Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Dirichlet problems for singular elliptic equations

Author: Chi Yeung Lo
Journal: Proc. Amer. Math. Soc. 39 (1973), 337-342
MSC: Primary 35J70
MathSciNet review: 0316895
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)

  • [1] P. Brousse and H. Poncin, Quelques résultats generaux concernant la détermination de solutions d'équations elliptiques par les conditions aux frontières, Mémoires sur la mécanique des fluides offerts à M. Dimitri P. Riabouchinsky, Publ. Sci. Tech. Ministère de l'Air, Paris, 1954. MR 16, 368.
  • [2] E. Hopf, Elementare Bemerkungen über die Lösungen partieller Differential gleichungen vom elliptischen Typus, S.-B. Preuss. Akad. Wiss. 19 (1927), 147-152.
  • [3] Alfred Huber, On the uniqueness of generalized axially symmetric potentials, Ann. of Math. (2) 60 (1954), 351–358. MR 0064284,
  • [4] Alfred Huber, Some results on generalized axially symmetric potentials, Proceedings of the conference on differential equations (dedicated to A. Weinstein), University of Maryland Book Store, College Park, Md., 1956, pp. 147–155. MR 0083050
  • [5] Martin Schechter, On the Dirichlet problem for second order elliptic equations with coefficients singular at the boundary, Comm. Pure Appl. Math. 13 (1960), 321–328. MR 0113031,
  • [6] J. Schauder, Über lineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z. 38 (1934), no. 1, 257–282 (German). MR 1545448,
  • [7] -, Numerische Abschätzungen in elliptischen linearen Differential-gleichungen, Studia Math. 5 (1934), 34-42.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35J70

Retrieve articles in all journals with MSC: 35J70

Additional Information

Keywords: Singular elliptic equations, boundary value problem, maximum principle, barrier function, Schauder lemma
Article copyright: © Copyright 1973 American Mathematical Society