Sobolev Spaces Research Articles

Existence and uniqueness of solutions of the Koopman–von Neumann equation on bounded domains

Abstract The Koopman--von Neumann equation describes the evolution of a complex-valued wavefunction corresponding to the probability distribution given by an associated classical Liouville equation. Typically, it is defined on the whole Euclidean space. The investigation of bounded domains, particularly in practical scenarios involving quantum-based simulations of dynamical systems, has received little attention so far. We consider the Koopman–von Neumann equation associated with an ordinary differential equation on a bounded domain whose trajectories are contained in the set's closure. Our main results are the construction of a strongly continuous semigroup together with the existence and uniqueness of solutions of the associated initial value problem. To this end, a functional-analytic framework connected to Sobolev spaces is proposed and analyzed. Moreover, the connection of the Koopman–von Neumann framework to transport equations is highlighted.

On One Point Singular Nonlinear Initial Boundary Value Problem for a Fractional Integro-Differential Equation via Fixed Point Theory

In this article, we focus on examining the existence, uniqueness, and continuous dependence of solutions on initial data for a specific initial boundary value problem which mainly arises from one-dimensional quasi-static contact problems in nonlinear thermo-elasticity. This problem concerns a fractional nonlinear singular integro-differential equation of order θ∈[0,1]. The primary methodology involves the application of a fixed point theorem coupled with certain a priori bounds. The feasibility of solving this problem is established under the context of data related to a weighted Sobolev space. Furthermore, an additional result related to the regularity of the solution for the formulated problem is also presented.

On stability and regularity for subdiffusion equations involving delays

We study a class of nonlocal evolution equations involving time-varying delays which is employed to depict subdiffusion processes. The global solvability, stability and regularity are shown by using the resolvent theory, nonlocal Halanay inequality, fixed point argument and embeddings of fractional Sobolev spaces. Our result is applied to the nonlocal Fokker–Planck model with nonlinear force fields.

Inverse initial‐value problems for time fractional diffusion equations in fractional Sobolev spaces

AbstractWe study the time fractional diffusion equation , , in a bounded domain with an elliptic operator and a locally Lipschitz nonlinearity on fractional Sobolev spaces, subjected to the homogeneous Dirichlet boundary condition. Data have not been measured at the initial time , but at a final time , that is, is given instead of . The problem is, therefore, called an inverse initial‐value problem. We first establish the well‐posedness of this problem on fractional Sobolev spaces and the regularity of the solution by assuming only the local Lipschitz continuity of . Second, an susceptible‐infected (shortly, SI) model with heterogeneity and a Navier–Stokes equation have been exemplified. Finally, a spatial ‐estimate for the solution and its gradient has been provided. The essential tools are asymptotic behaviours of Mittag–Leffler functions, fractional power spaces, fractional Sobolev spaces and embedding, weighted functional spaces, and estimates for heat semigroup.

Linear Statistics of Determinantal Point Processes and Norm Representations

Abstract We study the asymptotic behavior of the fluctuations of smooth and rough linear statistics for determinantal point processes on the sphere and on the Euclidean space. The main tool is the generalization of some norm representation results for functions in Sobolev spaces and in the space of functions of bounded variation.

Trace principle for Riesz potentials on Herz-type spaces and applications

We establish trace inequalities for Riesz potentials on Herz-type spaces and examine the optimality of conditions imposed on specific parameters. We also present some applications in the form of Sobolev-type inequalities, including the Gagliardo–Nirenberg–Sobolev inequality and the fractional integration theorem in the Herz space setting. In addition, we obtain a Sobolev embedding theorem for Herz-type Sobolev spaces.

Anisotropic obstacle Neumann problems in weighted Sobolev spaces and variable exponent

Abstract In this paper we focus on a certain class of anisotropic obstacle problems governed by a Leray–Lions operator. This problem is subject to homogeneous Neumann boundary conditions. By applying truncation techniques and the monotonicity method, we establish the existence of entropy solutions for the problem studied in the framework of anisotropic weighted Sobolev spaces with variable exponent.

Strongly quasilinear elliptic systems with perturbations through yong measures

We use the Galerkin method to obtain weak solutions for strongly quasilinear elliptic systems in the Sobolev space W1,p0(Ω; ℝm) of the form Gϖ(x) = f(x,ϖ,Dϖ) in Ω, with ϖ(x) = 0 on ∂Ω, where the term Gϖ(x) includes perturbations. Our approach involves the use of Young measures to obtain the desired results.

Mathematical analysis of a norm-conservative numerical scheme for the Ostrovsky equation

AbstractThe target of this study is a norm-conservative scheme for the Ostrovsky equation, as its mathematical analysis has not been addressed. First, the existence and uniqueness of its numerical solutions are demonstrated. Subsequently, a convergence estimate in the two-norm is established. This, in turn, implies a convergence in the first-order Sobolev space using a supplementary sup-norm boundedness argument. Finally, this conservative scheme can be implemented in a differential form, which is considerably better than the integral form in terms of computational cost-effectiveness.

On the modulus of continuity of fractional Orlicz-Sobolev functions

AbstractNecessary and sufficient conditions are presented for a fractional Orlicz-Sobolev space on $${\mathbb {R}}^n$$ R n to be continuously embedded into a space of uniformly continuous functions. The optimal modulus of continuity is exhibited whenever these conditions are fulfilled. These results pertain to the supercritical Sobolev regime and complement earlier sharp embeddings into rearrangement-invariant spaces concerning the subcritical setting. Classical embeddings for fractional Sobolev spaces into Hölder spaces are recovered as special instances. Proofs require novel strategies, since customary methods fail to produce optimal conclusions.

