# Variable Exponent Sobolev Spaces And Regularity Of Domains

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### LIOUVILLE THEOREMS FOR ELLIPTIC PROBLEMS IN VARIABLE EXPONENT

We deal with the variable exponent Lebesgue and Sobolev spaces, which recently have received more and more attention both from the theoretical and from the ap-plied point of view. We refer to books [10,16] for detailed information on the varia-ble exponent spaces and to the survey [31] summarising inter alia developments on

### Conference on recent developments in L -variational problems

TITLE: Regularity of Quasi-Minimizers for Non-Homogeneous Energies on Metric Spaces ABSTRACT: It is well-known that on Euclidean domains, minimizers of the p-Dirichlet energy integral enjoy a rich regularity theory. (For instance, they are Holder continuous and satisfy a Harnack inequality.) Though such functions are solutions to an associated

### export.arxiv.org

arXiv:2103.08928v1 [math.AP] 16 Mar 2021 A NEW CLASS OF DOUBLE PHASE VARIABLE EXPONENT PROBLEMS: EXISTENCE AND UNIQUENESS ANGEL CRESPO-BLANCO, LESZEK GASI´ NSKI, PETTERI HARJULEH

### Symmetry results for the p.x/-Laplacian equation

2 Recalls on variable exponent Sobolev spaces We recall here some deﬁnitions and basic properties of the variable exponent Lebesgue Sobolev spaces Lp. /. /, W1;p /and W1;p./ 0. /, where is a bounded domain in RN. We set CC. /D ° h2C. /Wmin h>1 ; and, for h2C. /, we denote h WDmin h and hCWDmax h: For p2CC. /, we introduce the variable

### Linear Nonlinear Analysis Algebra and its Applications

The paper is organized as follows. In Section2some notions about variable exponent spaces and Γ-convergence are recalled. Also preliminary results are proved in Section2, in view of the last section. Finally, Section3contains the proof ofTheorem 1.1and a more general result which can be useful for future developments. 2. Preliminary results

### (1.3) 3q0 > 0 : VÇ € R3, x € fi : a^KI2 > a0 < 2,

2. Regularity. In this section, we specify the precise regularity of solutions to problem (1.1)- (1.2) in terms of the anisotropically weighted Sobolev spaces of [6]. 2.1. Subdomains and weights. In the bounded Lipschitz polyhedron Ū C K3 with plane faces, we denote by C the set of corners c and by £ the set of (open) edges e of Í7, and we

### A L E J A N D R O V É LEZ- S A N T I A G O

9. A. Vélez-Santiago. Embedding and trace results for variable exponent Sobolev and Maz'ya spaces on non-smooth domains. Glasgow Mathematical Journal 58 (2016), 471-489. 10. A. Vélez-Santiago. Ambrosetti Prodi-type problems for quasi-linear elliptic equations with nonlocal boundary conditions. Calculus of Variations and

### HARDY S INEQUALITY IN A VARIABLE EXPONENT

Our aim is to generalize Hardy s inequality to variable exponent Sobolev spaces. Although variable exponent Lebesgue and Sobolev spaces have a long prehistory, the foundations of the theory in Rn were laid by Kov´aˇcik and R´akosn´ık [14] only in the early 1990 s. After Diening s breakthrough in the

### Local Hölder estimates for general elliptic p(x)-Laplacian

Inthis paper weobtain theinterior H¨older regularity ofthegradients ofweak solutions domains, J Funct. Anal., 250 P. HARJULEHTO, Variable exponent Sobolev

### QUT

Random elds of variable order on multifractal domains M.D. Ruiz-Medina1;V.V. Anh2 and J.M. Angulo1 1D ep artm nof S i s cd O R h University of Granada, Spain 2 S

### Regularity Problem For Quasilinear Elliptic And Parabolic

problem. WE study regularity properties of weak solutions and we prove the Harnack's inequality and continuity of solution. We show by proving a comparison principle that Keller-Osserman property is valid and we discuss the existence of Evans functions for solutions to the quasilinear elliptic equations in Sobolev spaces with variable exponent.

