Hölder-estimates for non-autonomous parabolic problems with rough data

Abstract

In this paper we establish Hölder estimates for solutions to nonautonomous parabolic equations on non-smooth domains which are complemented with mixed boundary conditions. The corresponding elliptic operators are of divergence type, the coefficient matrix of which depends only measurably on time. These results are in the tradition of the classical book of Ladyshenskaya et al. [40], which also serves as the starting point for our investigations. 2010 Mathematics Subject Classification. 35B65, 35K10, 35K15. Key words and phrases. Linear, second order parabolic equations, non-smooth data, Hölder continuity of the solution.147 148 HANNES MEINLSCHMIDT AND JOACHIM REHBERGdivergence of vector-valued L p -functions. As is well-known, such representations are highly non-unique; in particular the zero-functional may be represented as the divergence of a non-zero vector valued function. Lastly, it is not quite clear how broad the admissable geometric setting really is: on one hand "piecewise C 1 " is demanded, on the other the crucial "Condition A" ([40, pg. 9], compare also [39, Ch. II.B, Definition B.3]) -well-known from elliptic theory -comes into play.Our intention is to deliver a text which• clearly defines the underlying geometric concept for the domain Ω -thereby avoiding "Condition A", • incorporates mixed boundary conditions within an appropriately defined framework,• avoids a global Lipschitz condition for Ω -at least insofar the Dirichlet boundary part D is concerned, compare also [26], • gives a result in the formulation of classical Hölder spaces.The paper is organized as follows: In the next chapter we first introduce some terminology and our geometrical assumptions on the domain Ω and the Dirichlet boundary part D. Then, as a starting point, we quote the classical result on the existence and uniqueness of solutions for non-autonomous parabolic equations in a Hilbert spaces setting, namely the right-hand side f being taken from L 2 (J; V * ) yielding solutions from L 2 (J; V ) ∩ W 1,2 (J; V * ), where V → H → V * is a Gelfand triplet. Afterwards the main result is announced, cf. Theorem 2.13. In Ch. 3 we give the proof of this theorem, starting with the quotation of several classical results from the book [40]. The idea is to take these in a setting -namely the half cube and the ball and homogeneous Dirichlet boundary conditions -where the inherent technical difficulties are still not appearing: here it is clear that the general suppositions on the domain posed in [40] are fulfilled and, concerning the results, falls back to the classical Hölder spaces. After establishing some preliminaries in Ch. 3.2, in Ch. 3.3 we establish permanence principles, such as localization of the equation, bi-Lipschitz transformations and reflection which allow us to treat the problems in the geometric setting established above. This results in problems of the same complexity as before on a half-cube or a cube, respectively. Now employing the classical Ladyshenskaya results, one is enabled to deduce the required Hölder results for...

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