Anisotropic Sobolev Spaces Research Articles

Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions

Abstract We introduce a novel framework for embedding anisotropic variable exponent Sobolev spaces into spaces of anisotropic variable exponent Hölder-continuous functions within rectangular domains. We establish a foundational approach to extend the concept of Hölder continuity to anisotropic settings with variable exponents, providing deeper insight into the regularity of functions across different directions. Our results not only broaden the understanding of anisotropic function spaces but also open new avenues for applications in mathematical and applied sciences.

Nonlinear elliptic unilateral problems with measure data in the anisotropic Sobolev space

Abstract In this article, we consider a nonlinear elliptic unilateral equation whose model is − ∑ i = 1 N ∂ i σ i ( x , u , ∇ u ) + L ( x , u , ∇ u ) + N ( x , u , ∇ u ) = μ − div ϕ ( u ) in Ω . -\mathop{\sum }\limits_{i=1}^{N}{\partial }^{i}{\sigma }_{i}\left(x,u,\nabla u)+L\left(x,u,\nabla u)+N\left(x,u,\nabla u)=\mu -{\rm{div}}\phi \left(u)\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega . We prove the existence of entropy solutions for the aforementioned equation in the anisotropic Sobolev space, under the hypotheses, μ = f − div F \mu =f-{\rm{div}}F belongs to L 1 ( Ω ) + W − 1 , p ′ ( Ω ) {L}^{1}\left(\Omega )+{W}^{-1,{p}^{^{\prime} }}\left(\Omega ) . The nonlinear terms L ( x , s , ∇ u ) L\left(x,s,\nabla u) satisfy the sign and growth conditions, and N ( x , s , ∇ u ) N\left(x,s,\nabla u) verifies only the growth conditions.

Approximation of a two‐dimensional Gross–Pitaevskii equation with a periodic potential in the tight‐binding limit

AbstractThe Gross–Pitaevskii (GP) equation is a model for the description of the dynamics of Bose–Einstein condensates. Here, we consider the GP equation in a two‐dimensional setting with an external periodic potential in the ‐direction and a harmonic oscillator potential in the ‐direction in the so‐called tight‐binding limit. We prove error estimates which show that in this limit the original system can be approximated by a discrete nonlinear Schrödinger equation. The paper is a first attempt to generalize the results from [19] obtained in the one‐dimensional setting to higher space dimensions and more general interaction potentials. Such a generalization is a non‐trivial task due to the oscillations in the external periodic potential which become singular in the tight‐binding limit and cause some irregularity of the solutions which are harder to handle in higher space dimensions. To overcome these difficulties, we work in anisotropic Sobolev spaces. Moreover, additional non‐resonance conditions have to be satisfied in the two‐dimensional case.

Variable exponents anisotropic nonlinear elliptic systems with L p ′ → ( ⋅ ) -data

Our paper is devoted to studying the existence results of distributional solutions in the anisotropic Sobolev space with variable exponents and zero boundary W ˚ 1 , p → ( ⋅ ) ( Ω , R m ) for a family of boundary value problems, which is a variable exponent anisotropic nonlinear elliptic system with L p ′ → ( ⋅ ) data for the datum f (i.e. ∈ L p ′ → ( ⋅ ) ( Ω , R m ) ), and has a sum of two Carathéodory nonlinearities, one of which includes the solution u and the other its partial derivatives ∂ i u , i = 1 , … , N .

Existence and Uniqueness Solutions for Some Strongly Quasilinear Parabolic Problems in Anisotropic Sobolev Spaces

Existence and Uniqueness Solutions for Some Strongly Quasilinear Parabolic Problems in Anisotropic Sobolev Spaces

Convergence problem of the generalized Kadomtsev–Petviashvili II equation in anisotropic Sobolev space

Convergence problem of the generalized Kadomtsev–Petviashvili II equation in anisotropic Sobolev space

Horizontal p-Kirchhoff equation on the Heisenberg group

We study the existence and multiplicity of solutions both for the isotropic and the anisotropic horizontal p-Kirchhoff equations on the Heisenberg group Hn, via Krasnoselskii's genus. Finally, an interesting question for the anisotropic embedding in the subelliptic literature is presented and it is conjectured that the exponent p‾⁎:=QnQ∑i=1n1pi−n can be the effective critical exponent in the setting of anisotropic horizontal Sobolev space.

Author Correction: On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

Author Correction: On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

A semilinear non-homogeneous problem related to Korteweg-de Vries Equation

In this paper, we consider a non-homogeneous generalized Korteweg-de Vries problem with some hypotheses on the right-hand side, and we give a new regularity result of the solution in an anisotropic Sobolev space. Then we apply the obtained result to a non-homogeneous KdV problem. This work is an extension of solvability results for a right-hand side f in Lebesgue space.

On subordination conditions for systems of minimal di erential operators

In this paper, we provide a review of results on apriori estimates for systems of minimal differential operators in the scale of spaces \(L^p(\Omega),\) where \(p\in[1,\infty].\) We present results on the characterization of elliptic and \(l\)-quasielliptic systems using apriori estimates in isotropic and anisotropic Sobolev spaces \(W_{p,0}^l(\mathbb
 R^n),\)\(p\in[1,\infty].\) For a given set \(l=(l_1,\dots,l_n)\in\mathbb
 N^n\) we prove criteria for the existence of \(l\)-quasielliptic and weakly coercive systems and indicate wide classes of weakly coercive in \(W_{p,0}^l(\mathbb
 R^n),\) \(p\in[1,\infty],\) nonelliptic, and nonquasielliptic systems. In addition, we describe linear spaces of operators that are subordinate in the \(L^\infty(\mathbb R^n)\)-norm to the tensor product of two elliptic differential polynomials.

