Isoperimetric inequalities in nonlocal diffusion problems with integrable kernel

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameterδcharacterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part asδ → 0, the proposed Neumann-type boundary formulation recovers the local case asO(δ2) in theL∞(Ω) norm, which is optimal considering theO(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges withO(δ2) convergence.

ABSTRACTIn this study, we explore volume constrained nonlocal problem with integrable kernels. A central challenge addressed is the discontinuity of true solutions across the boundary, posing difficulties for designing efficient numerical methods. To address this issue, we introduce an auxiliary function which serves as the solution of an auxiliary problem. This approach effectively transforms the original problem to a new nonlocal problem with a true solution which is continuous across the boundary. We propose two numerical schemes to solve the new nonlocal problem: one is exclusively applicable to one‐dimensional case, while the other is suitable for general dimensional cases. Experimental results show that both schemes exhibit optimal convergence order under mild condition.

GENERAL MEASURE THEORY Measures and Measurable Functions Lusin's and Egoroff's Theorems Integrals and Limit Theorems Product Measures, Fubini's Theorem, Lebesgue Measure Covering Theorems Differentiation of Radon Measures Lebesgue Points Approximate continuity Riesz Representation Theorem Weak Convergence and Compactness for Radon Measures HAUSDORFF MEASURE Definitions and Elementary Properties Hausdorff Dimension Isodiametric Inequality Densities Hausdorff Measure and Elementary Properties of Functions AREA AND COAREA FORMULAS Lipschitz Functions, Rademacher's Theorem Linear Maps and Jacobians The Area Formula The Coarea Formula SOBOLEV FUNCTIONS. Definitions And Elementary Properties. Approximation Traces. Extensions. Sobolev Inequalities Compactness. Capacity Quasicontinuity Precise Representations of Sobolev Functions. Differentiability on Lines BV FUNCTIONS AND SETS OF FINITE PERIMETER Definitions and Structure Theorem Approximation and Compactness Traces. Extensions. Coarea Formula for BV Functions. Isoperimetric Inequalities. The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem. Pointwise Properties of BV Functions Essential Variation on Lines A Criterion for Finite Perimeter. DIFFERENTIABILITY AND APPROXIMATION BY C1 FUNCTIONS. Lp Differentiability a.e. Approximate Differentiability Differentiability A.E. for W1,P (P > N). Convex Functions Second Derivatives a.e. for convex functions Whitney's Extension Theorem Approximation by C1 Functions NOTATION REFERENCES

This paper studies two nonlocal diffusion problems with a free boundary and a fixed boundary in heterogeneous environment. The main goal is to understand how the evolution of the two species is affected by the heterogeneous environment. We first prove the existence and uniqueness of a global solution for such systems. Then, for models with Lotka–Volterra type competition or predator–prey growth terms, we establish the spreading‐vanishing dichotomy. Sharp criteria of spreading and vanishing are also obtained. Furthermore, we show that accelerated spreading occurs if and only if the kernel function violates the threshold condition.

In this paper, we consider a nonlocal problem. It is used in the rectangular area. For this problem, a difference problem is constructed, based on a rectangular grid. The error of theapproximatesolutionsfor a nonlocal problemis estimated. Many applied problems, in particular heat conductivity [1], [2], fluid mechanics [3], the theory of elasticity and shells [4], etc. are reduced to solving nonlocal boundary value problems for elliptical type equations. When solving nonlocal boundaryvalueproblems, nonlocal boundaryconditions complicate the justification of finite difference schemes due to the complexity of matrices obtained from systemsof equation.This complexity is especially evident for numerical methods, fornon-linear equations[5]. Here we consider a nonlocal boundary value problem for a quasi-linear equation. To solve the problem numerically we use the finite difference method. The error estimate of the approximate solution of the nonlocal problem is found.

A nonlocal convection diffusion problem

The aim of the paper is to prove theorems about the existence and uniqueness of mild and classical solutions of a nonlocal semilinear functional-differential evolution Cauchy problem. The method of semigroups, the Banach fixed-point theorem and theorems (see [2]) about the existence and uniqueness of the classical solutions of the first-order differential evolution problems in a not necessarily reflexive Banach space are used to prove the existence and uniqueness of the solutions of the problems considered. The results obtained are based on publications [1–6].

