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On an elliptic boundary value problem not in divergence form


Author: Nguyên Phuong Các
Journal: Proc. Amer. Math. Soc. 88 (1983), 47-52
MSC: Primary 35P15; Secondary 35J25
DOI: https://doi.org/10.1090/S0002-9939-1983-0691277-6
MathSciNet review: 691277
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Abstract: Let $ G$ be a bounded domain in $ {R^n}(n \geqslant 2)$ with smooth boundary $ \partial G$. We consider the boundary value problem $ Mu - cu = f$ on $ G$, $ u = 0$ on $ \partial G$, where $ M$ is an elliptic differential operator not in divergence form. We discuss the characterization of the first eigenvalue $ {\lambda _0}$ of $ M$ and the solvability of the boundary value problem in terms of the relationship between $ c( \cdot )$ and $ {\lambda _0}$.


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

  • [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620-709. MR 0415432 (54:3519)
  • [2] J. M. Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris Sér. A 265 (1967), 333-336. MR 0223711 (36:6759)
  • [3] M. Chicco, Solvability of the Dirichlet problem in $ {H^{2,p}}(\Omega )$ for a class of linear second order elliptic partial differential equations, Boll. Un. Mat. Ital. 4 (1971), 374-387. MR 0298209 (45:7261)
  • [4] M. A. Krasnosel'skii, Positive solutions of operator equations, Noordhoff, Groningen, 1964. MR 0181881 (31:6107)
  • [5] P. L. Lions, Problèmes elliptiques de 2ème ordre non sous forme divergence, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), 263-271. MR 559671 (80m:35023)
  • [6] J. Serrin, A remark on the preceding paper of Amann, Arch. Rational Mech. Anal. 44 (1972), 182-186. MR 0410080 (53:13830)
  • [7] H. Amann and M. G. Crandall, On some existence theorems for semilinear elliptic equations, Indiana Univ. Math. J. 27 (1978), 779-790. MR 503713 (80a:35047)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0691277-6
Keywords: First eigenvalue, solvability of boundary value problem
Article copyright: © Copyright 1983 American Mathematical Society

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