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Transactions of the American Mathematical Society

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

 
 

 

Dirichlet problem for degenerate elliptic equations


Authors: Avner Friedman and Mark A. Pinsky
Journal: Trans. Amer. Math. Soc. 186 (1973), 359-383
MSC: Primary 35J70; Secondary 60H15
DOI: https://doi.org/10.1090/S0002-9947-1973-0328345-2
MathSciNet review: 0328345
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Abstract: Let ${L_0}$ be a degenerate second order elliptic operator with no zeroth order term in an m-dimensional domain G, and let $L = {L_0} + c$. One divides the boundary of G into disjoint sets ${\Sigma _1},{\Sigma _2},{\Sigma _3};{\Sigma _3}$ is the noncharacteristic part, and on ${\Sigma _2}$ the “drift” is outward. When c is negative, the following Dirichlet problem has been considered in the literature: $Lu = 0$ in G, u is prescribed on ${\Sigma _2} \cup {\Sigma _3}$. In the present work it is assume that $c \leq 0$. Assuming additional boundary conditions on a certain finite number of points of ${\Sigma _1}$, a unique solution of the Dirichlet problem is established.


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Keywords: Dirichlet problem, degenerate elliptic operator, weak solution, stochastic differential equations, stochastic integrals, exit time
Article copyright: © Copyright 1973 American Mathematical Society