New perturbation methods for nonlinear parabolic problems

By Tuomo Kuusi, Giuseppe Mingione. 2012

73 citations recorded by the source corpus.

Abstract We develop methods aimed at deriving regularity results for solutions to nonlinear degenerate parabolic equations and systems via local perturbation; as a consequence we obtain, in a unified way, Lipschitz continuity of solutions under weak parabolicity assumptions, and gradient continuity results in borderline cases. Nonlinear Schauder estimates as those of Misawa (2002) [29] are recovered and extended to more general settings.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/211c931fa3752a64bd15e8b23acfb0ed07d3744b

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Regularity for Double Phase Variational Problems (2015, background): “We notice that functional appearing in the above display has standard polynomial growth of order 2q−p, therefore by standard regularity theory (see for instance [18], [25]) we know that vm ∈W 1,∞ loc (B1) .”
Riesz Potentials and Nonlinear Parabolic Equations (2013, methodology): “The starting point of the proof is the work in [27], where the continuity of the gradient of solutions to parabolic equations as in (1.”
The Wolff gradient bound for degenerate parabolic equations (2014, methodology): “1 continues to hold in the case of solutions to the p-Laplacean system, as is shown, starting from the techniques introduced here, in [28]; moreover, the singular case 1 < p < 2 is demonstrated in [25].”
A nonlinear Stein theorem (2014, methodology): “In this respect, we recall that in [17] we worked with intrinsic cylinders with vertex (x0, t0) of the type Q%(x0, t0) = B(x, %)×(t0−λ%(2), t0), for a number λ > 0 which is suitably related to the size of the gradient on the same cylinder Q%(x0, t0).”