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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Weak solutions of the Gellerstedt and the Gellerstedt-Neumann problems


Authors: A. K. Aziz and M. Schneider
Journal: Trans. Amer. Math. Soc. 283 (1984), 741-752
MSC: Primary 35M05
DOI: https://doi.org/10.1090/S0002-9947-1984-0737897-7
MathSciNet review: 737897
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Abstract: We consider the question of existence of weak and semistrong solutions of the Gellerstedt problem

$\displaystyle u{\vert _{{\Gamma _0} \cup {\Gamma _1} \cup {\Gamma _2}}} = 0$

and the Gellerstedt-Neumann problem

$\displaystyle ({d_n}u = k(y){u_x}dy - {u_y}dx{\vert _{{\Gamma _0}}} = 0,\qquad u{\vert _{{\Gamma _1} \cup {\Gamma _2}}} = 0)$

for the equation of mixed type

$\displaystyle L[u] \equiv k(y){u_{xx}} + {u_{yy}} + \lambda u = f(x,y),\qquad \lambda = \operatorname{const} < 0$

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).

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DOI: https://doi.org/10.1090/S0002-9947-1984-0737897-7
Article copyright: © Copyright 1984 American Mathematical Society

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