Sobolev Type Research Articles

Fractional Sobolev type spaces of functions of two variables via Riemann-Liouville derivatives

AbstractWe introduce and study the spaces of fractionally absolutely continuous functions of two variables of any order and the fractional Sobolev type spaces of functions of two variables. Our approach is based on the Riemann-Liouville fractional integrals and derivatives. We investigate relations between these spaces as well as between the Riemann-Liouville and weak derivatives.

A note on a NSFD discretization of a Sobolev type PDE

Sobolev type PDEs are used to model a broad range of physical phenomena. However, their discretizations by means of finite differences have generally been implicit schemes. We demonstrate that explicit, two level discrete-time representations can exist within the context of the NSFD methodology. One consequence of our explicit construction is that the step-sizes are expressible in terms of the parameters appearing in the original PDE.

On fractional inequalities on metric measure spaces with polar decomposition

Abstract In this paper, we prove the fractional Hardy inequality on polarisable metric measure spaces. The integral Hardy inequality for 1 < p ≤ q < ∞ 1<p\leq q<\infty is playing a key role in the proof. Moreover, we also prove the fractional Hardy–Sobolev type inequality on metric measure spaces. In addition, logarithmic Hardy–Sobolev and fractional Nash type inequalities on metric measure spaces are presented. In addition, we present applications on homogeneous groups and on the Heisenberg group.

Existence of a mild solution for a fractional impulsive differential equation of the Sobolev type including deviating argument

The impulsive fractional differential equation of the Sobolev type, including deviating arguments, is the subject of the study. The analytic semigroup and fixed point approaches serve the purpose of determining the existence of the approximations. The fractional power of a closed linear operator concept is used to show how the approximation converges. To arrive at a unique approach, an approximation strategy is used. Our main conclusions are defined using an example.

Weak monotonicity property of Korevaar-Schoen norms on nested fractals

In this paper, we study the weak monotonicity property of Korevaar-Schoen (p-energy) norms on nested fractals for 1<p<∞. Such property has many important applications on fractals and other metric measure spaces, such as constructing p-energies (when p=2 this is basically a Dirichlet form), generalizing the classical Sobolev type inequalities and the celebrated Bourgain-Brezis-Mironescu convergence.

On a Sobolev Type Theorem for the Generalized Riesz Potential Generated by the Generalized Shift Operator on Morrey Space

On a Sobolev Type Theorem for the Generalized Riesz Potential Generated by the Generalized Shift Operator on Morrey Space

Mixed Anisotropic and Nonlocal Sobolev Type Inequalities with Extremal

Mixed Anisotropic and Nonlocal Sobolev Type Inequalities with Extremal

Analysis on nonlinear differential equation with a deviating argument via Faedo–Galerkin method

This article focuses on the impulsive fractional differential equation (FDE) of Sobolev type with a nonlocal condition. Existence and uniqueness of the approximations are determined via analytic semigroup and fixed point method. Convergence’s approximation is demonstrated by the idea of fractional power of a closed linear operator. Using an approximation procedure, a novel approach is reached. An illustration is used to clarify our key findings.

Embeddings of a Multi-Weighted Anisotropic Sobolev Type Space

Parameters such as various integral and differential characteristics of functions, smoothness properties of regions and their boundaries, as well as many classes of weight functions cause complex relationships and embedding conditions for multi-weighted anisotropic Sobolev type spaces. The desire not to restrict these parameters leads to the development of new approaches based on the introduction of alternative definitions of spaces and norms in them or on special localization methods. This article examines the embeddings of multi-weighted anisotropic Sobolev type spaces with anisotropy in all the defining characteristics of the norm of space, including differential indices, summability indices, as well as weight coefficients. The applied localization method made it possible to obtain an embedding for the case of an arbitrary domain and weights of a general type, which is important in applications in differential operators’ theory, numerical analysis.

Nonlinear Elliptic Equations on the Sierpiński Carpet

This paper deals with a semilinear elliptic equation for a properly defined Laplace operator on the Sierpiński carpet. We investigate the existence of solutions, under Dirichlet boundary conditions, for the problem∆u+a(x)u=f(x,u).This work is motivated by previos studies established for such problems on the Sierpiński gasket by many authors: Kenneth J. Falconer, Jiaxin Hu, G.Bonanno and others (more details can be found in Introduction). The approach used in this study is based on variational methods, more specifically, the the mountain pass theorem of Ambrosetti and Rabinowit, and Sobolev type inequality, which was of principal use for achieving other results.

Study of Uniqueness and Ulam-Type Stability of Abstract Hadamard Fractional Differential Equations of Sobolev Type via Resolvent Operators

This paper focuses on studying the uniqueness of the mild solution for an abstract fractional differential equation. We use Banach’s fixed point theorem to prove this uniqueness. Additionally, we examine the stability properties of the equation using Ulam’s stability. To analyze these properties, we consider the involvement of Hadamard fractional derivatives. Throughout this study, we put significant emphasis on the role and properties of resolvent operators. Furthermore, we investigate Ulam-type stability by providing examples of partial fractional differential equations that incorporate Hadamard derivatives.

