Weak solutions of the Gellerstedt and the Gellerstedt-Neumann problems

Abstract:

We consider the question of existence of weak and semistrong solutions of the Gellerstedt problem \[ u{|_{{\Gamma _0} \cup {\Gamma _1} \cup {\Gamma _2}}} = 0\] and the Gellerstedt-Neumann problem \[ ({d_n}u = k(y){u_x}dy - {u_y}dx{|_{{\Gamma _0}}} = 0,\qquad u{|_{{\Gamma _1} \cup {\Gamma _2}}} = 0)\] for the equation of mixed type \[ L[u] \equiv k(y){u_{xx}} + {u_{yy}} + \lambda u = f(x,y),\qquad \lambda = \operatorname {const} < 0\] in a region $G$ bounded by a piecewise smooth curve ${\Gamma _0}$ lying in the half-plane $y > 0$ and intersecting the line $y = 0$ at the points $A( - 1,0)$ and $B(1,0)$. For $y < 0$, $G$ is bounded by the characteristic curves ${\gamma _1}(x < 0)$ and ${\gamma _2}(x > 0)$ of (1) through the origin and the characteristics ${\Gamma _1}$ and ${\Gamma _2}$ through $A$ and $B$ which intersect ${\gamma _1}$ and ${\gamma _2}$ at the points $P$ and $Q$, respectively. Using a variation of the energy integral method, we give sufficient conditions for the existence of weak and semistrong solutions of the boundary value problems (Theorems 4.1, 4.2, 5.1).