### ON SOME PROPERTIES OF LORENTZ-SOBOLEV SPACES WITH VARIABLE

theory of variable exponent Sobolev spaces is an important theoretical tool to study the variable exponent problems. The Sobolev embeddings theorems in the variable exponent Sobolev space 𝑊1, (.)(Ω) have been studied by many authors. Also Diening [10] proved the optimal Sobolev embedding

### EXTENSIONS IN SPACES WITH VARIABLE EXPONENTS : THE HALF SPACE

EXTENSIONS IN SPACES WITH VARIABLE EXPONENTS THE HALF SPACE L. DIENING, S. FRÖSCHL Abstract. In this article we study the HardyLittlewood max‐ imal operator in variable exp

### Local gradient estimates for the p(x)-Laplacian elliptic

[9] L. DIENING&M.RU˚ZIˇ CKAˇ , Calder´on-Zygmund operators on generalized Lebesgue spaces Lp( ) and problems related to ﬂuid dynamics, J. Reine Angew. Math. bf 563 (2003), 197 220. [10] X. FAN,Global C1,α regularity for variable exponent elliptic equations in divergence form,J.Differ-ential Equations bf 235 (2007), 397 417.

### Mellin analysis of weighted Sobolev spaces with

On domains with conical points, weighted Sobolev spaces with powers of the distance to the conical points as weights form a classical framework for describing the regularity of solutions of elliptic boundary value problems, cf. papers by Kondrat ev

### VARIABLE EXPONENT LEBESGUE SPACES

Variable exponent Lebesgue spaces on Euclidean spaces have attracted a steadily increasing interest over the last couple of years, but the variable exponent framework has not yet been applied to the metric measure space setting. Variable exponent spaces have been independently discovered by several investigators [7, 14, 20, 21].

### Weighted Sobolev Spaces By Alois Kufner

weighted sobolev spaces. weighted variable sobolev spaces and capacity. carleson measures for weighted hardy sobolev spaces. 110 726s09 topics in analysis mathematics. embeddings of weighted sobolev spaces and generalized. weighted fe math kobe 3 / 25

### CONTEMPORARY MATHEMATICS

Dynamics on bounded domains CHIARA FROSINI 99 On the rate of tangential convergence of functions from Hardy spaces, O

### The Hardy‐Littlewood maximal operator

examples showing the necessity of some regularity conditions on p for the boundedness. As an application of the auxiliary pointwise estimate for the Hardy‐Littlewood maximal operator we prove some density results for generalized Lebesgue and Sobolev spaces with variable exponent. Contents §1. Introduction §2. Weight class A {p} §3.

### Electronic Journal of Di erential Equations, Vol. 2014 (2014

Abstract. In this article we obtain a Pohozaev-type inequality for Sobolev spaces with variable exponents. This inequality is used for proving the nonex-istence of nontrivial weak solutions for the Dirichlet problem p(x)u = juj q(x) 2u; x 2 u(x) = 0; x [email protected]; with non-standard growth. Our results extend those obtained by Otani [16].^ 1

### Conformal Mappings and Isometric Immersions

necessity of the Sobolev exponent in the Iwaniec-Martin conjecture. In the second part, we prove the developability and C1,1/2 loc regularity of W 2,2 isometric im-mersions of n-dimensional domains into Rn+1 for n≥ 3. The result is sharp in the sense that W1,p,1 ≤ p≤ ∞ and W2,p,1 ≤ p<2 isometric immersions may not be developable.

### Electronic Journal of Di erential Equations, Vol. 2015 (2015

equations under the framework of variable exponent Sobolev spaces. Motivated by their works, we shall study the existence and boundedness of weak solutions to problem (1.1) with sublinear growth. When the variable exponent depend only on space variable x, evolution variational inequality without initial conditions has been studied in [4, 5, 24].

### Traces and Fractional Sobolev Extension Domains with Variable

Jan 04, 2018 Keywords: Variable exponent fractional Sobolev spaces, extension oper-ator, trace operator, complemented subspace problem 1 Introduction This paper is devoted to the problem of extendability in the fractional Sobolev spaces with variable exponent and its relation with the trace operator. The

### EXTENSIONS IN SPACES WITH VARIABLE Title EXPONENTS : THE HALF

These domains were introduced by Jones [Jon81] and are therefore also called Jones domains. They are the natural domains for the extension of Sobolev functions and it is therefore our aim to solve (EP) for the same type of domains. Once this problem is solved, the extension re-sult for variable exponent Sobolev spaces will follow immediately from

### Boundary determination for hybrid imaging from a single

variable exponent equation and after that state a regularity result. Before proceeding, we deﬁne the variable exponent Lebesgue space Lp(Ω), d a bounded open set and d≥1. The variable exponent Sobolev spaces are deﬁned in terms of Lp(Ω) in the usual way. Following the book of Diening, Harjulehto, H¨ast¨o and R˚uˇziˇcka [13

### Lecture Notes On Sobolev Spaces Department Of Mathematics

Sobolev spaces In this chapter we begin our study of Sobolev spaces. The Sobolev space is a vector space of functions that have weak derivatives. Motivation for studying these spaces is that solutions of partial differential equations, when they exist, belong naturally to Sobolev spaces. 1.1Weak derivatives Notation.

