A generalization of the flat cone condition for regularity of solutions of elliptic equations

Lieberman, Gary M.

doi:10.1090/S0002-9939-1987-0884468-6

A generalization of the flat cone condition for regularity of solutions of elliptic equations
HTML articles powered by AMS MathViewer

by Gary M. Lieberman PDF

Proc. Amer. Math. Soc. 100 (1987), 289-294 Request permission

Abstract:

Barrier arguments are used to prove regularity of boundary points for a large class of uniformly elliptic operators when the domain satisfies a geometric condition. The condition is that the exterior of the domain contains a suitable lower dimensional set.

References

Yu. A. Alkhutov, On the regularity of boundary points with respect to the Dirichlet problem for second-order elliptic equations, Mat. Zametki 30 (1981), no. 3, 333–342, 461 (Russian). MR 633981
A. Azzam, Smoothness properties of solutions of mixed boundary value problems for elliptic equations in sectionally smooth $n$-dimensional domains, Ann. Polon. Math. 40 (1981), no. 1, 81–93. MR 645800, DOI 10.4064/ap-40-1-81-93
Patricia Bauman, A Wiener test for nondivergence structure, second-order elliptic equations, Indiana Univ. Math. J. 34 (1985), no. 4, 825–844. MR 808829, DOI 10.1512/iumj.1985.34.34045
Kai Lai Chung, Lectures from Markov processes to Brownian motion, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 249, Springer-Verlag, New York-Berlin, 1982. MR 648601
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627

-capacity and its application to the study of solutions of a second order elliptic equation with discontinuous coefficients

5

Gary M. Lieberman, Mixed boundary value problems for elliptic and parabolic differential equations of second order, J. Math. Anal. Appl. 113 (1986), no. 2, 422–440. MR 826642, DOI 10.1016/0022-247X(86)90314-8
W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 43–77. MR 161019
Keith Miller, Barriers on cones for uniformly elliptic operators, Ann. Mat. Pura Appl. (4) 76 (1967), 93–105 (English, with Italian summary). MR 221087, DOI 10.1007/BF02412230
Keith Miller, Exceptional boundary points for the nondivergence equation which are regular for the Laplace equation and vice-versa, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 315–330. MR 229961
Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984. Corrected reprint of the 1967 original. MR 762825, DOI 10.1007/978-1-4612-5282-5
Carlo Pucci, Regolarità alla frontiera di soluzioni di equazioni ellittiche, Ann. Mat. Pura Appl. (4) 65 (1964), 311–328 (Italian, with English summary). MR 171990, DOI 10.1007/BF02418230