A priori estimates and precise regularity for parabolic systems with discontinuous data

Abstract

We deal with linear parabolic (in sense of Petrovskii) systems of order 2b with discontinuous principal coefficients. A'priori estimates in Sobolev and Sobolev-Morrey spaces are proved for the strong solutions by means of potential analysis and boundedness of certain singular integral operators with kernels of mixed homogeneity. As a byproduct, precise characterization of the Morrey, BM O and Hölder regularity is given for the solutions and their derivatives up to order 2b − 1.

Parabolic and Elliptic Systems with VMO Coefficients

“…See [16] for a discussion of these two conditions. However, it is still stronger than Petrovskii's condition used in [15], [9] and [26]- [29]. An interesting question is whether the results of Theorem 2.4 and 2.5 can be extended to operators satisfying the latter condition.…”

“…Another set of papers concerning the L p solvability of parabolic systems with discontinuous coefficients are [26] and [27], where the authors established the interior regularity of solutions to higher order parabolic systems in L p spaces and Sobolev-Morrey spaces, when the coefficients are VMO in both spatial and time variables. With continuous coefficients, very general mixed problems of parabolic systems in cylindrical and non-cylindrical regions were studied before in [30].…”

“…In this paper, we consider parabolic and elliptic operators in both divergence and non-divergence form. In comparison to [26], [27] and [17], the advantage of our results is that the leading coefficients are allowed to be merely measurable in the time variable and VMO in the spatial variables. This is the same class of coefficients used in [22] and [23], and is denoted as VMO x (see the definition in Section 3).…”

“…Our approach is based on a method from [22] and [23]. Unlike the arguments in [7]- [9], [17], [29], [26]- [28] and [31], which are based on the Calderon-Zygmund theorem for certain estimates of singular integrals and the Coifman-Rochberg-Weiss commutator theorem, our proofs rely on, as in [22], [23], pointwise estimates of sharp functions of spatial derivatives of solutions. * We note that this flexible method is also applicable to equations or systems with partially VMO coefficients (see, for instance, [19], [20] and [13]).…”

Schauder estimates for solutions of higher-order parabolic systems

“…and in R d+1 if T = ∞, m positive integer, u and f complex vector-valued functions, A γ complex matrix-valued function. Moreover the leading coefficients satisfy the so-called Legendre-Hadamard ellipticity condition, which is more general than the strong ellipticity condition considered, for example, in [3,15] and is still stronger than the uniform parabolicity condition in the sense of Petrovskii, which was used in [6,17,21].…”