Lu=x^u+kux=x f(x,y) Eliptik denklemi için genelleştirilmiş fonksiyon sınıflarında bazı problemler

Abstract

This work consists of six parts. The first part, which is the introduction part, organized to help, the other parts in the points of understanding. In the second part we have been given domain D and a solution related with the equation (1.1) given, on a part of the boundary of D. Under these conditions we examined the solution of the boundary value problem in domain D. We have also examined the properties of existency, uniticty and stability for the solution of the problem in some known spaces. In these examinations we used a priori, Galerkin and Carleman methods. In the third part we add the term K(x, y, %, tj) u (£, t|) dÇ dTj to the equation (1.1) and examined the solution of the new boundary value problem as in the second part. After this we obtained some results similar to the second part. In part four, we examined the mixed problem related with the equation (1.1 ). In part five, we examined the inverse problem of finding the pair of ( u, f ) related with the equation Lu = x f ( x, 'y ). In the last part, which is the sixth part, we examined the solution of Cauchy problem related with the equation of Lu = x f ( x, 'y ). After that we add the condition of u| = u 2 to the Cauchy problem examined the inverse problem of finding ( u, f ) pair. Key Words : Potential theory, elliptic type equation, inverse problem, mixed problem generalized function, Galerkin method.