On the Interior Smoothness of Solutions to Second-Order Elliptic Equations
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We study the interior smoothness properties of solutions to a linear second-order uniformly elliptic equation in selfadjoint form without lower-order terms and with measurable bounded coefficients. In terms of membership in a special function space we combine and supplement some properties of solutions such as membership in the Sobolev space W2, loc1 and Holder continuity. We show that the membership of solutions in the introduced space which we establish in this article gives some new properties that do not follow from Holder continuity and the membership in W2,loc1.
Keywordselliptic equation function spaces smoothness of solutions
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- 1.DeGiorgi E., “Sulla differenziabilita e l'analiticita delle estremali degli integrali multipli regolari,” Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. (5), 3, 25–43 (1957).Google Scholar
- 2.Nash J., “Continuity of solutions of parabolic and elliptic equations,” Amer. J. Math., 80, 931–954 (1958).Google Scholar
- 3.Moser J., “A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations,” Comm. Pure Appl. Math., 13, No.3, 457–468 (1960).Google Scholar
- 4.Ladyzhenskaya O. A. and Ural'tseva N. N., Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1973).Google Scholar
- 5.Gushchin A. K., “Interior smoothness of solution to a second-order elliptic equation,” Dokl. Ross. Akad. Nauk (to appear).Google Scholar
- 6.Gushchin A. K., “On the Dirichlet problem for a second-order elliptic equation,” Mat. Sb., 137 No.1, 19–64 (1988).Google Scholar
- 7.Gushchin A. K. and Mikhailov V. P., “On the existence of boundary values of solutions of an elliptic equation,” Mat. Sb., 182, No.6, 787–810 (1991).Google Scholar
- 8.Gushchin A. K. and Mikhailov V. P., “On solvability of nonlocal problems for a second-order elliptic equation,” Mat. Sb., 185, No.1, 121–160 (1994).Google Scholar
- 9.Gushchin A. K., “Some properties of the solutions of the Dirichlet problem for a second-order elliptic equation,” Mat. Sb., 189, No.7, 53–90 (1998).Google Scholar
- 10.Gushchin A. K., “A condition for the compactness of operators in a certain class and its application to studying solvability of nonlocal problems for elliptic equations,” Mat. Sb., 193, No.5, 17–36 (2002).Google Scholar
- 11.Gushchin A. K., “Carleson's type estimate of solutions of a second-order elliptic equation,” Dokl. Ross. Akad. Nauk, 396, No.1, 15–18 (2004).Google Scholar