Caccioppoli-type Inequality Research Articles

Variants of Bernstein’s theorem for variational integrals with linear and nearly linear growth

AbstractUsing a Caccioppoli-type inequality involving negative exponents for a directional weight we establish variants of Bernstein’s theorem for variational integrals with linear and nearly linear growth. We give some mild conditions for entire solutions of the equation $$\begin{aligned} {\text {div}} \Big [Df(\nabla u)\Big ] = 0 \,, \end{aligned}$$ div [ D f ( ∇ u ) ] = 0 , under which solutions have to be affine functions. Here f is a smooth energy density satisfying $$D^2 f>0$$ D 2 f > 0 together with a natural growth condition for $$D^2 f$$ D 2 f .

Convergence and local-to-global results for p-superminimizers on quasiopen sets

In this paper, several convergence results for fine p-(super)minimizers on quasiopen sets in metric spaces are obtained. For this purpose, we deduce a Caccioppoli-type inequality and local-to-global principles for fine p-(super)minimizers on quasiopen sets. A substantial part of these considerations is to show that the functions belong to a suitable local fine Sobolev space. We prove our results for a complete metric space equipped with a doubling measure supporting a p-Poincaré inequality with 1<p<∞. However, most of the results are new also for unweighted Rn.

Non-linear operators of divergence form on the Sierpinski gasket

We consider non-linear operators L of divergence type on the Sierpinski gasket. We prove the Harnack inequality for non-negative L-harmonic functions on the Sierpinski gasket. The proof is based on Moser’s method, and we construct a new measure λ by combining a Hausdorff measure and an energy measure, which plays an essential role at key steps. The Harnack inequality is achieved by involving the Caccioppoli type inequality and the weighted Sobolev inequality which give the local boundedness and the Weak Harnack inequality of solutions with respect to the measure λ.

A fractional Korn-type inequality for smooth domains and a regularity estimate for nonlinear nonlocal systems of equations

In this paper we prove a fractional analogue of the classical Korn's first inequality. The inequality makes it possible to show the equivalence of a function space of vector field characterized by a Gagliardo-type seminorm with 'projected difference' with that of a corresponding fractional Sobolev space. As an application, we will use it to obtain a Caccioppoli-type inequality for a nonlinear system of nonlocal equations, which in turn is a key ingredient in applying known results to prove a higher fractional differentiability result for weak solutions of the nonlinear system of nonlocal equations. The regularity result we prove will demonstrate that a well-known self-improving property of scalar nonlocal equations will hold for strongly coupled systems of nonlocal equations as well.

Regularity Theory for Mixed Local and Nonlocal Parabolic p-Laplace Equations

We investigate the mixed local and nonlocal parabolic p-Laplace equation $$\begin{aligned} \partial _t u(x,t)-\Delta _p u(x,t)+\mathcal {L}u(x,t)=0, \end{aligned}$$where \(\Delta _p\) is the usual local p-Laplace operator and \(\mathcal {L}\) is the nonlocal p-Laplace type operator. Based on the combination of suitable Caccioppoli-type inequality and Logarithmic Lemma with a De Giorgi–Nash–Moser iteration, we establish the local boundedness and Hölder continuity of weak solutions for such equations.

Sufficient Criteria for Obtaining Hardy Inequalities on Finsler Manifolds

We establish Hardy inequalities involving a weight function on complete, not necessarily reversible Finsler manifolds. We prove that the superharmonicity of the weight function provides a sufficient condition to obtain Hardy inequalities. Namely, if \(\rho \) is a nonnegative function and \(-\varvec{\Delta } \rho \ge 0\) in weak sense, where \(\varvec{\Delta }\) is the Finsler–Laplace operator defined by \( \varvec{\Delta } \rho = \mathrm {div}(\varvec{\nabla } \rho )\), then we obtain the generalization of some Riemannian Hardy inequalities given in D’Ambrosio and Dipierro (Ann Inst H Poincaré Anal Non Linéaire 31(3):449–475, 2014). By extending the results obtained, we prove a weighted Caccioppoli-type inequality, a Gagliardo–Nirenberg inequality and a Heisenberg–Pauli–Weyl uncertainty principle on complete Finsler manifolds. Furthermore, we present some Hardy inequalities on Finsler–Hadamard manifolds with finite reversibility constant, by defining the weight function with the help of the distance function. Finally, we extend a weighted Hardy inequality to a class of Finsler manifolds of bounded geometry.

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.

Regularity of quasi-minimizers for non-uniformly elliptic integrals

In this paper we consider a class of non-uniformly elliptic integral functionals and we prove the local boundedness of the quasi-minimizers. Our approach is based on a suitable adaptation of the celebrated De Giorgi proof, and it relies on an appropriate Caccioppoli-type inequality.

