Geodesic
On the best Hölder exponents for generalized solutions of the Dirichlet problem for a second order elliptic equation
Sbornik. Mathematics, Tome 59 (1988) no. 1, pp. 113-127 
V. A. Kondrat'ev; J. Kopáček; O. A. Oleinik. On the best Hölder exponents for generalized solutions of the Dirichlet problem for a second order elliptic equation. Sbornik. Mathematics, Tome 59 (1988) no. 1, pp. 113-127. http://geodesic.mathdoc.fr/item/SM_1988_59_1_a6/
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     title = {On the best {H\"older} exponents for generalized solutions of the {Dirichlet} problem for a~second order elliptic equation},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The authors study the behavior of generalized solutions of the Dirichlet problem for a second order elliptic equation in a neighborhood of a boundary point. Under certain assumptions on the structure of the boundary of the domain in such a neighborhood, and on the coefficients of the equation, a power modulus of continuity is obtained at the boundary point for generalized solutions of the Dirichlet problem, the exponent being best possible for domains with the indicated structure of the boundary near that point. The assumptions on the coefficients of the equation are essential, as an example shows. With the help of the indicated results on the modulus of continuity at boundary points, it is then shown that generalized solutions belong to Hölder spaces in the closure of the domain, the Hölder exponent again being best possible for the class of domains under consideration. Bibliography: 8 titles.

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[2] Verzhbinskii G. M., Mazya V. G., “Asimptoticheskoe povedenie reshenii ellipticheskikh uravnenii vtorogo poryadka vblizi granitsy. I”, Sib. matem. zhurn., 12:6 (1971), 1217–1249 ; “Асимптотическое поведение решений эллиптических уравнений второго порядка вблизи границы. II”, Сиб. матем. журн., 13:6 (1972), 1239–1271 | Zbl | Zbl

[3] De Dzhordzhi E., “O differentsiruemosti i analitichnosti ekstremalei kratnykh regulyarnykh integralov”, Matematika (sb. perevodov), 4:6 (1960), 23–38

[4] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[5] Littman W., Stampacchia G., Weinberger H., “Regular points for elliptic equations with discontinuous coefficients”, Ann. Scuola Norm. Sup. di Pisa, ser. 3, 17 (1963), 43–77 | MR | Zbl

[6] Agmon S., Duglas A., Nirenberg L., Otsenki reshenii ellipticheskikh uravnenii vblizi granitsy, IL, M., 1962

[7] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Izd-vo LGU, L., 1950

[8] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka, M., 1977 | MR | Zbl