We study energy minimizing properties of the function u = limj!1+ uj , where uj is the solution to the pj (·)- Laplacian Dirichlet problem with prescribed boundary values. Here p: ! (1,1) is a variable exponent and pj (x) = max{p(x),�j} forj > 1. This problem leads in a natural way to a mixture of Sobolev and total variation norms. The main results are obtained under the assumption that p is strongly log-Hcontinuous and bounded. To motivate our approach we also consider the one-dimensional case and give examples which justify our assumptions. The results can be applied in the analysis of a model for image restoration combining total variation and isotropic smoothing. Resume

We prove the local boundedness and the local Hölder continuity of weak solutions to nonlocal equations with variable orders and exponents under sharp assumptions.

We prove regularity theorems for minimizers of integral functionals of the Calculus of Variations ∫Ωf(x,Du(x))dx, with non-standard growth conditions of (p,q) type |z|p⩽f(x,z)⩽L(|z|q+1),p<q. In particular, we find that a sufficient condition for minimizers to be regular is qp<n+αn, where the function f(x,z) is α-Holder continuous with respect to the x-variable and x∈Ω⊂Rn; this condition is also sharp. We include results in the setting of Orlicz spaces; moreover, we treat certain relaxed functionals too. Finally, we address a problem posed by Marcellini in [43], showing a minimizer with an isolated singularity.

We prove Lipschitz continuity for local minimizers of integral functionals of the Calculus of Variations in the vectorial case, where the energy density depends explicitly on the space variables and has general growth with respect to the gradient. One of the models is

We define and study variable exponent Sobolev spaces with zero boundary values. This allows us to prove that the Dirichlet energy integral has a minimizer in the variable exponent case. Our results are based on a Poincaré-type inequality, which we prove under a certain local jump condition for the variable exponent.

Using sub-super solutions method, we study the existence of weak positive solutions for a new class of the system of differential equations with respect to the symmetry conditions. Our results are natural extensions from the previous ones in Fan (J Math Anal Appl 330:665–682 2007), Zhang (Nonlinear Anal 70:305–316 2009).

Using Moser's iteration method, we investigate the local boundedness of solutions for p(x)-Laplacian equations and derive a Harnack type inequality important for applications.

This article is concerned with Lp(x) estimates of the gradient of p(x)-harmonic maps. It is known that p(x)-harmonic maps are the weak solutions of a system with natural growth conditions, but it is difficult to use the classical elliptic techniques to find gradient estimates. In this article, we use the monotone inequality to show that the minimum p(x)-energy can be expressed by the Lp(x) norm of a gradient of a function Φ, which is a weak solution of a single equation.

Let $(g^{\alpha\beta}(x))$ and $(h_{ij}(u))$ be uniformly elliptic symmetric matrices, and assume that $h_{ij}(u)$ and $p(x) \, (\, \geq 2)$ are sufficiently smooth. We prove partial regularity of minimizers for the functional [ {\mathcal F}(u) = \int_\Omega (g^{\alpha \beta}(x) h_{ij}(u) D_\alpha u^iD_\beta u^j)^{p(x)/2} dx, \] under the non-standard growth conditions of $p(x)$-type. If $g^{\alpha\beta}(x)$ are in the class $VMO$, we have partial H\"older regularity. Moreover, if $g^{\alpha\beta}$ are H\"older continuous, we can show partial $C^{1,\alpha}$-regularity.

In this paper, we study local regularity properties of minimizers of nonlocal variational functionals with variable exponents and weak solutions to the corresponding Euler–Lagrange equations. We show that weak solutions are locally bounded when the variable exponent p is only assumed to be continuous and bounded. Furthermore, we prove that bounded weak solutions are locally Hölder continuous under some additional assumptions on p. On the one hand, the class of admissible exponents is assumed to satisfy a log-Höldertype condition inside the domain, which is essential even in the case of local equations. On the other hand, since we are concerned with nonlocal problems, we need an additional assumption on p outside the domain.

