Sbornik. Mathematics, Tome 11 (1970) no. 1, pp. 89-99
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N. V. Krylov. Bounded inhomogeneous nonlinear elliptic and parabolic equations in the plane. Sbornik. Mathematics, Tome 11 (1970) no. 1, pp. 89-99. http://geodesic.mathdoc.fr/item/SM_1970_11_1_a6/
@article{SM_1970_11_1_a6,
author = {N. V. Krylov},
title = {Bounded inhomogeneous nonlinear elliptic and parabolic equations in~the plane},
journal = {Sbornik. Mathematics},
pages = {89--99},
year = {1970},
volume = {11},
number = {1},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1970_11_1_a6/}
}
TY - JOUR
AU - N. V. Krylov
TI - Bounded inhomogeneous nonlinear elliptic and parabolic equations in the plane
JO - Sbornik. Mathematics
PY - 1970
SP - 89
EP - 99
VL - 11
IS - 1
UR - http://geodesic.mathdoc.fr/item/SM_1970_11_1_a6/
LA - en
ID - SM_1970_11_1_a6
ER -
%0 Journal Article
%A N. V. Krylov
%T Bounded inhomogeneous nonlinear elliptic and parabolic equations in the plane
%J Sbornik. Mathematics
%D 1970
%P 89-99
%V 11
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1970_11_1_a6/
%G en
%F SM_1970_11_1_a6
A study is made of equations of the form $F\bigl(x,D_{ij}u-d\delta_{ij}\frac{\partial u}{\partial t},D_iu,u\bigr)=0$ in a bounded smooth domain in the plane $(d=0)$ or in a smooth cylinder above the plane $(d=1)$ with Dirichlet data on the boundary, and also of the problem with a free boundary for these equations. It is proved that if the function $tF\bigl(x,\frac\xi t\bigr)$ satisfies an ellipticity condition with respect to $\xi_{ij}$, a boundedness condition for the “coefficients” of $\xi$ and $t$ and a negative condition for the “coefficient” of $u$, then all the problems have a solution in the corresponding Sobolev–Slobodetskii space which is unique. Bibliography: 6 titles.
[1] N. V. Krylov, “Ob odnom klasse nelineinykh uravnenii v prostranstve izmerimykh funktsii”, Matem. sb., 80(122) (1969), 253–265 | MR | Zbl
[2] N. V. Krylov, “Ob uravneniyakh minimaksnogo tipa v teorii ellipticheskikh i parabolicheskikh uravnenii na ploskosti”, Matem. sb., 81(123) (1970), 3–20 | MR
[3] N. V. Krylov, “Ob ogranichenno neodnorodnykh nelineinykh ellipticheskikh i parabolicheskikh uravneniyakh na ploskosti”, Uspekhi matem. nauk, XXIV:4(148), 201–202 | MR
[4] R. Kurant, Uravneniya s chastnymi proizvodnymi, Mir, Moskva, 1964 | MR
[5] O. A. Ladyzhenskaya, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, Moskva, 1964 | MR
[6] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, Moskva, 1967
