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Unilateral problems with convex constraints for parabolic quasilinear operators. (English. Russian original) Zbl 0677.35060
Sib. Math. J. 29, No. 5, 846-857 (1988); translation from Sib. Mat. Zh. 29, No. 5(171), 198-211 (1988).
A priori estimates of solutions are obtained for a quasilinear parabolic problem with unilateral constraints imposed on the solution’s gradient. A set that defines these constraints may be unbounded and (in case of an operator with a linear principal part) non-strictly convex. The derived estimates provide the solvability of the problem studied in \(L_{\infty}(0,T;W^ 2_{p,loc}(\Omega)),\) \(\Omega \subset R^ n\), \(0<t<T\). Also are established conditions under which the solution belongs to \(L_{\infty}(0,T;W^ 2_ p(\Omega))\cap C^{0,1}({\bar \Omega}),\) \(1<p<\infty\) or \(L_{\infty}(0,T;W^ 2_{\infty}(\Omega))\).
Reviewer: I.Zino
MSC:
35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35K20 Initial-boundary value problems for second-order parabolic equations
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References:
[1] L. C. Evans, ?A second order elliptic equation with gradient constraint,? Commun. Part. Different. Equat.,4, No. 5, 361-371 (1981).
[2] T. N. Rozhkovskaya, ?Unilateral problems for elliptic operators with convex constraints on the gradient of the solution,? Sib. Mat. Zh.,26, No. 3, 134-147;26, No. 5, 150-158 (1986).
[3] T. N. Rozhkovskaya, ?Unilateral problems for parabolic quasilinear operators,? Dokl. Akad. Nauk SSSR,290, No. 3, 549-552 (1986).
[4] H. Ishii and S. Koike, ?Boundary regularity and uniqueness for an elliptic equation with gradient constraint,? Commun. Part. Different. Equat.,8, No. 4, 317-346 (1983). · Zbl 0538.35012 · doi:10.1080/03605308308820271
[5] O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967). · Zbl 0164.12302
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