Geodesic
On smoothness up to the boundary of solutions of parabolic equations with a degenerate operator
Sbornik. Mathematics, Tome 82 (1995) no. 2, pp. 271-292 
A. V. Babin; S. Zh. Kabakbaev. On smoothness up to the boundary of solutions of parabolic equations with a degenerate operator. Sbornik. Mathematics, Tome 82 (1995) no. 2, pp. 271-292. http://geodesic.mathdoc.fr/item/SM_1995_82_2_a1/
@article{SM_1995_82_2_a1,
     author = {A. V. Babin and S. Zh. Kabakbaev},
     title = {On smoothness up to the~boundary of solutions of parabolic equations with a~degenerate operator},
     journal = {Sbornik. Mathematics},
     pages = {271--292},
     year = {1995},
     volume = {82},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1995_82_2_a1/}
}
TY  - JOUR
AU  - A. V. Babin
AU  - S. Zh. Kabakbaev
TI  - On smoothness up to the boundary of solutions of parabolic equations with a degenerate operator
JO  - Sbornik. Mathematics
PY  - 1995
SP  - 271
EP  - 292
VL  - 82
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1995_82_2_a1/
LA  - en
ID  - SM_1995_82_2_a1
ER  - 
%0 Journal Article
%A A. V. Babin
%A S. Zh. Kabakbaev
%T On smoothness up to the boundary of solutions of parabolic equations with a degenerate operator
%J Sbornik. Mathematics
%D 1995
%P 271-292
%V 82
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1995_82_2_a1/
%G en
%F SM_1995_82_2_a1

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

Parabolic equations $\partial_tu=-Au+f_0$, $ u|_{t=0}=f_1$ are considered that are of second order with a nonnegative quadratic form $a(x,\zeta)$ corresponding to the space variables. This form degenerates on the boundary: $a(x,\nu)=0$, where $\nu$ is the normal vectors, which corresponds to the condition of impermeability of the boundary. Special function spaces $E^s$ with weight are introduced. Semiboundedness of the operator $A$ in these spaces with arbitrary $s$ is proved: $(Av,v)_{E^s}\geqslant -C\|v\|_{E^s}^2$. On this basis theorems on the smoothness of solutions for $f_0,f_1\in E^s$ are proved. Theorems on the smoothness of solutions $u(x)$ of the elliptic equation $Au + \lambda u = f_0$ are also obtained.

[1] Babin A. V., “O svyazi analiticheskikh svoistv operatornykh funktsii i gladkosti reshenii vyrozhdayuschikhsya differentsialnykh uravnenii”, Funktsion. analiz i ego pril., 22:1 (1988), 60–61 | MR | Zbl

[2] Babin A. V., “O gladkosti reshenii differentsialnykh uravnenii v osobykh tochkakh granitsy oblasti”, Izv. AN SSSR. Ser. matem., 54:6 (1990), 1134–1154 | MR

[3] Kon Dzh. Dzh., Nirenberg L., “Nekoertsitivnye kraevye zadachi”, Psevdodifferentsialnye operatory, Mir, M., 1967, 88–165 | MR

[4] Oleinik O. A., Radkevich E. V., “Uravneniya vtorogo poryadka s neotritsatelnoi kharakteristicheskoi formoi”, Itogi nauki i tekhniki. Matematicheskii analiz. 1969, VINITI, M., 1971, 7–252 | MR

[5] Babin A. B., Iterations of Differential Operators, Gordon and Breach, N.Y., 1989 | MR | Zbl

[6] Baonendi M. S., Gonlaonic C., “Regulatrite analitique et iteres d'operators elliptiques degeneres”, J. Funct. Anal., 9 (1972), 208–248 | DOI | MR