Geodesic
H\"older estimates for solutions to degenerate nondivergence elliptic and parabolic equations
Algebra i analiz, Tome 21 (2009) no. 4, pp. 174-195.

Voir la notice de l'article provenant de la source Math-Net.Ru

We deal with a class of nondivergence type elliptic and parabolic equations degenerating at the coordinate hyperplanes. Assuming that the degeneration is coordinatewise and varies regularly, we prove the Hölder continuity of solutions. Also, the approximative solutions are considered. 
@article{AA_2009_21_4_a3,
     author = {A. I. Nazarov},
     title = {H\"older estimates for solutions to degenerate nondivergence elliptic and parabolic equations},
     journal = {Algebra i analiz},
     pages = {174--195},
     publisher = {mathdoc},
     volume = {21},
     number = {4},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AA_2009_21_4_a3/}
}
TY  - JOUR
AU  - A. I. Nazarov
TI  - H\"older estimates for solutions to degenerate nondivergence elliptic and parabolic equations
JO  - Algebra i analiz
PY  - 2009
SP  - 174
EP  - 195
VL  - 21
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AA_2009_21_4_a3/
LA  - ru
ID  - AA_2009_21_4_a3
ER  - 
%0 Journal Article
%A A. I. Nazarov
%T H\"older estimates for solutions to degenerate nondivergence elliptic and parabolic equations
%J Algebra i analiz
%D 2009
%P 174-195
%V 21
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AA_2009_21_4_a3/
%G ru
%F AA_2009_21_4_a3
A. I. Nazarov. H\"older estimates for solutions to degenerate nondivergence elliptic and parabolic equations. Algebra i analiz, Tome 21 (2009) no. 4, pp. 174-195. http://geodesic.mathdoc.fr/item/AA_2009_21_4_a3/

[1] Krylov N. V., Safonov M. V., “Nekotoroe svoistvo reshenii parabolicheskikh uravnenii s izmerimymi koeffitsientami”, Izv. AN SSSR. Ser. mat., 44:1 (1980), 161–175 | MR | Zbl

[2] Safonov M. V., “Neravenstvo Kharnaka dlya ellipticheskikh uravnenii i gëlderovost ikh reshenii”, Zap. nauch. semin. LOMI, 96, 1980, 272–287 | MR

[3] Ladyzhenskaya O. A., Uraltseva N. N., “Otsenki konstanty Gëldera dlya funktsii, udovletvoryayuschikh ravnomerno ellipticheskomu ili ravnomerno parabolicheskomu kvazilineinomu neravenstvu s neogranichennymi koeffitsientami”, Zap. nauch. semin. LOMI, 147, 1985, 72–94 | MR | Zbl

[4] Fabes E. B., Stroock D. W., “The $L_p$-integrability of Green's functions and fundamental solutions for elliptic and parabolic equations”, Duke Math. J., 51:4 (1984), 997–1016 | DOI | MR | Zbl

[5] Krylov N. V., “Ogranichenno neodnorodnye ellipticheskie i parabolicheskie uravneniya v oblasti”, Izv. AN SSSR. Ser. mat., 47:1 (1983), 75–108 | MR | Zbl

[6] Kohn J. J., Nirenberg L., “Degenerate elliptic-parabolic equations of second order”, Comm. Pure Appl. Math., 20 (1967), 797–872 | DOI | MR | Zbl

[7] Daskalopoulos P., Lee K., “Free-boundary regularity on the focusing problem for the Gauss curvature flow with flat sides”, Math. Z., 237:4 (2001), 847–874 | DOI | MR | Zbl

[8] Hamilton R., “Worn stones with flat sides”, A Tribute to Ilya Bakelman (College Station, TX, 1993), Discourses Math. Appl., 3, Texas A and M Univ., College Station, TX, 1994, 69–78 | MR | Zbl

[9] Daskalopoulos P., Hamilton R., “The free boundary in the Gauss curvature flow with flat sides”, J. Reine Angew. Math., 510 (1999), 187–227 | MR | Zbl

[10] Daskalopoulos P., Lee K., “Worn stones with flat sides all time regularity of the interface”, Invent. Math., 156 (2004), 445–493 | DOI | MR | Zbl

[11] Daskalopoulos P., Lee K., “Hölder regularity of solutions of degenerate elliptic and parabolic equations”, J. Funct. Anal., 201:2 (2003), 341–379 | DOI | MR | Zbl

[12] Alkhutov Yu. A., “O gëlderovoi nepreryvnosti reshenii vyrozhdayuschikhsya ellipticheskikh uravnenii vtorogo poryadka nedivergentnogo vida”, Dokl. RAN, 413:3 (2007), 295–300 | MR | Zbl

[13] Alkhutov Yu. A., “Neravenstvo Kharnaka dlya reshenii odnogo klassa vyrozhdayuschikhsya ellipticheskikh uravnenii”, Mezhdunar. konf. po differentsialnym uravneniyam i dinamicheskim sistemam, Tez. dokl. (Suzdal, 10–15 iyulya 2006 g.), 22–24

[14] Seneta E., Regularly varying functions, Lecture Notes in Math., 508, Springer-Verlag, Berlin–New York, 1976 | MR | Zbl

[15] Lib E., Loss M., Analiz, Nauch. kn., Novosibirsk, 1998

[16] Aleksandrov A. D., “Usloviya edinstvennosti i otsenki resheniya zadachi Dirikhle”, Vestn. Leningr. un-ta. Ser. mat., mekh., astronom., 1963, no. 3, 5–29 | Zbl

[17] Krylov N. V., “Posledovatelnosti vypuklykh funktsii i otsenki maksimuma resheniya parabolicheskogo uravneniya”, Sib. mat. zh., 17:2 (1976), 290–303 | MR | Zbl

[18] Nazarov A. I., Uraltseva N. N., “Vypuklo-monotonnye obolochki i otsenka maksimuma resheniya parabolicheskogo uravneniya”, Zap. nauch. semin. LOMI, 147, 1985, 95–109 | MR | Zbl

[19] Tso K., “On an Aleksandrov–Bakel'man type maximum principle for second-order parabolic equations”, Comm. Partial Differential Equations, 10:5 (1985), 543–553 | DOI | MR | Zbl

[20] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, 2-e izd., Nauka, M., 1973 | MR

[21] Nadirashvili N. S., “Nonuniqueness in the martingale problem and the Dirichlet problem for uniformly elliptic equations”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24:3 (1997), 537–549 | MR

[22] Safonov M. V., “Nonuniqueness for second-order elliptic equations with measurable coefficients”, SIAM J. Math. Anal., 30:4 (1999), 879–895 | DOI | MR | Zbl