Ein Randwertproblem für eine nichtlineare Gleichung gemischtenTyps im ${\mathbb{R}}^3$

Abstract

A boundary value problem for the equation $Tu=Lu—u|u{|}^p=f(x,u),p>0$, is studied in a simply connected bounded domain $G$ of ${\mathbb{R}}^3$. Here $L$ denotes a linear second order differential operator which is elliptic, parabolic or hyperbolic if $x_3>0,x_3=0$ or $x_3<0$, respectively. The boundary of 0 is formed by a non-characteristic and by two characteristic surfaces. The boundary value problem to be solved is to find a solution of the equation in 0 which assumes zero data on the non-characteristic and on one of the characteristic boundary surfaces. It is proved that this problem has a generalized solution belonging to $L^{p+2}$ and to a Sobolew space with weight. Using apriori estimates the solubility of a sequence of approximate problems is shown whose solutions turn out to converge towards a solution of the boundary value problem in question.