Annali di Matematica Pura ed Applicata

, Volume 188, Issue 1, pp 61–122

Partial regularity and singular sets of solutions of higher order parabolic systems

Authors

    • Department MathematikUniversität Erlangen–Nürnberg
Article

DOI: 10.1007/s10231-008-0067-4

Cite this article as:
Bögelein, V. Annali di Matematica (2009) 188: 61. doi:10.1007/s10231-008-0067-4

Abstract

In the present paper we provide a broad survey of the regularity theory for non-differentiable higher order parabolic systems of the type
$$ \int \limits_{\Omega_T} u\cdot \varphi_t - A(z,u,Du,\dots,D^m u) \cdot D^m \varphi \, {\rm d}z =\int \limits_{\Omega_T} \sum_{k=0}^{m-1} B^k(z,u,Du,\dots,D^m u) \cdot D^k\varphi \, {\rm d}z.$$
Initially, we prove a partial regularity result with the method of A-polycaloric approximation, which is a parabolic analogue of the harmonic approximation lemma of De Giorgi. Moreover, we prove better estimates for the maximal parabolic Hausdorff-dimension of the singular set of weak solutions, using fractional parabolic Sobolev spaces. Thereby, we also consider different situations, which yield a better dimension reduction result, including the low dimensional case and coefficients A(z, Dmu), independent of the lower order derivatives of u.

Keywords

Partial regularitySingular setHigher order parabolic systems

Mathematics Subject Classification (2000)

35D1035G2035K55

Copyright information

© Springer-Verlag 2008