Maximal function estimates and self-improvement results for Poincaré inequalities

Our main result is an estimate for a sharp maximal function, which implies a Keith–Zhong type self-improvement property of Poincaré inequalities related to differentiable structures on metric measure spaces. As an application, we give structure independent representation for Sobolev norms and universality results for Sobolev spaces. peerReviewed

Maximal regularity via reverse Hölder inequalities for elliptic systems of n-Laplace type involving measures

In this note, we consider the regularity of solutions of the nonlinear elliptic systems of n-Laplacian type involving measures, and prove that the gradients of the solutions are in the weak Lebesgue space Ln,∞. We also obtain the a priori global and local estimates for the Ln,∞-norm of the gradients of the solutions without using BMO-estimates. The proofs are based on a new lemma on the higher integrability of functions.

Bonnesenʼs inequality for John domains in Rn

Abstract We prove sharp quantitative isoperimetric inequalities for John domains in R n . We show that the Bonnesen-style inequalities hold true in R n under the John domain assumption which rules out cusps. Our main tool is a proof of the isoperimetric inequality for symmetric domains which gives an explicit estimate for the isoperimetric deficit. We use the sharp quantitative inequalities proved in Fusco et al. (2008) [7] and Fuglede (1989) [4] to reduce our problem to symmetric domains.

Gradient regularity for elliptic equations in the Heisenberg group

Abstract 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, C 1 , α -regularity for p-harmonic functions in the Heisenberg group for …

A short proof of the self-improving regularity of quasiregular mappings

. The theoryof quasiregular mappings is a central topic in modern analysis withimportant connections to a variety of topics as elliptic partial differen-tial equations, complex dynamics, differential geometry and calculus ofvariations [13] [10].A remarkable feature of quasiregular mappings is the self-improvingregularity. In 1957 [2], Bojarski proved that for planar quasiregularmappings, there exists an exponent

On strong solutions of the differential equations modeling the steady flow of certain incompressible generalized Newtonian fluids

In this paper we discuss a system of partial differential equations describing the steady flow of an incompressible fluid and prove the existence of a strong solution under suitable assumptions on the data. In the 2D-case this solution turns out to be of class C^{1,\alpha}.

De Giorgi–Nash–Moser Theory

We consider the second-order, linear, elliptic equations with divergence structure $$\mathrm{div} (\mathbb{A}(x)\nabla u(x))\;=\;\sum\limits^n_{i,j=1}\;\partial_{x_{i}}(a_{ij}(x)\partial_{x_{j}}u(x))\;=\;0.$$

Hardy’s inequality and the boundary size

We establish a self-improving property of the Hardy inequality and an estimate on the size of the boundary of a domain supporting a Hardy inequality.

Discontinuous solutions of linear, degenerate elliptic equations

Abstract We give examples of discontinuous solutions of linear, degenerate elliptic equations with divergence structure. These solve positively conjectures of De Giorgi.

Variational integrals with a wide range of anisotropy

Continuity of solutions of linear, degenerate elliptic equations

We consider the simplest form of a second order, linear, degenerate, divergence structure equation in the plane. Under an integrability condition on the degenerate function, we prove that the solutions are continuous.

Mappings of finite distortion: Sharp Orlicz-conditions

We establish continuity, openness and discreteness, and the condition $(N)$ for mappings of finite distortion under minimal integrability assumptions on the distortion.

A note on mappings of finite distortion: The sharp modulus of continuity

Geometric rigidity of conformal matrices

We provide a geometric rigidity estimate a la Friesecke-James-Muller for conformal matrices. Namely, we replace SO(n) by a arbitrary compact subset of conformal matrices, bounded away from 0 and invariant under SO(n), and rigid motions by Mobius transformations.

The Poincaré inequality is an open ended condition

Let p > 1 and let (X,d,µ) be a complete metric measure space with µ Borel and doubling that admits a (1,p)-Poincare inequality. Then there exists e > 0 such that (X,d,µ) admits a (1,q)-Poincare inequality for every q > p - e, quantitatively.

Mappings of finite distortion: Reverse inequalities for the Jacobian

Let f be a nonconstant mapping of finite distortion. We establish integrability results on 1/Jf by studying weights that satisfy a weak reverse Holder inequality where the associated constant can depend on the ball in question. Here Jf is the Jacobian determinant of f.

Removable sets for continuous solutions of quasilinear elliptic equations

We show that sets of n − p + α ( p − 1 ) n-p+\alpha (p-1) Hausdorff measure zero are removable for α \alpha -Hölder continuous solutions to quasilinear elliptic equations similar to the p p -Laplacian. The result is optimal. We also treat larger sets in terms of a growth condition. In particular, our results apply to quasiregular mappings.

Mappings of finite distortion: the degree of regularity

This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let K(x)⩾1 be a measurable function defined on a domain Ω⊂Rn,n⩾2, and such that exp(βK(x))∈Lloc1(Ω), β>0. We show that there exist two universal constants c1(n),c2(n) with the following property: Let f be a mapping in Wloc1,1(Ω,Rn) with |Df(x)|n⩽K(x)J(x,f) for a.e. x∈Ω and such that the Jacobian determinant J(x,f) is locally in L1log−c1(n)βL. Then automatically J(x,f) is locally in L1logc2(n)βL(Ω). This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite disto…

Mappings of finite distortion: a new proof for discreteness and openness

We give a new and elementary proof of the known result: a non-constant mapping of finite distortion f : Ω ⊂ ℝn → ℝn is discrete and open, provided that its distortion function if n = 2 and that for some p > n − 1 if n ≥ 3.

C1,α-regularity for variational problems in the Heisenberg group

We study the regularity of minima of scalar variational integrals of $p$-growth, $1<p<\infty$, in the Heisenberg group and prove the H\"older continuity of horizontal gradient of minima.