Weighted Sobolev Type Inequalities in a Smooth Metric Measure Space

In this paper, we obtain weighted Sobolev type inequalities with explicit constants that extend the inequalities obtained by Guo et al. (Math Res Lett 28(5):1419–1439, 2021) in the Riemannian setting. As an application, we prove some new logarithmic Sobolev type inequalities in some smooth metric measure spaces.

Studies over the existence of a certain impulse-based fuzzy integrodifferential equations of the Sobolev type

The present investigation employs impulses and a non-local constraint to prove the existence are some various types of abstract differential and integrodifferential equations related to the Sobolev type. Semigroup theory, specifically variants of constant formula, is utilized to get the analytical results for those equations. Furthermore, findings using the Banach fixed point approach were examined using fuzzy numbers with values spanning the En\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {E}_n$$\\end{document} range, which includes the normal, convex, upper semi-continuous, and compactly supported interval. A description is given for each situation to illustrate the principle.

FUNCTION SPACES AND APPROXIMATION THEORY DIFFERENTIAL PROPERTIES OF FUNCTIONS FROM SPACES

Abstract. In this abstract we establish an embedding theorem in Sobolev type spaces of functions of several group variables determined in the domain G  En and satysfying "

Existence and controllability results of semilinear Sobolev type difference equation

Existence and controllability results of semilinear Sobolev type difference equation

ASYMPTOTIC BEHAVIOUR OF THE JACOBI SOBOLEV-TYPE ORTHOGONAL POLYNOMIALS. A NON DIAGONAL CASE

Asymptotic behaviour of the Jacobi Sobolev-type orthogonal polynomials. A non-diagonal case. Rev. Acad. Colomb. Cienc. 34 (133): 529-539, 2010. ISSN 0370-3908. MATEMÁTICAS 1 Universidad Nacional de Colombia, Bogotá, Colombia. Correo electrónico: haduenasr@unal.edu.co 2 Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo No. 340, Colima, Colima, México. Correo electrónico: garzaleg@gmail.com AMS Classification 2000: 33C47. Consider the following Sobolev type inner product (p, q) = ∫ p'(x)q(x)(1 - x)θ(1 + x)θdx + i[p'(1)q(1) - p(1)q'(1)], (1) where p and q are polynomials with real coefficients, 0 &lt; θ &gt; -1, ξ(x) = (p(x), q(x))', and A = [M0, M1; M1, M2] is a positive semidefinite matrix, with M0M1 ≥ 0, and A ∈ ℝ. The family of polynomials orthogonal with respect to (1), P_n^{(θ,θ)}, are called Jacobi Sobolev-type orthogonal polynomials. An expression that relates this family of polynomials with P_n^{(0,0)}, the usual Jacobi orthogonal polynomials, was obtained in [8]. Here, we obtain the outer relative asymptotic for P_n^{(0,0)}, as well as the corresponding Mehlér-Heine formula.

A Nyström Method for 2D Linear Fredholm Integral Equations on Curvilinear Domains

This paper is devoted to the numerical treatment of two-dimensional Fredholm integral equations, defined on general curvilinear domains of the plane. A Nyström method, based on a suitable Gauss-like cubature formula, recently proposed in the literature is proposed. The convergence, stability and good conditioning of the method are proved in suitable subspaces of continuous functions of Sobolev type. The cubature formula, on which the Nyström method is constructed, has an error that behaves like the best polynomial approximation of the integrand function. Consequently, it is also shown how the Nyström method inherits this property and, hence, the proposed numerical strategy is fast when the involved known functions are smooth. Some numerical examples illustrate the efficiency of the method, also in comparison with other methods known in the literature.

Trigonometric and hyperbolic Poincare, Sobolev and Hilbert-Pachpatte type inequalities

In this article based on trigonometric and hyperbolic type Taylor formulae we establish Poincare, Sobolev and Hilbert-Pachpatte type inequalities of different kinds specific and general.

BOUNDARY VALUE PROBLEMS WITH AN INTEGRO-DIFFERENTIAL NON-LOCAL CONDITION FOR COMPOSITE TYPE DIFFERENTIAL EQUATIONS OF THE FOURTH ORDER

The paper studies new nonlocal boundary value problems with an integro-differential boundary condition for unsteady differential equations of the Sobolev type of the fourth order. The peculiarity of the studied problems is that they contain derivatives both in spatial variables and derivatives in time variables in the boundary condition. For the problems under study, the existence and uniqueness theorems of regular solutions are proved – solutions having all derivatives generalized by S.L. Sobolev included in the corresponding equations.

On the completeness of a system of decreasing elementary solutions for one class of fourth-order operator-differential equations on the semi-axis

In this work, we study a homogeneous fourth-order elliptic operator-differential equation with an unbounded operator under non-homogeneous boundary conditions on the semi-axis. We obtain sufficient conditions for the completeness of decreasing elementary solutions of the equation under study in the space of all its solutions from a fourth-order Sobolev type space.