Dirichlet problems for singular elliptic equations. II
HTML articles powered by AMS MathViewer
- by Chi Yeung Lo PDF
- Proc. Amer. Math. Soc. 58 (1976), 201-204 Request permission
Abstract:
Consider an elliptic equation \[ ( \ast )\quad L[u] = \sum \limits _{i,j = 1}^n {{a_{ij}}\frac {{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}} + \sum \limits _{i = 1}^n {{b_i}\frac {{\partial u}}{{\partial {x_i}}} + cu = f} } \] 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
-
E. Hopf, Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen vom elliptischen Typus, S.-B. Preuss. Akad. Wiss. 19 (1927), 147-152.
- Chi Yeung Lo, Dirichlet problems for singular elliptic equations, Proc. Amer. Math. Soc. 39 (1973), 337–342. MR 316895, DOI 10.1090/S0002-9939-1973-0316895-X
- J. Schauder, Über lineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z. 38 (1934), no. 1, 257–282 (German). MR 1545448, DOI 10.1007/BF01170635 —, Numerische Abschätzungen in elliptischen linearen Differentialgleichungen, Studia Math. 5 (1934), 34-42.
- N. S. Hall, D. W. Quinn, and R. J. Weinacht, Poisson integral formulas in generalized bi-axially symmetric potential theory, SIAM J. Math. Anal. 5 (1974), 111–118. MR 338408, DOI 10.1137/0505012
Additional Information
- © Copyright 1976 American Mathematical Society
- 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