Regularity of minima: An invitation to the dark side of the calculus of variations

I am presenting a survey of regularity results for both minima of variational integrals, and solutions to non-linear elliptic, and sometimes parabolic, systems of partial differential equations. I will try to take the reader to the Dark Side... This work has been partially supported by MIUR via the project “Calcolo delle Variazioni” (Cofin 2004), and by GNAMPA via the project “Studio delle singolarità in problemi geometrici e variazionali”.     

Regularity Results for a Class of Functionals with Non-Standard Growth

We consider the integral functional

Regularity for minimizers of functionals with p – q growth

We prove higher integrability for the gradient of vector-valued minimizers of some integral functionals with p – q growth. Received February 9, 1998     

Gradient regularity for elliptic equations in the Heisenberg group

We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised in [J.J. Manfredi, G. Mingione, Regularity results for quasilinear elliptic equations in the Heisenberg group, Math. Ann. 339 (2007) 485-544], where only dimension dependent bounds for the growth exponent are given. We also obtain explicit a priori local regularity estimates, and cover the case of the horizontal p-Laplacean operator, extending some regularity proven in [A. Domokos, J.J. Manfredi, C1,α-regularity for p-harmonic functions in the Heisenberg group for p near 2, in: Contemp. Math., vol. 370, 2005, pp. 17-23]. In turn, using some recent techniques of Caffarelli and Peral [L. Caffarelli, I. Peral, On W1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998) 1-21], the a priori estimates found are shown to imply the suitable local Calderón-Zygmund theory for the related class of non-homogeneous, possibly degenerate equations involving discontinuous coefficients. These last results extend to the sub-elliptic setting a few classical non-linear Euclidean results [T. Iwaniec, Projections onto gradient fields and Lp-estimates for degenerated elliptic operators, Studia Math. 75 (1983) 293-312; E. DiBenedetto, J.J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, Amer. J. Math. 115 (1993) 1107-1134], and to the non-linear case estimates of the same nature that were available in the sub-elliptic setting only for solutions to linear equations.     

Linear Potentials in Nonlinear Potential Theory

Pointwise gradient bounds via Riesz potentials, such as those available for the linear Poisson equation, actually hold for general quasilinear degenerate equations of p-Laplacean type. The regularity theory of such equations completely reduces to that of the classical Poisson equation up to the C ¹-level.     

Riesz Potentials and Nonlinear Parabolic Equations

The spatial gradient of solutions to nonlinear degenerate parabolic equations can be pointwise estimated by the caloric Riesz potential of the right hand side datum, exactly as in the case of the heat equation. Heat kernels type estimates persist in the nonlinear case.     

Full C1,α-regularity for free and constrained local minimizers of elliptic variational integrals¶with nearly linear growth

We consider local minimizers of elliptic variational integrals with integrand f of nearly linear growth. In the scalar case N= 1 a side condition of the type u≥Φ may be incorporated, for N > 1 u is an unconstrained minimizer and f is required just to depend on the modulus of Du. We show in both cases that u has H?lder continuous first derivatives in the interior of the domain Ω.     

The Singular Set of Minima of Integral Functionals

In this paper we provide upper bounds for the Hausdorff dimension of the singular set of minima of general variational integrals where F is suitably convex with respect to Dv and Hölder continuous with respect to (x,v). In particular, we prove that the Hausdorff dimension of the singular set is always strictly less than n, where .     

Gradient continuity estimates

We consider degenerate elliptic equations of p-Laplacean type

$$-{\rm{div}}\, (\gamma(x)|Du|^{p-2}Du)=\mu\,,$$

and give a sufficient condition for the continuity of Du in terms of a natural non-linear Wolff potential of the right-hand side measure μ. As a corollary we identify borderline condition for the continuity of Du in terms of the data: namely μ belongs to the Lorentz space L(n, 1/(p ? 1)), and γ(x) is a Dini continuous elliptic coefficient. This last result, together with pointwise gradient bounds via non-linear potentials, extends to the non homogeneous p-Laplacean system, thereby giving a positive answer in the vectorial case to a conjecture of Verbitsky. Continuity conditions related to the density of μ, or to the decay rate of its L ⁿ -norm on small balls, are identified as well as corollaries of the main non-linear potential criterium.

     

Second order parabolic systems, optimal regularity, and singular sets of solutions

We present a new, complete approach to the partial regularity of solutions to non-linear, second order parabolic systems of the form

u_t−divA(x,t,u,Du)=0.