Inverse problem for wave equation of memory type with acoustic boundary conditions

In this article, for the direct problem, which consists of finding the velocity potential and the displacement of boundary points in the wave equation of memory type with initial and boundary acoustic conditions, the inverse problem of determining the memory kernel by the integral overdetermination condition is studied. By introducing a new function, the problem is reduced to a problem with homogeneous boundary conditions. Using the technique of estimating integral equations and the contraction mappings principle in Sobolev spaces, the existence and uniqueness theorem for the inverse problem is proved.

New weighted Trudinger-Moser inequality for functions not necessarily radially symmetric and applications

In this paper, we prove some new inequality of Trudinger-Moser' type defined in some weighted Sobolev space whose norm is a combination of the norms of the N-Laplacian and the p-Laplacian with 1<p<N in RN,N≥2. The presence and the behavior of the weights prevent us to use of the symmetrization arguments. This inequality holds for functions which are not required to be radially symmetric. The strategy used in order to establish our inequality mainly consists of a proper change of variables which permits to return to the non-weighted case. Various improvements of this inequality are also proved. Finally, one of these improvements is used to study some elliptic equation involving a weighted (N,p)-Laplacian operator.

Boundary Domain Integral Equations for Variable Coefficient Mixed BVP in 2D Unbounded Domain

In this paper, the direct segregated Boundary Domain Integral Equations (BDIEs) for the Mixed Boundary Value Problems (MBVPs) for a scalar second order elliptic Partial Differential Equation (PDE) with variable coefficient in unbounded (exterior) 2D domain is considered. Otar Chkadua, Sergey Mikhailov and David Natroshvili formulated both the interior and exterior 3D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order elliptic PDE with a variable coefficients. On the other hand Sergey Mikhailov and Tamirat Temesgen formulated only the interior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. However, in this paper we formulated the exterior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. The aim of this work is to reduce the MBVPs to some direct segregated BDIEs with the use of an appropriate parametrix (Levi function). We examine the characteristics of corresponding parametrix-based integral volume and layer potentials in some weighted Sobolev spaces, as well as the unique solvability of BDIEs and their equivalence to the original MBVPs. This analysis is based on the corresponding properties of the MBVPs in weighted Sobolev spaces that are proved as well.

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.

A compact embedding result for nonlocal Sobolev spaces and multiplicity of sign-changing solutions for nonlocal Schrödinger equations

A compact embedding result for nonlocal Sobolev spaces and multiplicity of sign-changing solutions for nonlocal Schrödinger equations

Asymptotic behavior of the two‐phase flow around the planar Couette flow in three‐dimensional space

This paper is concerned with the nonlinear stability of planar Couette flow for the three‐dimensional NS‐NS equations, which are used to model the motion of two‐phase flow. Our result shows that the planar Couette flow is asymptotically stable for the initial perturbations sufficiently small in some Sobolev space if the background velocity and the Reynolds number are sufficiently small. Moreover, it is proved that the solution converges to the stationary state at an algebraic time decay rate, and we also give the decay rate of the relative velocity.

Korevaar–Schoen–Sobolev spaces and critical exponents in metric measure spaces

We survey, unify and present new developments in the theory of Korevaar–Schoen–Sobolev spaces on metric measure spaces. While this theory coincides with those of Cheeger and Shanmugalingam if the space is doubling and supports a Poincaré inequality, it offers new perspectives in the context of fractals for which the approach by weak upper gradients is inadequate.

A strong maximum principle for quasilinear elliptic differential equations in a variable exponent Sobolev space

A strong maximum principle is an important property for elliptic and parabolic equations. In this paper, we consider a strong maximum principle for a class of quasilinear elliptic equations containing p(⋅)-Laplacian and mean curvature operator.

Analysis and approximation of elliptic problems with Uhlenbeck structure in convex polytopes

We prove the well posedness in weighted Sobolev spaces of certain linear and nonlinear elliptic boundary value problems posed on convex domains and under singular forcing. It is assumed that the weights belong to the Muckenhoupt class Ap with p∈(1,∞). We also propose and analyze a convergent finite element discretization for the nonlinear elliptic boundary value problems mentioned above. As an instrumental result, we prove that the discretization of certain linear problems are well posed in weighted spaces.

Non-constant functions with zero nonlocal gradient and their role in nonlocal Neumann-type problems

This work revolves around properties and applications of functions whose nonlocal gradient, or more precisely, finite-horizon fractional gradient, vanishes. Surprisingly, in contrast to the classical local theory, we show that this class forms an infinite-dimensional vector space. Our main result characterizes the functions with zero nonlocal gradient in terms of two simple features, namely, their values in a layer around the boundary and their average. The proof exploits recent progress in the solution theory of boundary-value problems with pseudo-differential operators. We complement these findings with a discussion of the regularity properties of such functions and give illustrative examples. Regarding applications, we provide several useful technical tools for working with nonlocal Sobolev spaces when the common complementary-value conditions are dropped. Among these, are new nonlocal Poincaré inequalities and compactness statements, which are obtained after factoring out functions with vanishing nonlocal gradient. Following a variational approach, we exploit the previous findings to study a class of nonlocal partial differential equations subject to natural boundary conditions, in particular, nonlocal Neumann-type problems. Our analysis includes a proof of well-posedness and a rigorous link with their classical local counterparts via Γ-convergence as the fractional parameter tends to 1.