In this paper, we consider the system of differential equations $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l@{\quad }l} -\Delta _{p(x)} u=\lambda ^{p(x)}[\lambda _{1}f(v)+\mu _{1}h(u)] &{} \mathrm{in }\, \Omega , \\ -\Delta _{p(x)} v=\lambda ^{p(x)}[\lambda _{2}g(u)+\mu _{2}\tau (v)] &{} \mathrm{in }\,\Omega , \\ u=v=0 &{} \mathrm{on }\,\partial \Omega ,\\ \end{array}\right. \end{aligned}$$-Δp(x)u=λp(x)[λ1f(v)+μ1h(u)]inΩ,-Δp(x)v=λp(x)[λ2g(u)+μ2τ(v)]inΩ,u=v=0on∂Ω,where $$\Omega \subset \mathbb R ^N$$Ω⊂RN is a bounded domain with $$C^2$$C2 boundary $$\partial \Omega $$∂Ω and $$1<p(x)\in C^{1}(\overline{\Omega })$$1<p(x)∈C1(Ω¯) is a function. The operator $$-\Delta _{p(x)} u = -\mathrm{div}(|\nabla u|^{p(x)-2}\nabla u)$$-Δp(x)u=-div(|∇u|p(x)-2∇u) is called the $$p(x)$$p(x)-Laplacian, $$\lambda ,\,\lambda _{1},\,\lambda _{2},\,\mu _{ 1}$$λ,λ1,λ2,μ1 and $$\mu _{2}$$μ2 are positive parameters. Using the sub-supersolutions method, we prove the existence of positive solutions if $$\begin{aligned} \lim _{u \rightarrow +\infty }\frac{f[M(g(u))^\frac{1}{p^--1}]}{u^{p^--1}} = 0, \quad \forall M > 0, \end{aligned}$$limu→+∞f[M(g(u))1p--1]up--1=0,∀M>0,$$\begin{aligned} \lim _{u \rightarrow +\infty } \frac{h(u)}{u^{p^-- 1}} = 0, \quad \lim _{u \rightarrow +\infty } \frac{\tau (u)}{u^{p^- - 1}} = 0. \end{aligned}$$limu→+∞h(u)up--1=0,limu→+∞τ(u)up--1=0.In particular, we do not assume any symmetry condition, and we do not assume any sign condition on $$f(0), g(0), h(0)$$f(0),g(0),h(0) or $$\tau (0)$$τ(0).

In this paper we study the following p(x)-Laplacian problem: -div(a(x)VD uV(p(x)-2)D u)+b(x)V uV(p(x)-2)u = f(x, u), x e U, u = 0, on P U, where 1< p(1) L p(x) L p(2) < n, U S R-n is a bounded domain and applying the mountain pass theorem we obtain the existence of solutions in W-0(1,p(x)) for the p(x)-Laplacian problems in the superlinear and sublinear cases. © 2004 Elsevier Inc. All rights reserved.

In this paper we prove that the concept of Lebesgue points generalizes naturally to the setting of variable exponent Lebesgue and Sobolev spaces. We assume that the variable exponent is log-Holder continuous, which, although restrictive, is a common assumption in variable exponent spaces.

In this note we consider the Dirichlet energy integral in the variable exponent case under minimal assumptions on the exponent. First we show that the Dirichlet energy integral always has a minimizer if the boundary values are in $L^\infty$. Second, we give an example which shows that if the so-called "jump-condition", known to be sufficient, is violated, then a minimizer need not exist for unbounded boundary values.

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...

Abstract: We consider the integral functional under non-standard growth assumptions that we call p(x) type: namely, we assume that a relevant model case being the functional Under sharp assumptions on the continuous function p(x)>1 we prove regularity of minimizers. Energies exhibiting this growth appear in several models from mathematical physics.

Abstract We study the global C 1 , α regularity of the bounded generalized solutions of the variable exponent elliptic equations in divergence form with both Dirichlet and Neumann boundary conditions. Our results are a generalization of the corresponding results in the constant exponent case.

Abstract We prove regularity results for weak solutions to systems modelling electro-rheological fluids in the stationary case, as proposed in [27, 31]; a particular case of the system we consider is where ℰ(u) is the symmetric part of the gradient Du and the variable growth exponent p(x) is a Hölder continuous function larger than 3n/(n+2).

Abstract We prove Calderón and Zygmund type estimates for a class of elliptic problems whose model is the non-homogeneous p (x )-Laplacean system Under optimal continuity assumptions on the function p (x ) > 1 we prove that Our estimates are motivated by recent developments in non-Newtonian fluidmechanics and elliptic problems with non-standard growth conditions, and are the natural, ‘‘non-linear’’ counterpart of those obtained by Diening and Růžička [L. Diening and M. Růžička, Calderón-Zygmund operators on generalized Lebesgue spaces L p (‧) and problems related to fluid dynamics, J. reine angew. Math. 563 (2003), 197–220] in the linear case.

Abstract We mainly consider the system { − Δ p ( x ) u = λ f ( x , v ) in Ω , − Δ p ( x ) v = λ g ( x , u ) in Ω , u = v = 0 on ∂ Ω , where Ω ⊂ R N is a bounded domain, p ( x ) is a function which satisfies some conditions, − Δ p ( x ) u = − div ( | ∇ u | p ( x ) − 2 ∇ u ) is called p ( x ) -Laplacian. We give the existence of positive solutions under some conditions. In particular, we do not assume radial symmetric conditions on the system.