Duke Mathematical Journal

Gradient estimates for a class of parabolic systems

Emilio Acerbi and Giuseppe Mingione

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We establish local Calderón-Zygmund-type estimates for a class of parabolic problems whose model is the nonhomogeneous, degenerate/singular parabolic p-Laplacian system ut-div(|Du|p-2Du)=div(|F|p-2F), proving that FLlocqDuLqloc,qp. We also treat systems with discontinuous coefficients of vanishing mean oscillation (VMO) type

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Duke Math. J., Volume 136, Number 2 (2007), 285-320.

First available in Project Euclid: 21 December 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35K65: Degenerate parabolic equations


Acerbi, Emilio; Mingione, Giuseppe. Gradient estimates for a class of parabolic systems. Duke Math. J. 136 (2007), no. 2, 285--320. doi:10.1215/S0012-7094-07-13623-8. https://projecteuclid.org/euclid.dmj/1166711371

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  • E. Acerbi and G. Mingione, Gradient estimates for the $\px$-Laplacean system, J. Reine Angew. Math. 584 (2005), 117--148.
  • B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in $\er^n$, Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), 257--324.
  • M. Bramanti and M. C. Cerutti, $W\sb p\sp 1,2$ solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients, Comm. Partial Differential Equations 18 (1993), 1735--1763.
  • S.-S. Byun, Parabolic equations with BMO coefficients in Lipschitz domains, J. Differential Equations 209 (2005), 229--265.
  • L. A. Caffarelli and I. Peral, On $W\sp 1,p$ estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998), 1--21.
  • F. Chiarenza, M. Frasca, and P. Longo, $W\sp 2,p$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 336 (1993), 841--853.
  • G. Cupini, N. Fusco, and R. Petti, Hölder continuity of local minimizers, J. Math. Anal. Appl. 235 (1999), 578--597.
  • E. Dibenedetto, Degenerate Parabolic Equations, Universitext, Springer, New York, 1993.
  • E. Dibenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems. J. Reine Angew. Math. 357 (1985), 1--22.
  • E. Dibenedetto and J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, Amer. J. Math. 115 (1993), 1107--1134.
  • F. Duzaar and A. Gastel, Nonlinear elliptic systems with Dini continuous coefficients, Arch. Math. (Basel) 78 (2002), 58--73.
  • L. Escauriaza, Weak type-$(1,1)$ inequalities and regularity properties of adjoint and normalized adjoint solutions to linear nondivergence form operators with VMO coefficients, Duke Math. J. 74 (1994), 177--201.
  • E. Giusti, Direct Methods in the Calculus of Variations, World Sci., River Edge, N.J., 2003.
  • T. Iwaniec, Projections onto gradient fields and $L\spp$-estimates for degenerated elliptic operators, Studia Math. 75 (1983), 293--312.
  • T. Iwaniec and C. Sbordone, Riesz transforms and elliptic PDEs with VMO coefficients, J. Anal. Math. 74 (1998), 183--212.
  • J. Kinnunen and J. L. Lewis, Higher integrability for parabolic systems of $p$-Laplacian type, Duke Math. J. 102 (2000), 253--271.
  • —, Very weak solutions of parabolic systems of $p$-Laplacian type, Ark. Mat. 40 (2002), 105--132.
  • J. Kinnunen and P. Lindqvist, Summability of semicontinuous solutions to a quasilinear parabolic equation, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (5) 4 (2005), 59--78.
  • J. Kinnunen and S. Zhou, A local estimate for nonlinear equations with discontinuous coefficients, Comm. Partial Differential Equations 24 (1999), 2043--2068.
  • —, A boundary estimate for nonlinear equations with discontinuous coefficients, Differential Integral Equations 14 (2001), 475--492.
  • J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969.
  • M. Misawa, $L^q$ estimates of gradients for evolutional $p$-Laplacian systems, J. Differential Equations 219 (2006), 390--420.
  • D. K. Palagachev, Quasilinear elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 347 (1995), 2481--2493.
  • I. Peral and F. Soria, ``A note on estimates for quasilinear parabolic equations'' in Proceedings of the 2001 Luminy Conference on Quasilinear Elliptic and Parabolic Equations and System, Electron. J. Differ. Equ. Conf. 8, Southwest Texas State Univ., San Marcos, 2002, 121--131.
  • D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391--405.