Variable Spaces Research Articles

Weighted Rellich and Hardy inequalities in L p(·) spaces

Abstract In this note we prove weighted Rellich–Sobolev and Hardy–Sobolev inequalities in variable exponent Lebesgue spaces L p ⁢ ( ⋅ ) ⁢ ( 𝔾 ) {L^{p(\,\cdot\,)}(\mathbb{G})} defined on stratified homogeneous groups 𝔾 {\mathbb{G}} . To derive the main results, we rely on weighted estimates for the Riesz potential operators in L p ⁢ ( ⋅ ) ⁢ ( 𝔾 ) {L^{p(\,\cdot\,)}(\mathbb{G})} , where 𝔾 {{\mathbb{G}}} is a general homogeneous group. The results are new even for the Abelian (Euclidean) case 𝔾 = ( ℝ d , + ) {\mathbb{G}=(\mathbb{R}^{d},+)} and the Heisenberg groups 𝔾 = ℍ n {\mathbb{G}={\mathbb{H}}^{n}} . The main statements are obtained for variable exponents satisfying the condition that the Hardy–Littlewood maximal operator is bounded in appropriate variable exponent Lebesgue spaces. We also give some quantitative estimates for the norms of integral operators involved in derived estimates.

(α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces

Abstract In this study, we generalize fuzzy metric-like, non-Archimedean fuzzy metric-like, and all the variants of fuzzy metric spaces. We propose the idea of fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike, respectively. We also propose the idea of ( α , F ) \left(\alpha ,F) -Geraghty-type generalized F F -contraction mappings utilizing fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike spaces. We investigate the presence of unique fixed points using the recently introduced contraction mappings. In order to complement our study, we consider an application to dynamic market equilibrium.

P(x)-biharmonic equations involving (h,r(x))-Hardy singular coefficients with no-flux boundary conditions

In this article, we investigate $p(x)$-biharmonic equations involving two kinds of different Hardy potentials, in which the $r(x)$-Hardy potentials are seldom mentioned in previous papers. New criteria for the existence of generalized solutions are reestablished when the nonlinear terms satisfying appropriate assumptions. The results are based on variational methods and the theory of variable exponent Sobolev spaces.

Infinitely of solutions for fractional κ(ξ)$$ \kappa \left(\xi \right) $$‐Kirchhoff equation in Hκ(ξ)ϖ,ν;μ(Λ)$$ {\mathcal{H}}_{\kappa \left(\xi \right)}^{\varpi, \nu; \mu}\left(\Lambda \right) $$

This work aims to develop the variational framework for some Kirchhoff problems involving the ‐Hilfer operator. Precisely, we use the symmetric mountain pass theorem to prove the existence of unfairly of nontrivial solutions. Further, we research the results from the theory of variable exponent Sobolev spaces and from the theory of ‐fractional space .

Existence and multiplicity of solutions for a p(x)-Choquard-Kirchhoff problem involving critical growth and concave-convex nonlinearities

This paper is devoted to studying a kind of p(x)-Choquard-Kirchhoff problems involving critical growth and concave-convex nonlinearities. Combining the concentration-compactness principle for weighted variable exponent spaces, the calculus of variations, genus theory and the Hardy-Littlewood-Sobolev type inequality, we obtain the existence of nontrivial solutions for this kind of p(x)-Choquard-Kirchhoff problems. Our results improve the related ones in the literature.

Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (lq(·)(Lp(·))). Moreover, it was developed using classical Lebesgue space (Lp) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not.

Triple weak solution for p(x)-Laplacian like problem involving no flux boundary condition

Abstract In this paper, we consider a class of p ⁢ ( x ) {p(x)} -Laplacian like problems with indefinite weight involving no flux boundary condition. Using variational techniques and the critical point theorem of Bonanno and Marano [4], we prove the existence of at least three weak solutions to the problem in Sobolev variable exponent spaces.

Experimental and numerical investigations of the mixing processes of pollutants at notched weirs in rivers

This study experimentally and numerically investigated the mixing processes at a notched weir to examine the relationship between the mass exchange coefficient and geo-hydraulic factors of notched weirs. Laboratory experiments on idealized notched weirs with variable notch spaces, notch heights, and hydraulic factors were performed to measure the flow field dynamics and depth-averaged concentration fields of the solute tracer. The experiment results demonstrated that flow separations occurred upstream of the notched weir. These flow structures controlled the horizontal storage zone and tracer mixing between the main flow and storage zone. The tracer cloud exhibited a longer retention time as the storage zone area increased. The normalized value of measured mass exchange coefficients by mean flow velocity, ke /Umean ranged from of 0.11 to 0.25, approximately one order larger than the values obtained in previous studies on lateral transient storage zones, such as lateral groin fields, and cavities, which implied that the notched weir in this study generated a storage zone with a fast exchange process and relatively weak storage effect. Numerical simulations with expanded geometric and hydraulic conditions revealed that the mass exchange coefficient was strongly correlated with the geometric factors, such as the notch space and height, and hydraulic factors, such as the Froude and Reynolds numbers. An empirical relationship applicable to rivers with notched weir structures under low-flow conditions (Fr < 0.1) was proposed based on the experiment and numerical results. The performance of the proposed empirical relationship showed a satisfactory level of R2 of 0.87. This relationship can be applied as a tool for estimating the mass exchange coefficient of notched weirs using easily measurable field data in rivers.

