## Higher integrability for parabolic systems of $$p$$-Laplacian type.(English)Zbl 0994.35036

From the introduction: In this work, we study regularity of solutions to second-order parabolic systems: ${\partial u_i\over \partial t}= \text{div} A_i(x,t, \nabla u)+B_i(x,t,\nabla u), \quad i=1,\dots,N. \tag{1}$ In particular, we are interested in systems of $$p$$-Laplacian type. The principal prototype that we have in mind is the $$p$$-parabolic system $${\partial u_i\over \partial t}=\text{div}(|\nabla u|^{p-2} \nabla u_i)$$, $$i=1,\dots,N$$, with $$1<p <\infty$$. The purpose of this work is to obtain higher integrability results in the $$p$$-parabolic setting. We prove that the gradient of a weak solution to (1) satisfies a reverse Hölder inequality for $$p>2n/(n+2)$$. One of the difficulties in proving our main result is that a solution does not remain a solution under multiplication by a constant that is neither 0 nor 1. Since reverse Hölder inequalities are invariant under multiplication by a constant, we have to choose a class of cylinders whose side lengths depend on the size of the function in order to obtain a reverse Hölder inequality and then higher integrability.

### MSC:

 35D10 Regularity of generalized solutions of PDE (MSC2000) 35K55 Nonlinear parabolic equations 35K40 Second-order parabolic systems

### Keywords:

reverse Hölder inequality
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### References:

 [1] H. J. Choe, On the regularity of parabolic equations and obstacle problems with quadratic growth nonlinearities , J. Differential Equations 102 (1993), 101–118. · Zbl 0817.35048 [2] E. DiBenedetto, Degenerate Parabolic Equations , Universitext, Springer-Verlag, New York, 1993. · Zbl 0794.35090 [3] F. W. Gehring, The $$L^p$$-integrability of the partial derivatives of a quasiconformal mapping , Acta Math. 130 (1973), 265–277. · Zbl 0258.30021 [4] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems , Ann. of Math. Stud. 105 , Princeton University Press, Princeton, 1983. · Zbl 0516.49003 [5] M. Giaquinta and G. Modica, Regularity results for some classes of higher order nonlinear elliptic systems , J. Reine Angew. Math. 311 / 312 (1979), 145–169. · Zbl 0409.35015 [6] M. Giaquinta and M. Struwe, On the partial regularity of weak solutions of nonlinear parabolic systems , Math. Z. 179 (1982), 437–451. · Zbl 0469.35028 [7] N. G. Meyers and A. Elcrat, Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions , Duke Math. J. 42 (1975), 121–136. · Zbl 0347.35039 [8] E. W. Stredulinsky, Higher integrability from reverse Hölder inequalities , Indiana Univ. Math. J. 29 (1980), 407–413. · Zbl 0442.35064
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