On the obstacle problem associated to the Kolmogorov–Fokker–Planck operator with rough coefficients

This work is devoted to the study of the obstacle problem associated to the Kolmogorov–Fokker–Planck operator with rough coefficients through a variational approach. In particular, after the introduction of a proper anisotropic Sobolev space and related properties, we prove the existence and uniqueness of a weak solution for the obstacle problem by adapting a classical perturbation argument to the convex functional associated to the case of our interest. Finally, we conclude this work by providing a one-sided associated variational inequality, alongside with an overview on related open problems.

On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

We obtain critical embeddings and the concentration-compactness principle for the anisotropic variable exponent Sobolev spaces. As an application of these results,we confirm the existence of and find infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real parameters. With the groundwork laid in this work, there is potential for future extensions, particularly in extending the concentration-compactness principle to anisotropic fractional order Sobolev spaces with variable exponents in bounded domains. This extension could find applications in solving the generalized fractional Brezis–Nirenberg problem.

Existence and regularity of solutions of nonlinear anisotropic elliptic problem with Hardy potential

In this paper, we are interested in the existence and regularity of solutions for some anisotropic elliptic equations with Hardy potential and L m ( Ω ) data in appropriate anisotropic Sobolev spaces. The aim of this work is to get natural conditions to show the existence and regularity results for the solutions, that is related to an anisotropic Hardy inequality.

A Jacobi spectral method for calculating fractional derivative based on mollification regularization

In this article, we construct a Jacobi spectral collocation scheme to approximate the Caputo fractional derivative based on Jacobi–Gauss quadrature. The convergence analysis is provided in anisotropic Jacobi-weighted Sobolev spaces. Furthermore, the convergence rate is presented for solving Caputo fractional derivative with noisy data by invoking the mollification regularization method. Lastly, numerical examples illustrate the effectiveness and stability of the proposed method.

Anisotropic Sobolev Spaces with Weights

We study anisotropic weighted Sobolev spaces defined on a half-space. These spaces are important in the study of singular elliptic operators which degenerate at the boundary. We prove trace properties and density of smooth functions. Hardy inequalities are studied in detail.

Anisotropic parabolic-elliptic systems with degenerate thermal conductivity

In this paper, we investigate the existence and regularity of a capacity solution for a coupled nonlinear anisotropic parabolic-elliptic system, where the elliptic component of the parabolic equation involves thermal conductivities a i ( u ) that satisfy lim s → + ∞ a i ( s ) = 0 for all i = 1 , … , N . We work with anisotropic Sobolev spaces and use Schauder's fixed-point theorem to find weak solutions to approximate problems. In addition, we validate the convergence of a subsequence of approximate solutions to a capacity solution.

Korn’s inequality in anisotropic Sobolev spaces

Abstract Korn’s inequality has been at the heart of much exciting research since its first appearance in the beginning of the 20th century. Many are the applications of this inequality to the analysis and construction of discretizations of a large variety of problems in continuum mechanics. In this paper, we prove that the classical Korn inequality holds true in anisotropic Sobolev spaces. We also prove that an extension of Korn’s inequality, involving non-linear continuous maps, is valid in such spaces. Finally, we point out that another classical inequality, namely Poincaré’s inequality, also holds in anisotropic Sobolev spaces.

Anisotropic elliptic system with variable exponents and degenerate coercivity with Lm data

In a bounded open domain , where , with Lipschitz boundary , we consider the Dirichlet problem for the elliptic system given by here, , , represents a vector-valued function, denotes the partial derivative of u with respect to , and the vector fields and are Carathédory functions. In this paper, we focus on nonlinear degenerate anisotropic elliptic systems with variable growth and data, where m is small. Specifically, we consider the case where the right-hand side term f belongs to with 1<m<N. To analyze this problem, we work with an appropriate functional setting that involves anisotropic Sobolev spaces with variable exponents and weak Lebesgue (Marcinkiewicz) spaces with variable exponents.

Propagation of regularity for solutions to a Kawahara equation in two spatial dimensions

We prove a propagation of regularity result for solutions of an IVP for a Kawahara equation in two spatial dimensions. We show that if the initial data has sufficient smoothness and belongs to an appropriate Sobolev space when restricted to a half-plane {(x,y)|ax+by≥c0} for some a>0, b,c0∈R, then the solution u(t) lies in the same Sobolev space when restricted to the half-plane {(x,y)|ax+by≥c} for any real number c and any t such that 0<t≤T where T is the existence time of the solution. In addition, we prove a propagation of regularity result for initial data belonging to an anisotropic Sobolev space on a half-plane. These propagation of regularity results show that regularity of the initial data is transferred to the solution with infinite speed.

Traveling wave solutions to the inclined or periodic free boundary incompressible Navier-Stokes equations

This paper concerns the construction of traveling wave solutions to the free boundary incompressible Navier-Stokes system. We study a single layer of viscous fluid in a strip-like domain that is bounded below by a flat rigid surface and above by a moving surface. The fluid is acted upon by a bulk force and a surface stress that are stationary in a coordinate system moving parallel to the fluid bottom. We also assume that the fluid is subject to a uniform gravitational force that can be resolved into a sum of a vertical component and a component lying in the direction of the traveling wave velocity. This configuration arises, for instance, in the modeling of fluid flow down an inclined plane. We also study the effect of periodicity by allowing the fluid cross section to be periodic in various directions. The horizontal component of the gravitational field gives rise to stationary solutions that are pure shear flows, and we construct our solutions as perturbations of these by means of an implicit function argument. An essential component of our analysis is the development of some new functional analytic properties of a scale of anisotropic Sobolev spaces, including that these spaces are an algebra in the supercritical regime, which may be of independent interest.