### Intrinsic atomic characterization of 2-microlocal spaces on

Besov-type spaces of variable smoothness on rough domains, namely bounded Lipschitz domains in Rn, epigraph of Lipschitz functions or (ǫ,δ)-domains. Our aim is to extend the results already known in order to obtain an intrinsic characterization for the scale of 2-microlocal Besov and Triebel-Lizorkin spaces Bw p( ),q(Ω) and F w p( ),q

### Journal of Mathematical Analysis and Applications

(1.1) are variable exponent Lebesgue and Sobolev spaces (for its deﬁnition, see Section 2). For 3D bounded domains, in [20], the existence of local strong solutions to ERF s steady motions with Dirichlet boundary conditions has been obtained if 1.8

### Curriculum Vitae - Weebly

with variable exponent. Asymp. Anal.93 (2015) 161-185. 9. J. Fern andez Bonder, N. Saintier and A. Silva. On the Sobolev trace Theorem for variable exponent spaces in the critical range.Ann. Mat. Pura Appl.(4)193 (2014) 1607-1628. 10. J. Fern andez Bonder, N. Saintier and A. Silva.Existence of solution to a critical equation with variable

### Geometric problems for parabolic and elliptic PDE s VI

If the exponent q() is essentially subcritical in the sense of Sobolev, then the existence of time-global bounds is a standard fact. On the other hand, the existence of bounds is highly nontrivial for the case where the exponent touches the critical Sobolev exponent as above.

### Analysis Of Partial Differential Equations

obtained many astonishing and frequently cited results in the theory of harmonic potentials on non-smooth domains, potential and capacity theories, spaces of functions with bounded variation, maximum principle for higher-order elliptic equations, Sobolev multipliers, approximate approximations, etc.

### Regularity of p(.)-superharmonic functions, the Kellogg

of variable exponent Sobolev spaces, as well as potential theory. We also observe that some of the characterizations of the. p( )-Sobolev spaces with zero boundary data discussed in [13] can be improved, and these improvements turn out useful for our later results. We also discuss the squeezing Lemma 2.6 for variable exponent Sobolev

### Pietro Poggi-Corradini

A condition suﬃcient for the partial regularity of minimizers in two-dimensional nonlinear elasticity M. Foss Dynamics on bounded domains C. Frosini On the rate of tangential convergence of functions from Hardy spaces, 0 < p < 1 K. Hare and A. Stokolos Counterexamples of regularity in variable exponent Sobolev spaces P. Hasto

### List of talks

S˘erban Costea: Regularity of capacities associated to nonre exive Sobolev-Lorentz spaces in the Euclidean setting Luminit˘a-Ioana Cotirla: A new class of harmonic functions de ned by an operator Mihai Cristea: Some conditions of lightness, openess and discreteness and local and global in-vertibility of Sobolev mappings

### Variational Analysis In Sobolev And Bv Spaces Applications To

BV Spaces Elliptic Problems in Nonsmooth Domains Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis provides researchers and graduate students with a thorough introduction to the theory of nonlinear partial differential equations (PDEs) with a variable exponent, particularly those of elliptic type.

### [hal-00402645, v1] Mellin analysis of weighted Sobolev spaces

On domains with conical points, weighted Sobolev spaces wi th powers of the distance to the conical points as weights form a classica l framework for describing the regularity of solutions of elliptic boundary value prob lems, cf. papers by Kondrat'ev and Maz'ya-Plamenevskii. Two classes of weighted norms are usually considered: Ho-

### Workshop Nonstandard Growth Analysis and its Applications 2017

Compact embeddings of variable Sobolev spaces on complete Riemannian manifolds One can see that Riemannian manifold is a good setting for deﬁning variable exponent Sobolev spaces. After introducing such space we start to study Sobolev embedding. For non compact manifolds they not hold. Inspired by result from Euclidean setting and space

### Characterization of Riesz and Bessel potentials on variable

if one assumes that the exponent satisﬁes natural regularity conditions. As a consequence of this characterization, we describe a relation between the spaces of Riesz or Bessel potentials and the variable Sobolev spaces. 1. Introduction The Lebesgue spaces L p( ) with variable exponent and the corresponding Sobolev spaces Wm p( ) have been