Caccioppoli-type Estimates Research Articles

Regularity Theory for Nonlocal Equations with General Growth in the Heisenberg Group

Abstract We deal with a wide class of generalized nonlocal $p$-Laplace equations, so-called nonlocal $G$-Laplace equations, in the Heisenberg framework. Under natural hypotheses on the $N$-function $G$, we provide a unified approach to investigate in the spirit of De Giorgi-Nash-Moser theory, some local properties of weak solutions to such kind of problems, involving boundedness, Hölder continuity and Harnack inequality. To this end, an improved nonlocal Caccioppoli-type estimate as the main auxiliary ingredient is exploited several times.

Hölder Continuity and Boundedness Estimates for Nonlinear Fractional Equations in the Heisenberg Group

We extend the celebrate De Giorgi-Nash-Moser theory to a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p-Laplacian operator on the Heisenberg-Weyl group mathbb {H}^n. Among other results, we prove that the weak solutions to such a class of problems are bounded and Hölder continuous, by also establishing general estimates as fractional Caccioppoli-type estimates with tail and logarithmic-type estimates.

Boundary Lebesgue mixed-norm estimates for non-stationary Stokes systems with VMO coefficients

We consider Stokes systems with measurable coefficients and Lions-type boundary conditions. We show that, in contrast to the Dirichlet boundary conditions, local boundary mixed-norm -estimates hold for the spatial second-order derivatives of solutions, assuming the smallness of the mean oscillations of the coefficients with respect to the spatial variables in small cylinders. In the un-mixed norm case with the result is still new and provides local boundary Caccioppoli-type estimates. The main challenges in the work arise from the lack of regularity of the pressure and time derivatives of the solutions and from interaction of the boundary with the nonlocal structure of the system. To overcome these difficulties, our approach relies heavily on several newly developed regularity estimates for both divergence and non-divergence form parabolic equations with coefficients that are only measurable in the time variable and in one of the spatial variables.

A global fractional Caccioppoli-type estimate for solutions to nonlinear elliptic problems with measure data

We prove a global fractional differentiability result via the fractional Caccioppoli-type estimate for solutions to nonlinear elliptic problems with measure data. This work is inspired by the recent paper [B. Avelin, T. Kuusi and G. Mingione, Arch. Ration

The Liouville-type theorem for problems with nonstandard growth derived by Caccioppoli-type estimate

Let u be a nonnegative solution to the PDI -,mathrm{div} mathcal {A}(x, u, nabla u)geqslant mathcal {B}(x,u, nabla u) in Omega , where mathcal {A} and mathcal {B} are differential operators with p(x)-type growth. As a consequence of the Caccioppoli-type inequality for the solution u, we obtain the Liouville-type theorem under some integral condition. We simplify the assumptions on functions mathcal {A} and mathcal {B}, and we do not restrict the range of p(x) by the dimension n, therefore we can cover quite general family of problems.

Caccioppoli-type estimates and Hardy-type inequalities derived from weighted p-harmonic problems

We obtain Caccioppoli-type estimates for nontrivial and nonnegative solutions to anticoercive partial differential inequalities of elliptic type involving weighted p-Laplacian: -Delta _{p,a} u:= -{mathrm {div}}(a(x)|{nabla }u|^{p-2}{nabla }u)ge b(x)Phi (u)chi _{{u>0}}, where u is defined in a domain Omega . Using Caccioppoli-type estimates, we obtain several variants of Hardy-type inequalities in weighted Sobolev spaces.

Caccioppoli-Type Estimates for a Class of Nonlinear Differential Operators

We establish Caccioppoli-type estimates for a class of nonlinear differential equations with the help of a differential identity that generalizes the well-known multidimensional Picone formula. In special cases, these estimates give the Finsler p-Laplacian, the p-Laplacian, and the pseudo-p-Laplacian.

Interior regularity of space derivatives to an evolutionary, symmetric $$\varphi $$ φ -Laplacian

We consider the Orlicz-growth generalization to the evolutionary p-Laplacian and to the evolutionary symmetric p-Laplacian. We derive the spatial second-order Caccioppoli-type estimate for a local weak solution to these systems. Our result is new even for the p-case.

Analysis of [formula omitted]-Laplace thermistor models describing the electrothermal behavior of organic semiconductor devices

We study a stationary thermistor model describing the electrothermal behavior of organic semiconductor devices featuring non-Ohmic current–voltage laws and self-heating effects. The coupled system consists of the current-flow equation for the electrostatic potential and the heat equation with Joule heating term as source. The self-heating in the device is modeled by an Arrhenius-like temperature dependency of the electrical conductivity. Moreover, the non-Ohmic electrical behavior is modeled by a power law such that the electrical conductivity depends nonlinearly on the electric field. Notably, we allow for functional substructures with different power laws, which gives rise to a p(x)-Laplace-type problem with piecewise constant exponent.We prove the existence and boundedness of solutions in the two-dimensional case. The crucial point is to establish the higher integrability of the gradient of the electrostatic potential to tackle the Joule heating term. The proof of the improved regularity is based on Caccioppoli-type estimates, Poincaré inequalities, and a Gehring-type Lemma for the p(x)-Laplacian. Finally, Schauder’s fixed-point theorem is used to show the existence of solutions.

