The Dirichlet Problem for a Class of Elliptic Difference Equations

Under suitable assumptions on the order of nonlinearity we prove existence and uniqueness theorems for difference Dirichlet problems of divergence type. We also show that the discrete solutions converge to a solution of the continuous problem. We do not assume that our equation comes from a variational problem. Some of our results are constructive or allow for the application of constructive methods. Introduction. Let Í2 be a plane bounded region such that the boundary of S2, dQ, is of class C1. If P is a point in the plane, then we denote it by (xu x2). Place a square grid on the plane of grid width h. All points of the form (mh, nh), with m and n integers, are called mesh points. Let P0 = (x0i, x02) be a mesh point. Then a neighborhood of P0 is the set of points 9l(P0) = {(x0i, x02), (x0i + h, x02), (x01 + h, x02 + h), (xou x02 + h), (Xoi — h, Xo2 ~r h), (Xoi h, Xo2), (Xoi n, Xo2 n), (x0i, Xo2 h), (Xoi -r h, Xo2 h)\. We define tih as that set of mesh points P such that 3l(P) C Û, and we define the boundary of Qh as those mesh points P in Ö such that at least one element of 9l(P) is in the exterior of Q = Ü + dfl. Let V(P) be any function which is everywhere finite for P G £2* + 3Í2A = Üh; such a function will be called a mesh function. Let P = (xu x2) he a mesh point. Then, for any mesh function we define forward difference quotients by V.AP) = { V(xi + h, x2) V(P))/h, VX.(P) = { V(xi, x2 + h) V(P))/h and backward difference quotients by Vtt(P) = { V(P) Vtex h, X2))/h, Vt,(P) = Í V(P) Vtei, x2 h)}/h. The vector (VZl(P), Vt,(P)) is denoted by V„K(P) and the vector (V,AP), Vt,(P)) is denoted by VhV(P). We denote by Oj; (and Q'h') the set of points P £_&» such that, for any mesh function V(P) defined on Ùh, the vector V»F(P) (and \/kV(P)) is defined using only mesh points in Qh. If Dh is any set of mesh points in the plane, then we define mh(Dh) to be h2 times the number of points in Dh. A set of mesh points will be called connected iff one can go from any mesh point in the set to any other mesh point in the set along line segments of length h connecting only elements of the set. We assume all mesh sets are connected. A function u(P) G C"c(ü) iff the support of u(P) is a compact subset of ü and all pth order partial derivatives of u(P) are continuous over 0. The set £m(0) denotes all Received December 2, 1969, revised December 3, 1970. AMS 1970 subject classifications. Primary 39A10; Secondary 35J60.