Self-improving property of degenerate parabolic equations of porous medium-type

Abstract:We show that the gradient of solutions to degenerate parabolic equations of porous medium-type satisfies a reverse H\"older inequality in suitable intrinsic cylinders. We modify the by-now classical Gehring lemma by introducing an intrinsic Calder\'on-Zygmund covering argument, and we are able to prove local higher integrability of the gradient of a proper power of the solution $u$.

[1]  L. Diening,et al.  BMO estimates for the p-Laplacian , 2012 .

[2]  T. Kuusi,et al.  Riesz Potentials and Nonlinear Parabolic Equations , 2013, 1302.0266.

[3]  Christopher van der Heide.  Regularity results for nonlinear elliptic systems , 2016 .

[4]  Sebastian Schwarzacher,et al.  On Regularity of the Time Derivative for Degenerate Parabolic Systems , 2015, SIAM J. Math. Anal..

[5]  B. Dacorogna.  Direct methods in the calculus of variations , 1989 .

[6]  N. Meyers,et al.  Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions , 1975 .

[7]  T. Kuusi,et al.  A nonlinear Stein theorem , 2014 .

[8]  Lars Diening,et al.  Partial Regularity for Minimizers of Quasi-convex Functionals with General Growth , 2012, SIAM J. Math. Anal..

[9]  F. Gehring.  TheLp-integrability of the partial derivatives of A quasiconformal mapping , 1973 .

[10]  A. Friedman,et al.  Addendum to “Holder estimates for nonlinear degenerate parabolic Systems”. , 1985 .

[11]  John L. Lewis,et al.  Higher integrability for parabolic systems of p-Laplacian type , 2000 .

[12]  M. Giaquinta,et al.  On the partial regularity of weak solutions of nonlinear parabolic systems , 1982 .

[13]  E. DiBenedetto.  Degenerate Parabolic Equations , 1993 .

[14]  Timothy S. Murphy,et al.  Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , 1993 .

[15]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[16]  E. Acerbi,et al.  Gradient estimates for a class of parabolic systems , 2007 .

[17]  M. Parviainen,et al.  Self-improving property of nonlinear higher order parabolic systems near the boundary , 2010 .

[18]  V. Bögelein.  HIGHER INTEGRABILITY FOR WEAK SOLUTIONS OF HIGHER ORDER DEGENERATE PARABOLIC SYSTEMS , 2008 .

[19]  H\"older-Zygmund Estimates for Degenerate Parabolic Systems , 2013, 1307.5293.

[20]  G. Mingione,et al.  The Regularity of General Parabolic Systems With Degenerate Diffusion , 2012 .

[21]  M. Parviainen.  Global gradient estimates for degenerate parabolic equations in nonsmooth domains , 2009 .

[22]  U. Gianazza,et al.  Harnack's Inequality for Degenerate and Singular Parabolic Equations , 2011 .

[23]  M. Parviainen.  Reverse Hölder inequalities for singular parabolic equations near the boundary , 2009 .

[24]  T. Kuusi,et al.  Pointwise Calderón–Zygmund gradient estimates for the p -Laplace system , 2015, Journal de Mathématiques Pures et Appliquées.