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Dirichlet problems for singular elliptic equations. II


Author: Chi Yeung Lo
Journal: Proc. Amer. Math. Soc. 58 (1976), 201-204
MSC: Primary 35J70
DOI: https://doi.org/10.1090/S0002-9939-1976-0412606-0
MathSciNet review: 0412606
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Abstract: Consider an elliptic equation

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

in a bounded domain $ G$ in the half space $ {x_n} > 0$ with boundary $ \partial G = {S_1} \cup {S_2}$ of class $ {C^{2 + \alpha }}$ where $ {S_1}$ is contained in the hyperplane $ {x_n} = 0$ and $ {S_2}$ lies entirely in $ {x_n} > 0$. The coefficient $ {b_n}$ possesses certain type of singularity at $ {x_n} = 0$. Let $ {b_n} = h/k$ where $ h \in {C^\alpha }(\bar G)$ and $ k \to 0$ as $ {x_n} \to 0$. It is found that the solvability of the Dirichlet problem of $ L[u] = f$ in $ G$ depends on the nature of singularity of $ {b_n}$ and also the value of $ h$ at $ {x_n} = 0$.

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

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0412606-0
Keywords: Singular elliptic equations, boundary value problem, barrier function
Article copyright: © Copyright 1976 American Mathematical Society

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