Giuseppe Mingione

Gradient estimates for a class of parabolic systems

We establish local Calderon-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 F ∈ Lqloc =⇒ Du ∈ Lqloc, ∀ q ≥ p. We also treat systems with discontinuous coefficients of vanishing mean oscillation (VMO) type.

Hölder continuity of the gradient of p(x)-harmonic mappings†

Abstract We prove that local minimizers u : R n → R N of the functional ∫ ∣D u ( x )∣ p(r) d x are of class C 1, α for some α > 0, provided p(x) > 1 is Holder continuous.

Parabolic Systems with Polynomial Growth and Regularity

The authors establish a series of optimal regularity results for solutions to general non-linear parabolic systems

Nonlocal Equations with Measure Data

u_t- \mathrm{div} \ a(x,t,u,Du)+H=0,

Degenerate problems with irregular obstacles

under the main assumption of polynomial growth at rate

Local Lipschitz regularity for degenerate elliptic systems

p

Guide to nonlinear potential estimates

i.e.

The Wolff gradient bound for degenerate parabolic equations

|a(x,t,u,Du)|\leq L(1+|Du|^{p-1}), p \geq 2.

Regularity results for electrorheological fluids: the stationary case

They give a unified treatment of various interconnected aspects of the regularity theory: optimal partial regularity results for the spatial gradient of solutions, the first estimates on the (parabolic) Hausdorff dimension of the related singular set, and the first Calderon-Zygmund estimates for non-homogeneous problems are achieved here.

The existence of regular boundary points for non-linear elliptic systems

We develop an existence, regularity and potential theory for nonlinear integrodifferential equations involving measure data. The nonlocal elliptic operators considered are possibly degenerate and cover the case of the fractional p-Laplacean operator with measurable coefficients. We introduce a natural function class where we solve the Dirichlet problem, and prove basic and optimal nonlinear Wolff potential estimates for solutions. These are the exact analogs of the results valid in the case of local quasilinear degenerate equations established by Boccardo and Gallouët (J Funct Anal 87:149–169, 1989, Partial Differ Equ 17:641–655, 1992) and Kilpeläinen and Malý (Ann Scuola Norm Sup Pisa Cl Sci (IV) 19:591–613, 1992, Acta Math 172:137–161, 1994). As a consequence, we establish a number of results that can be considered as basic building blocks for a nonlocal, nonlinear potential theory: fine properties of solutions, Calderón–Zygmund estimates, continuity and boundedness criteria are established via Wolff potentials. A main tool is the introduction of a global excess functional that allows us to prove a nonlocal analog of the classical theory due to Campanato (Ann Mat Pura Appl (IV) 69:321–381, 1965). Our results cover the case of linear nonlocal equations with measurable coefficients, and the one of the fractional Laplacean, and are new already in such cases.