Sobolev Spaces Research Articles

The Rellich-Kondrachov Theorem for Gelfand Pairs Over Hypergroups

Embedding results play important rôles in mathematical analysis. This paper addresses some embedding theorems in the context of Sobolev spaces theory on Gelfand pairs over hypergroups. Mainly, the analogue of the Rellich-Kondrachov theorem is proved.

Entropy solutions for some nonlinear and noncoercive parabolic problems in the anisotropic Sobolev spaces

In this paper, we studied the existence of solutions for a class parabolic problem involving a non-coercive Leray-Lions operator with a Hardy potential. Using an approximate method and some a priori estimation techniques, we show the existence of entropy solutions for the studied problem within the framework of anisotropic Sobolev spaces.

A new approach to weighted Sobolev spaces

A new approach to weighted Sobolev spaces

Convergence analysis of deep Ritz method with over-parameterization.

The deep Ritz method (DRM) has recently been shown to be a simple and effective method for solving PDEs. However, the numerical analysis of DRM is still incomplete, especially why over-parameterized DRM works remains unknown. This paper presents the first convergence analysis of the over-parameterized DRM for second-order elliptic equations with Robin boundary conditions. We demonstrate that the convergence rate can be controlled by the weight norm, regardless of the number of parameters in the network. To this end, we establish novel approximation results in Sobolev spaces with norm constraints, which have independent significance.

The generalized fractional KdV equation in weighted Sobolev spaces

The generalized fractional KdV equation in weighted Sobolev spaces

The basis property of generalized eigenfunctions for one boundary value problem with discontinuities at two interior points

In this study, we consider a spectral problem for one boundary value problem with discontinuities at two interior points. The boundary conditions involve a spectral parameter. We consider some compact, positive, self-adjoint operators to reduce the spectral problem to an operator-pencil equation. Then, it was proven that this operator-pencil is positive definite, the spectrum is discrete, and the system of weak eigenfunctions forms a Riesz basis of the appropriate Sobolev space.

Stability on 3D Boussinesq system with mixed partial dissipation

Abstract In the article, we are concerned with the three-dimensional anisotropic Boussinesq equations with the velocity dissipation in x 2 {x}_{2} and x 3 {x}_{3} directions and the thermal diffusion in only x 3 {x}_{3} direction. When the spatial domain is the whole space R 3 {{\mathbb{R}}}^{3} , the global well-posedness and stability problem for the partially dissipated Boussinesq system remain the extremely challenging open problems. Attention here focuses on the periodic domain Ω = R × T 2 \Omega ={\mathbb{R}}\times {{\mathbb{T}}}^{2} . We aim at establishing the stability for the problem of perturbations near hydrostatic equilibrium and the large-time behavior of the perturbed solution. We first obtain the global existence of some symmetric fluids in H 2 ( Ω ) {H}^{2}\left(\Omega ) for small initial data. Then the exponential decay rates for the oscillations u ˜ \widetilde{u} and θ \theta in H 1 ( Ω ) {H}^{1}\left(\Omega ) and the homogeneous Sobolev space H v 2 ˙ ( Ω ) \dot{{H}_{v}^{2}}\left(\Omega ) are also shown. The proof is based on a key observation that we can decompose the velocity u u into the average u ¯ \overline{u} on T 2 {{\mathbb{T}}}^{2} and the corresponding oscillation u ˜ \widetilde{u} . This enables us to establish the strong Poincaré-type inequalities on u ˜ \widetilde{u} , u 3 , θ {u}_{3},\theta and some anisotropic inequalities, which ensure the establishment of the closed priori estimates. In addition, we also prove the oscillations in one direction u ˜ ( 2 ) , u ˜ ( 3 ) {\widetilde{u}}^{\left(2)},{\widetilde{u}}^{\left(3)} in H 1 ( Ω ) {H}^{1}\left(\Omega ) decay to zero exponentially.

Nonlinear Partial Differential Equations in Fractional Sobolev Spaces: Existence, Regularity, and Stability

This paper explores the theory and application of fractional Sobolev spaces Ws,p(Ω) in the analysis of nonlinear partial differential equations (PDEs). Specifically, we examine the existence, uniqueness, regularity, and blow-up phenomena of solutions to fractional PDEs, such as the fractional Laplace equation. The paper presents several original theorems related to the embedding properties of fractional Sobolev spaces, energy minimization problems, and the maximum principle for fractional operators. We also investigate the stability of solutions under perturbations and establish the fractional Poincare inequality and trace theorem. Our results contribute to the understanding of the interplay between fractional Sobolev norms and nonlinear variational problems, offering insights into energy minimization, regularity results, and nonlocal effects in PDEs. These findings have significant implications for the study of nonlocal problems, fractional variational calculus, and fractional diffusion equations in applied mathematics and physics.

Multidimensional Quasi-Monte Carlo Integration in Weighted Anchored Sobolev Spaces

In this article, an exact formula for the mean square worst-case error of the integration in the weighted anchored Sobolev spaces $$H_{\mathrm{Sob},s,\gamma,\mathbf{w}}$$ presented in terms of the functions of the system $$\Gamma_{\mathcal{B}_s}$$ is delivered. The notion of the so-called weighted anchored diaphony is introduced and is shown that it is a quantitative measure for the irregularity of the distribution of sequences. The relationship that exists between the mean square worst-case error and this type of the diaphony is established.

Well-posedness results for a family of dispersive–dissipative Benjamin–Ono equations

This paper is devoted to studying the Cauchy problem for a family of dispersive–dissipative BO equations u t + H u xx − ( D x α − D x β ) u + u u x = 0 . For a wide class of parameters β > 1 and 0 < α < β , taking into account dispersive and dissipative effects, we establish sharp well-posedness results in Sobolev spaces H s ( R ) and H s ( T ) which yield new well-posedness conclusions for some physical relevant models.