In the paper, a reduced basis (RB) method for time-dependent nonlocal problems with a special parameterized fractional Laplace kernel function is proposed. Because of the lack of sparsity of discretized nonlocal systems compared to corresponding local partial differential equation (PDE) systems, model reduction for nonlocal systems becomes more critical. The method of snapshots and greedy (MOS-greedy) algorithm of RB method is developed for nonlocal problems with random inputs, which provides an efficient and reliable approximation of the solution. A major challenge lies in the excessive influence of the time domain on the model reduction process. To address this, the Fourier transform is applied to convert the original time-dependent parabolic equation into a frequency-dependent elliptic equation, where variable frequencies are independent. This enables parallel computation for approximating the solution in the frequency domain. Finally, the proposed MOS-greedy algorithm is applied to the nonlocal diffusion problems. Numerical results demonstrate that it provides an accurate approximation of the full order problems and significantly improves computational efficiency.

In the paper we investigate asymptotics of solutions for nonlocal elliptic problems in plane angles and in ℝ2 \ {0}. These problems arise in studying nonlocal problems in bounded domains in the case where support of nonlocal terms intersects with a boundary of a domain. We obtain explicit formulas for the asymptotic coefficients in terms of eigenvectors and associated vectors of both adjoint nonlocal operators acting in spaces of distributions and formally adjoint (with respect to the Green formula) nonlocal transmission problems.

The author consider initial-boundary value problem for a system of differential equations in partial derivatives of third order. We study questions of existence and uniqueness of classical solution of the problem, and ways for its evaluation. The introduction of a new desired function enables us to reduce the problem to equivalent nonlocal problem with integral condition for a system of integro-differential euqtions of hyperbolic type with a functional relation. We establish conditions for unique solvability of the nonlocal problem by means of the method of functional parameters, propose algorithms for solving of the equivalent problem and prove their convergence. As a result, there are obtained conditions for existence of unique classical solution of initial-boundary value problem for the considered system of differential equations of third order in terms of initial data.

We prove an energy inequality for nonlocal diffusion operators of the following type, and some of its generalisations:\begin{document}\begin{equation*} Lu (x) := \int_{\mathbb{R}^N} K(x, y) (u(y) -u(x)) \,\mathrm{d} y, \end{equation*}\end{document}where $L$ acts on a real function $u$ defined on $\mathbb{R}^N$, and we assume that $K(x, y)$ is uniformly strictly positive in a neighbourhood of $x=y$. The inequality is a nonlocal analogue of the Nash inequality, and plays a similar role in the study of the asymptotic decay of solutions to the nonlocal diffusion equation $\partial_t u = L u$ as the Nash inequality does for the heat equation. The inequality allows us to give a precise decay rate of the $L^p$ norms of $u$ and its derivatives. As compared to existing decay results in the literature, our proof is perhaps simpler and gives new results in some cases.

Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection

We consider the problem to reconstruct the mass distribution of a string where the mass is concentrated in a finite number of points, or, equivalently, the problem to reconstruct a simply connected mass spring system with unknown masses and stiffness parameters if the following data are given. Problem 1: The spectra of the string and of a modification of the string, or. Problem 2: The spectra of two different modifications of the string. Here a modification of the string is a string which appears if we link the unknown string with another string of known mass distribution. The paper contains a necessary condition for the existence of a solution of Problem 1, and explicit formulas and an algorithm for the solutions of the Problems 1 and 2 under the condition that there exists a solution. For the case that the mass distribution of the unknown string is not discrete we consider the problem to find discrete approximations of this distribution from the respective spectral data. The methods are based on the spectral theory of generalized second order differential operators as developed by M. G. Krein

A pressure vessel made of anisotropic material is held together against the gas pressure tanks to tensile forces within the walls. The optimization problem consists in searching the shape and the thickness distribution of membrane shells, made of anisotropic material. The closed form solution of the optimization problem is obtained and corresponding isoperimetric inequality is proved for anisotropic Hill yield criteria.

From Grushin to Heisenberg via an isoperimetric problem