Efficient Bayesian inference using physics-informed invertible neural networks for inverse problems

This paper presents an innovative approach to tackle Bayesian inverse problems using physics-informed invertible neural networks (PI-INN). Serving as a neural operator model, PI-INN employs an invertible neural network (INN) to elucidate the relationship between the parameter field and the solution function in latent variable spaces. Specifically, the INN decomposes the latent variable of the parameter field into two distinct components: the expansion coefficients that represent the solution to the forward problem, and the noise that captures the inherent uncertainty associated with the inverse problem. Through precise estimation of the forward mapping and preservation of statistical independence between expansion coefficients and latent noise, PI-INN offers an accurate and efficient generative model for resolving Bayesian inverse problems, even in the absence of labeled data. For a given solution function, PI-INN can provide tractable and accurate estimates of the posterior distribution of the underlying parameter field. Moreover, capitalizing on the INN’s characteristics, we propose a novel independent loss function to effectively ensure the independence of the INN’s decomposition results. The efficacy and precision of the proposed PI-INN are demonstrated through a series of numerical experiments.

Continuous imbedding theorems in Musielak spaces

We provide continuous Sobolev-type imbedding results for Musielak spaces. In the first result the target space is a Musielak space bigger than the considered one, while the second is that of bounded and continuous functions. Continuous imbeddings into a particular Musielak space built upon Sobolev conjugate functions generate serious difficulties and does not holds in general for arbitrary Musielak functions, as it is well known in variable exponent Sobolev spaces. We overcome this problem by imposing only a regularity assumption on the Musielak functions without resorting to other restrictions such as the Δ2/∇2 conditions. Nevertheless, the continuous embedding in the space of bounded continuous functions is obtained by using the convex envelope, which then allows us to use a result already obtained in Orlicz spaces. The results we obtain extend and unify those obtained in classical Sobolev spaces, variable exponent Sobolev spaces as well as those obtained by Donaldson-Trudinger in the setting of Orlicz spaces.

Existence of solutions for singular elliptic equations with mixed boundary conditions

In this article, we investigate generalized solutions for elliptic equations involving double-phase ( p ( x ) , q ( x ) ) -Laplacian operators with Hardy potential in variable exponent spaces. ( p ( x ) , q ( x ) ) -Laplacian operators include p-Laplacian, q-Laplacian, p ( x ) -Laplacian and q ( x ) -Laplacian operators. Additionally, ( p ( x ) , q ( x ) ) -Laplacian elliptic equations with singular coefficients under mixed boundary conditions are seldom mentioned in previous work. Some new theorems of the existence on the generalized solutions are reestablished for such equations via variational methods when the nonlinearity satisfies suitable hypotheses in variable exponent Lebesgue spaces.

Generalized Hölder Inequality in Herz-Morrey Spaces with Variable Exponent

The paper investigates the conditions for the generalized Hölder’s inequality with a variable exponent in Herz-Morrey spaces. The main results are based on the exponent functions p(·) and α(·) . The proof of the first main result using the generalized Hölder’s inequality in Lebesgue spaces. The second main result of the paper is related to the weak space of the generalized Hölder’s inequality with a variable exponent in Herz-Morrey spaces. The theorems state the equivalence of certain conditions for the inequality. Mathematical proofs and analysis are providing to support the presented results for findings contribute to the understanding of Hölder’s inequalities in variable exponent spaces and their applications in Herz-Morrey spaces.

Fractional type Marcinkiewicz integral and its commutator on grand variable Herz-Morrey spaces

Fractional type Marcinkiewicz integral and its commutator on grand variable Herz-Morrey spaces

Extrapolation results on variable exponent grand Lebesgue space with Bp(·)$B_{p(\cdot)}$ weights

AbstractIn this paper, we study Rubio de Francia extrapolation theorems in the framework of the variable grand Lebesgue spaces with weights. As an application of the extrapolation theorems, we prove the boundedness of the Hardy averaging operator and the fractional Riemann Liouville transform for nonnegative and nonincreasing measurable functions. Some structural properties of the weighted grand Lebesgue spaces with variable exponent are also investigated.

