Second order parabolic systems, optimal regularity, and singular sets of solutions |
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| Authors: | Frank Duzaar Giuseppe Mingione |
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| Affiliation: | aMathematisches Institut der Friedrich-Alexander-Universität, zu Nürnberg-Erlangen, Bismarckstr. 1 1/2, 91054 Erlangen, Germany;bDipartimento di Matematica, Università di Parma, Via D'Azeglio 85/A, 43100 Parma, Italy |
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| Abstract: | We present a new, complete approach to the partial regularity of solutions to non-linear, second order parabolic systems of the form | In the first part we introduce the A-caloric approximation lemma, a parabolic analogue of the harmonic approximation lemma of De Giorgi [Sem. Scuola Normale Superiore Pisa (1960–1961); Lectures in Math., ETH Zürich, Birkhäuser, Basel, 1996] in the version of Simon. This allows to prove optimal partial regularity results for solutions in an elementary way, under minimal and natural assumptions. In the second part we provide estimates for the parabolic Hausdorff dimension of the singular sets of solutions; the proof makes use of parabolic fractional Sobolev spaces.
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| Keywords: | Partial regularity Parabolic systems Singular sets |
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