A second order, non-linear elliptic boundary value problem with generalized Goursat data

Abstract

In this paper we show that under certain conditions a unique {complex valued) solution exists for the non-linear elliptic equation (1.1) AU = [(x, y, u, u,~, uu) , which satisfies a generalized boundary condition on each of the analytic curves I'~=-{(x,y) ly=f~(x), o~x~_a} and F2=-{(x,y) lx=f~(y}, o~y~'b} namely u~(~, y) = ~o(X) u(x, y) + ~(~ uAx, y) + ~(x~) o, r~ (1.2) u~(x, y)= ~o(Y)u(x, y) -~ ~:(y)u,~(x, y) -~ h(y) on F2(1.3) u(0, 0) : y (0, 0)= F, A F2.The coefficients ~a, ~a, g, h and y are assumed to be complex valued. By separating real and imaginary parts in equations {1.1)-(1.3) our results may be seen to apply to a system of real partial differential equations.

The existence for the solution of the elliptic Cauchy problem

“…The above method gives the integral representation for the solution in a symmetric domain (or conformai symmetric domain), and information of continuous dependence. This theorem can be considered as a continuation of Hadamard [4], Garabedian [3], Henrici [5], Colton [2], and Aziz, Gilbert, Howard [1].…”

“…Also let A(z 9 £), B(z 9 £) be holomorphic functions for z, £ e D\JOKJD 9 where 5={z|z e D). The aim of this note is to announce some recent results on the global existence for the Cauchy problem of the first order linear elliptic equations (in complex normal form) : (1) dW/dz = A(z 9 z)W+ B(z, z)W where djdz = \(djdx + id/dy) • W = u + iv.…”

Non-linear initial data for second and higher order semi-linear elliptic equations.

“…Such problems, it was noted, occur when one considers inclusion phenomena in elasticity [8]. Mathematically, the approach taken there and in the present paper, are extensions and developments of the technique initiated in [1], [2]. The spirit of the paper is in function theoretic style initiated by Vekua [9] for the treatment of higher order equations.…”

“…1) with generalized, non-linear data (2. The linear case had been investigated by Colton [4], [5], [6] using a function theoretic approach similar to [1], [2]. 3).…”

Generalized Initial Value Problem for Fourth Order Equation of Elliptic Type

“…Problems of such a type were studied by many authors, among others A. K. Aziz and R. P. Gilbert [1], [2], D. L. Colton [3], [4] for the second order elliptic equations in two independent variables. In [5] D. L. Colton was studying the Cauchy problem for a class of fourth-order elliptic equations.…”