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Global gradient estimates for degenerate parabolic equations in nonsmooth domains. (English) Zbl 1179.35080

The author studies the global regularity of the solutions of degenerate parabolic equations of the form
\[ u_t=\operatorname{div}A(x,t,\nabla u), \]
where \(A(x,t,\nabla u)\) satisfies the Carathédory-type conditions and the \(p\)-growth conditions. It is shown that the weak solutions belong to a higher Sobolev space than assumed a priori if the complement of the domain satisfies a capacity density condition and if the boundary values are sufficiently smooth. Integrability estimates for the gradient are obtained. The results are extended to parabolic systems as well.

MSC:

35B45 A priori estimates in context of PDEs
35K55 Nonlinear parabolic equations
49N60 Regularity of solutions in optimal control
35K65 Degenerate parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
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