The weighted reverse Poincaré-type estimates for the difference of two convex vectors

In this paper we develop Caccioppoli-type estimates for arbitrary convex vectors and the vectors having both convex and concave arguments. To do this, we first develop these estimates for smooth convex vectors and then, through mollification, extend the results for arbitrary convex vectors. These types of estimates are valuable in the problems of financial mathematics for the establishment of the optimal investment strategies.

Fractional elliptic equations, Caccioppoli estimates and regularity

Let L=−divx(A(x)∇x) be a uniformly elliptic operator in divergence form in a bounded domain Ω. We consider the fractional nonlocal equations{Lsu=f,inΩ,u=0,on∂Ω,and{Lsu=f,inΩ,∂Au=0,on∂Ω. Here Ls, 0<s<1, is the fractional power of L and ∂Au is the conormal derivative of u with respect to the coefficients A(x). We reproduce Caccioppoli type estimates that allow us to develop the regularity theory. Indeed, we prove interior and boundary Schauder regularity estimates depending on the smoothness of the coefficients A(x), the right hand side f and the boundary of the domain. Moreover, we establish estimates for fundamental solutions in the spirit of the classical result by Littman–Stampacchia–Weinberger and we obtain nonlocal integro-differential formulas for Lsu(x). Essential tools in the analysis are the semigroup language approach and the extension problem.

The Liouville theorem under second order differentiability assumption

Iwaniec and Martin proved that in even dimensions n≥3, Wloc1,n/2 conformal mappings are Möbius transformations and they conjectured that it should also be true in odd dimensions. We prove this theorem for a conformal map f∈Wloc1,1 in dimension n≥3 under one additional assumption that the norm of the first order derivative |Df| satisfies |Df|p∈Wloc1,2 for p≥(n−2)/4. This is optimal in the sense that if |Df|p∈Wloc1,2 for p<(n−2)/4, it may not be a Möbius transform. This result shows the necessity of the Sobolev exponent in the Iwaniec–Martin conjecture. Meanwhile, we show that the Iwaniec–Martin conjecture can be reduced to a conjecture about the Caccioppoli type estimate.

On the Regularity of Weak Contact -Harmonic Maps

We prove Caccioppoli type estimates and consequently establish local Hölder continuity for a class of weak contact -harmonic maps from the Heisenberg group into the sphere .

Two-weight Caccioppoli-type estimates and weak reverse Hölder inequalities for 𝒜-harmonic tensors

In this paper, we first introduce a new weight-A λ3 r (λ1,λ2,Ω)-weight, and then prove the two-weight Caccioppoli-type estimates and the two-weight weak reverse Holder inequalities for A -harmonic tensors, which can be regarded as generalizations of the classical results.

Schauder type estimates of linearized Mullins-Sekerka problem

In this paper we obtain a Caccioppoli type estimate for the model of the linearized Mullins-Sekerka equations bya new technique, then we use this estimate to derive it's Schauder type estimates by polynomial approximation method.

A Caccioppoli-type estimate for very weak solutions to obstacle problems with weight

This paper gives a Caccioppoli-type estimate for very weak solutions to obstacle problems of the A-harmonic equation div A ( x , ∇ u ) =0 with |A ( x , ξ ) |≈w ( x ) |ξ | p - 1 , where 1 < p < ∞ and w(x) be a Muckenhoupt A1 weight.Mathematics Subject Classification (2000) 35J50, 35J60

Weighted Decomposition Estimates for Differential Forms

After introducing the definition of -weights, we establish the -weighted decomposition estimates and -weighted Caccioppoli-type estimates for -harmonic tensors. Furthermore, by Whitney covering lemma, we obtain the global results in domain . These results can be used to study the integrability of differential forms and to estimate the integrals for differential forms.

Removable Singularities of Solutions of A-harmonic Type Equations

A Caccioppoli type estimate is established for a class of second order PDEs of divergence type, and its removable singularities of Hausdorff dimension greater than zero is obtained.

Weighted integral inequalities for differential forms

In this paper, we obtain some weighted integral inequalities for differential forms, which can be considered as generalizations of the Poincare inequality, the Caccioppoli-type estimate, and the weak reverse Holder inequality, respectively. These results can be used to study the integrability of differential forms and to estimate the integrals of differential forms. We also give some applications of the above results.

Estimates of Weighted Integrals for Differential Forms

In this paper we obtain some weighted integral estimates for differential forms which are generalizations of the Poincaré inequality, the Caccioppoli-type estimate, and the weak reverse Hölder inequality, respectively. These results can be used to study the integrability of differential forms and to estimate the integrals for differential forms.