Differential systems in Sobolev spaces with generic inhomogeneous boundary conditions

The paper contains a review of results on linear systems of ordinary differential equations of an arbitrary order on a finite interval with the most general inhomogeneous boundary conditions in Sobolev spaces. The character of the solvability of such problems is investigated, their Fredholm properties are established, and their indexes and the dimensions of their kernels and co-kernels are found. In addition, necessary and sufficient conditions of continuity in the parameter of the solutions of the introduced classes of boundary-value problems in Sobolev spaces of an arbitrary order are obtained.

On Weighted Fractional Calculus With Respect to Functions

ABSTRACTThis paper aims to further explore the existing theory of weighted fractional operators with respect to functions. This theory extends some fundamental results of classical Riemann–Liouville and Caputo fractional derivatives to their weighted counterparts involving fractional differentiation and integration with respect to functions. By investigating the fundamental principles of these operators, we establish mean value theorems, Taylor's theorems, and integration by parts formulae. The Leibniz rule is extended for weighted Riemann–Liouville derivatives with respect to functions. Also, we present necessary conditions for the existence and uniqueness of solutions for a class of initial value problems, involving weighted Caputo fractional derivatives with respect to functions, in a Sobolev space.

CONDITIONS FOR SOLVABILITY OF THE CAUCHY PROBLEM FOR ONE PSEUDOHYPERBOLIC SYSTEM

The Cauchy problem for one system, not resolved with respect to the highest derivative with respect to time, is considered. The system under study belongs to the class of pseudohyperbolic systems. The system describes transverse bending-torsional vibrations of an elastic rod. Conditions for the solvability of the Cauchy problem in Sobolev spaces and estimates of the solution are obtained.

A higher-order quadratic NLS equation on the half-line

The well-posedness of the initial-boundary value problem for higher-order quadratic nonlinear Schrödinger equations on the half-line is studied by utilizing the Fokas solution formula for the corresponding linear problem. Using this formula, linear estimates are derived in Bourgain spaces for initial data in spatial Sobolev spaces on the half-line and boundary data in temporal Sobolev spaces suggested by the time regularity of the linear initial value problem. Then, the needed bilinear estimates are derived and used for showing that the iteration map defined via the Fokas solution formula is a contraction in appropriate solution spaces. Finally, well-posedness is established for optimal Sobolev exponents in a way analogous to the case of the initial value problem on the whole line with solutions in classical Bourgain spaces.

SOLVABILITY OF NONLINEAR BOUNDARY VALUE PROBLEMS FOR DIFFERENTIAL EQUILIBRIUM EQUATIONS OF NON-FLAT TIMOSHENKO TYPE SHELLS OF NON-ZERO GAUSSIAN CURVATURE IN ISOMETRIC COORDINATES

We study the solvability of a boundary value problem for a system of five nonlinear second-order partial differential equations under given nonlinear boundary conditions, which describes the equilibrium state of elastic non-flat inhomogeneous isotropic shells with loose edges in the framework of the Timoshenko shear model, referred to isometric coordinates.The boundary value problem is reduced to a nonlinear operator equation with respect to generalized displacements in Sobolev space, the solvability of which is established using the principle of compressed mappings.

INITIAL BOUNDARY VALUE PROBLEM FOR THE NONLINEAR MODIFIED BOUSSINESQ EQUATION

The problem in Sobolev spaces is investigated for a modified Boussinesq equation with a homogeneous Neumann boundary condition and with classical initial conditions. Based on the compactness method, it is shown that the approximate analytical solution, constructed in the form of Galerkin’s sum over the system of eigenfunctions of the Neumann boundary value problem, *-weakly converges to the exact solution.

EXISTENCE OF A RENORMALIZED SOLUTION OF A QUASI-LINEAR ELLIPTIC EQUATION WITHOUT THE SIGN CONDITION ON THE LOWEST TERM

The paper considers a second-order quasilinear elliptic equation with an integrable right-hand side. Restrictions on the structure of the equation are formulated in terms of the generalized 𝑁-function. Unlike the author’s previous works, there is no sign condition for the low-order term of the equation. In non-reflexive Musielak–Orlicz–Sobolev spaces in an arbitrary unbounded strictly Lipschitz domain, the existence of a renormalized solution to the Dirichlet problem of this equation is proven.

SPHERICAL SPLINE SOLUTIONS OF THE INHOMOGENEOUS BIHARMONIC EQUATION

An inhomogeneous biharmonic equation is considered on the unit sphere in three-dimensional space. The solution of this equation, belonging to the Sobolev space on the sphere, is approximated by a sequence of solutions of the same equation but with specific right-hand sides, represented as linear combinations of shifts of the Dirac delta function. It is proven that, given specified nodes on the sphere determining the shifts, special solutions of the equation — spherical biharmonic splines — exist, and the weights corresponding to each are solutions of an associated non-degenerate system of linear algebraic equations. The connection between the approximation quality of the differential problem solution by spherical biharmonic splines and the problem of the convergence rate of optimal weighted spherical cubature formulas is established.

Representation theorems for normed modules

In this paper we study the structure theory of normed modules, which have been introduced by Gigli. The aim is twofold: to extend von Neumann’s theory of liftings to the framework of normed modules, thus providing a notion of precise representative of their elements; to prove that each separable normed module can be represented as the space of sections of a measurable Banach bundle. By combining our representation result with Gigli’s differential structure, we eventually show that every metric measure space (whose Sobolev space is separable) is associated with a cotangent bundle in a canonical way.

Nonlocal problems with Neumann and Robin boundary condition in fractional Musielak–Sobolev spaces

Nonlocal problems with Neumann and Robin boundary condition in fractional Musielak–Sobolev spaces