New a priori estimates for nondiagonal strongly nonlinear parabolic systems

Abstract

We consider nondiagonal elliptic and parabolic systems of equations with quadratic nonlinearities in the gradient. We discuss a new description of regular points of solutions of such systems. For a class of strongly nonlinear parabolic systems, we estimate locally the Hölder norm of a solution. Instead of smallness of the oscillation, we assume local smallness of the Campanato seminorm of the solution under consideration. Theorems about quasireverse Hölder inequalities proved by the author are essentially used. We study systems under the Dirichlet boundary condition and estimate the Hölder norm of a solution up to the boundary (up to the parabolic boundary of the prescribed cylinder in the parabolic case).

L^{\infty} a priori bounds for gradients of solutions to quasilinear inhomogenous fast-growing parabolic systems

“…unbounded or discontinous) functions, which solve homogenous parabolic systems. For n > 2 it suffices for irregularity that the coefficients A(x, t) of the main part are discontinous (and still bounded) or that there is a relevant nondiagonality of the main part -for details, consult Arkhipova [1]. Nevertheless, there are many classes of main parts, which allow for higher regularity (even C 1,α ) in the homogenous case; these are: having structure close to Laplacian or p-Laplacian, like those studied in Ladyzhenskaya et al [19] or DiBenedetto [8], respectively, or having main part depending solely on ∇u: see well-known paper by Nečas and Šverak [22] or more extensive research done by Choe and Bae [7].…”

“…There are several approaches to answering this question: some authors relax the notion of regularity by resorting to partial regularity -see for example classical papers of Italian school: Campanato [4], Giaquinta and Struwe [13] and newer ones: Fanciullo [11], Frehse and Specovius-Neugebauer [12], Misawa [20], Duzaar and Mingione [10]; or by demanding a high integrability-type regularity 2 , like in Naumann [21], Kinnunen and Lewis [18] or Bensoussan and Frehse [3] (in the last paper the growth of the right-hand-side may be polynomially arbitrarily large!). Certain systems with peculiar structure or two-dimensional ones (or at least close to them in some sense) enjoy also high regularity, even if they are much more general than a Stokes-type system; for results in this direction compare papers of Seregin, Arkhipova, Frehse, Kaplicky (and many others), for instance: Arkhipova [1], J. Naumann and Wolff [16], Kaplicky [17], Zaja ¸czkowski and Seregin [27].…”