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Weak Solutions of The Gellerstedt and the Gellerstedt-Neumann Problems

A. K. Aziz and M. Schneider
Transactions of the American Mathematical Society
Vol. 283, No. 2 (Jun., 1984), pp. 741-752
DOI: 10.2307/1999159
Stable URL: http://www.jstor.org/stable/1999159
Page Count: 12
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Weak Solutions of The Gellerstedt and the Gellerstedt-Neumann Problems
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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 $$\Big(d_nu = k(y)u_x dy - u_y dx|_{\Gamma_0} = 0, u|_{\Gamma_1\cup\Gamma_2} = 0\Big)$$ for the equation of mixed type $$L\lbrack u \rbrack \equiv k(y) u_{xx} + u_{yy} + \lambdau = f(x,y), \lambda = \operatorname{const} < 0$$ 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).

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