convexity and injectivity radii, let u0:M→R be a bounded, uniformly continuous function, and let F:[0,∞)×T*M→R be an intrinsically uniformly continuous function. Assume that v,ωare bounded viscosity solutions of Then v=ω.We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations defined on a finite-dimensional Riemannian manifold M.Now we can prove the main comparison result for the Dirichlet problem.Theorem 5 LetΩbe a bounded open subset of a complete finite-dimensional Rieman-nian manifold M, and F:χ→R be proper, continuous, and satisfy: (1)There existsγ>0, so that for r≤s; and(2)There exists a function cu:[0,+∞]→0,+∞] with (?)ω(t)=0 and so that for all x,y∈Ω, r∈R,P∈T2,s(M)x, Q∈T2,s(M)y, with where Aa is the second derivative of the functionφα(x, y)=(?)d(x, y)2 at point (x, y)∈M×M, and the points x, y are assumed to be close enough to each other so that d(x, y)0;(A2) The functionf:Ω×[0, T]×RN×R×Rk→R is a continuous random field such that for fixed (x, y, p),f(·,·, x, y,σ*(x)p) is{(?)tB}-progressively measurable; and there exists some constant K>0 such that for P-a.e.ω∈Ω,(A3) The function u0:RN→R is continuous and, such that for some constants K,P>(A4) The function g∈Cb0,2,3([0, T]×RN×R; Rd); (A4') g satisfies (A4), and for anyε>0, there exists a function Gε∈C1,2,2,2([0, T]×Rd×RN×R), such that(A5) There exists an increasing, F-adapted process (?)={(?); t≥0}, such thatWe are now ready to prove the existence and uniquess of the stochastic viscosity solu-tions.Theorem 6 Assume(A1)-(A4). Then the random field v is a stochastic viscosity solu-tion of SPDE(f,0); and u is a stochastic viscosity solution to S PDE(f, g) respectively.Finally, if in addition (A4') holds and u0 is uniformly bounded, then the random fields u and v are both stochastically uniformly bounded.Corollary 7 Assume (A1)-(A5). Then(1) If v1∈C(FB, [0, T]×RN) is a stochastic viscosity solution and v2∈C(FB, [0, T]×RN) is anω-wise viscosity solution of S PDE(f,0), and both are uniformly stochastically bounded, then v1(t, x)≡v2(t, x) for all (t, x)∈[0, T]×RN P-a.s.;(2) The uniformly stochastically boundedω-wise viscosity solution to SPDE(f,0) is unique. In particular, if f is deterministic, then the uniformly bounded, deterministic viscosity solution of S PDE(f,0) is unique;(3) If in addition (A4') also holds, then the stochastic viscosity solution to SPDE(f, g) is unique among uniformly stochastically bounded random fields in C(FB, [0, T]×RN).