On the Dirichlet problem for the elliptic equations with boundary values in variable Morrey-Lorentz spaces

Under a metric measure space setting, we show that a solution to the elliptic equation with a non-negative potential, defined on the upper half-space, is in the essentially-bounded-Morrey-Lorentz space with variable exponent if and only if it can be represented as the Poisson integral of a variable Morrey-Lorentz function, where the doubling property, the pointwise upper bound on the heat kernel, the mean value property and the Liouville property are assumed. This extends the known result of Stein-Weiss in 1971 from the Euclidean space to the metric measure space, and from the Lebesgue function to the Morrey-Lorentz function with variable exponent.

Variational Model with Nonstandard Growth Condition in Image Restoration and Contrast Enhancement

AbstractWe propose a new variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to the simultaneous contrast enhancement and denoising of color images. The characteristic feature of the proposed model is that we deal with a constrained non-convex minimization problem that lives in variable Sobolev-Orlicz spaces where the variable exponent is unknown a priori and it depends on a particular function that belongs to the domain of the objective functional. In contrast to the standard approach, we do not apply any spatial regularization to the image gradient. We discuss the consistency of the variational model, give the scheme for its regularization, derive the corresponding optimality system, and propose an iterative algorithm for practical implementations.

Fixed Point of α-Modular Nonexpanive Mappings in Modular Vector Spaces 𝓁p(·)

Let C denote a convex subset within the vector space 𝓁p(·), and let T represent a mapping from C onto itself. Assume α=(α1,⋯,αn) is a multi-index in [0,1]n such that ∑i=1nαi=1, where α1&gt;0 and αn&gt;0. We define Tα:C→C as Tα=∑i=1nαiTi, known as the mean average of the mapping T. While every fixed point of T remains fixed for Tα, the reverse is not always true. This paper examines necessary and sufficient conditions for the existence of fixed points for T, relating them to the existence of fixed points for Tα and the behavior of T-orbits of points in T’s domain. The primary approach involves a detailed analysis of recurrent sequences in R. Our focus then shifts to variable exponent modular vector spaces 𝓁p(·), where we explore the essential conditions that guarantee the existence of fixed points for these mappings. This investigation marks the first instance of such results in this framework.

A p(x)-Kirchhoff Type Problem Involving the p(x)-Laplacian-Like Operators With Dirichlet Boundary Condition

This paper deals with a class of p(x)-Kirchhoff type problems involving the p(x)-Laplacian-like operators, arising from the capillarity phenomena, depending on two real parameters with Dirichlet boundary conditions. Using a topological degree for a class of demicontinuous operators of generalized (S+), we prove the existence of weak solutions of this problem. Our results extend and generalize several corresponding results from the existing literature. Keywords: p(x)-Kirchhoff type problems, p(x)-Laplacian-like operators, weak solutions, variable exponent Sobolev spaces.

Estimating Position-Dependent and Anisotropic Diffusivity Tensors from Molecular Dynamics Trajectories: Existing Methods and Future Outlook.

Confinement can substantially alter the physicochemical properties of materials by breaking translational isotropy and rendering all physical properties position-dependent. Molecular dynamics (MD) simulations have proven instrumental in characterizing such spatial heterogeneities and probing the impact of confinement on materials' properties. For static properties, this is a straightforward task and can be achieved via simple spatial binning. Such an approach, however, cannot be readily applied to transport coefficients due to lack of natural extensions of autocorrelations used for their calculation in the bulk. The prime example of this challenge is diffusivity, which, in the bulk, can be readily estimated from the particles' mobility statistics, which satisfy the Fokker-Planck equation. Under confinement, however, such statistics will follow the Smoluchowski equation, which lacks a closed-form analytical solution. This brief review explores the rich history of estimating profiles of the diffusivity tensor from MD simulations and discusses various approximate methods and algorithms developed for this purpose. Besides discussing heuristic extensions of bulk methods, we overview more rigorous algorithms, including kernel-based methods, Bayesian approaches, and operator discretization techniques. Additionally, we outline methods based on applying biasing potentials or imposing constraints on tracer particles. Finally, we discuss approaches that estimate diffusivity from mean first passage time or committor probability profiles, a conceptual framework originally developed in the context of collective variable spaces describing rare events in computational chemistry and biology. In summary, this paper offers a concise survey of diverse approaches for estimating diffusivity from MD trajectories, highlighting challenges and opportunities in this area.

Non-Differentiable Loss Function Optimization and Interaction Effect Discovery in Insurance Pricing Using the Genetic Algorithm

Non-Differentiable Loss Function Optimization and Interaction Effect Discovery in Insurance Pricing Using the Genetic Algorithm