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Dirichlet problems for singular elliptic equations


Author: Chi Yeung Lo
Journal: Proc. Amer. Math. Soc. 39 (1973), 337-342
MSC: Primary 35J70
DOI: https://doi.org/10.1090/S0002-9939-1973-0316895-X
MathSciNet review: 0316895
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Abstract: Boundary value problems are formulated for the equation

$\displaystyle ( \ast )\quad L[u] = \sum\limits_{i,j = 1}^n {{a_{ij}}\frac{{{\pa... ...tial {x_i}}}} + \frac{h}{{{x_n}}}\frac{{\partial u}}{{\partial {x_n}}} + cu = f$

in a bounded domain $ G$ in $ {E_n}$ with boundary $ \partial G = {S_1} \cup {S_2}$ where $ {S_1}$ is in $ {x_n} = 0$ and $ {S_2}$ is in $ {x_n} > 0$. A uniqueness theorem is established for $ ( \ast )$ when boundary data is only given on $ {S_2}$ for

$\displaystyle h({x_1}, \cdots ,{x_{n - 1}},0) \geqq 1;$

; whereas an existence and uniqueness theorem for the Dirichlet problem is proved for $ h({x_1},{x_2}, \cdots ,{x_{n - 1}},0) < 1$.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0316895-X
Keywords: Singular elliptic equations, boundary value problem, maximum principle, barrier function, Schauder lemma
Article copyright: © Copyright 1973 American Mathematical Society