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



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.


Partial regularity Singular set Higher order parabolic systems 

Mathematics Subject Classification (2000)

35D10 35G20 35K55 


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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department MathematikUniversität Erlangen–NürnbergErlangenGermany

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