Local boundedness of Quasi-minimizers of fully anisotropic scalar variational problems

Some properties of the fully anisotropic Sobolev space $$W^{1,(\Phi _1,\ldots ,\Phi _N)}$$ are investigated. In particular extension and embedding theorems for functions in $$W^{1,(\Phi _1,\ldots ,\Phi _N)}$$ are deduced. Thanks to a Caccioppoli type inequality it is possible to carry out the De Giorgi procedure to prove the local boundedness of quasi-minimizers of the classical integral functional of the Calculus of Variations satisfying fully anisotropic growth conditions.

Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure

We prove a Caccioppoli type inequality for the solution of a parabolic system related to the nonlinear Stokes problem. Using the method of Caccioppoli type inequality, we also establish the existence of weak solutions satisfying a local energy inequality without pressure for the non-Newtonian Navier-Stokes equations.

Generalizing a Caccioppoli-type inequality

We generalize the case p=2 of the Caccioppoli-type inequality from [5] to multidimensional domains. The argument also provides a simpler proof of the case p=2 of the original inequality.

On partial Hölder continuity and a Caccioppoli inequality for minimizers of asymptotically convex functionals between Riemannian manifolds

We initiate a low-order regularity theory for vectorial minimizers of the functional Open image in new window where \(\varvec{\textit{w}}:\mathcal {M}\rightarrow \mathcal {N}\) and \(\mathcal {M}\) and \(\mathcal {N}\) are orientable Riemannian manifolds, in the case where G is only asymptotically convex. In particular, we prove three interrelated regularity results. The first result establishes partial Morrey regularity estimates, and Holder continuity, for minimizers. For the second result by utilizing the obtained partial Holder continuity, we prove a partial Caccioppoli-type inequality for minimizers. Finally, this result then allows us to deduce a higher integrability result for the gradient of minimizers of this functional. Since we work in the asymptotically convex setting, our results apply to far more general integrands than previously available in the literature for this type of problem.

A Note on the Caccioppoli Inequality for Biharmonic Operators

In this note, we establish a Caccioppoli-type inequality for biharmonic operators. We employ Picone’s identity for biharmonic operators to establish Caccioppoli-type inequality.

A global regularity result for some degenerate elliptic systems

We establish a Caccioppoli-type inequality for a second-order degenerate elliptic systems of p-Laplacian type. As a direct consequence, exploiting classical Campanato’s approach and the hole-filling technique due to Widman, we are able to prove a global regularity result on Morrey’s and Lq spaces, with q>p.

Dirac-harmonic equations for differential forms

In this paper, we introduce the Dirac-harmonic equation for differential forms and prove some basic norm inequalities, including the Caccioppoli-type inequality, Poincaré-type inequality and the weak reverse Hölder inequality for the solutions of this kind of differential equations.

Local boundedness property for parabolic BVP's and the Gaussian upper bound for their Green functions

In the present note, we give a concise proof for the equivalence between the local boundedness property for parabolic Dirichlet BVP's and the gaussian upper bound for their Green functions. The parabolic equations we consider are of general divergence form and our proof is essentially based on the gaussian upper bound by Daners [2] and a Caccioppoli's type inequality. We also show how the same analysis enables us to get a weaker version of the local boundedness property for parabolic Neumann BVP's assuming that the corresponding Green functions satisfy a gaussian upper bound.

L q harmonic functions on graphs

We prove an analogue of Yau’s Caccioppoli-type inequality for nonnegative subharmonic functions on graphs. We then obtain a Liouville theorem for harmonic or nonnegative subharmonic functions of class Lq, 1 ≤ q 1. Also, we provide counterexamples for Liouville theorems for 0 < q < 1.

OPTIMAL PARTIAL REGULARITY FOR NONLINEAR SUB-ELLIPTIC SYSTEMS RELATED TO HÖRMANDER' S VECTOR FIELDS

This paper is concerned with partial regularity for weak solutions to nonlinear sub-elliptic systems related to Hormander' s vector fields. The method of A-harmonic approximation introduced by Simon and developed by Duzaar and Grotowski is adapted to our context, and then a Caccioppoli-type inequality and partial regularity with optimal local Holder exponent for gradients of weak solutions to the systems under super-quadratic natural growth conditions is established.

Hardy-Littlewood and Caccioppoli-Type Inequalities for -Harmonic Tensors

We prove the new versions of the weighted Hardy-Littlewood inequality and Caccioppoli-type inequality for -harmonic tensors. We also explore applications of our results to -quasiregular mappings and -harmonic functions in .

CACCIOPPOLI TYPE INEQUALITY FOR DEGENERATE WEAKLY QUASIREGULAR MAPPINGS

The Caccioppoli type inequality for degenerate weakly quasiregular mappings is obtained,which can be used to derive the self-improving regularity for them.