scholarly journals The existence of minimizers for an isoperimetric problem with Wasserstein penalty term in unbounded domains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Qinglan Xia ◽  
Bohan Zhou
Keyword(s):  
Unbounded Domain ◽  
Minimization Problem ◽  
Open Problem ◽  
Isoperimetric Problem ◽  
Unbounded Domains ◽  
Wasserstein Distance ◽  
Penalty Term ◽  
Singular Measures ◽  
Existence Of Minimizers

Abstract In this article, we consider the (double) minimization problem min ⁡ { P ⁢ ( E ; Ω ) + λ ⁢ W p ⁢ ( E , F ) : E ⊆ Ω , F ⊆ R d , | E ∩ F | = 0 , | E | = | F | = 1 } , \min\{P(E;\Omega)+\lambda W_{p}(E,F):E\subseteq\Omega,\,F\subseteq\mathbb{R}^{d},\,\lvert E\cap F\rvert=0,\,\lvert E\rvert=\lvert F\rvert=1\}, where λ ⩾ 0 \lambda\geqslant 0 , p ⩾ 1 p\geqslant 1 , Ω is a (possibly unbounded) domain in R d \mathbb{R}^{d} , P ⁢ ( E ; Ω ) P(E;\Omega) denotes the relative perimeter of 𝐸 in Ω and W p W_{p} denotes the 𝑝-Wasserstein distance. When Ω is unbounded and d ⩾ 3 d\geqslant 3 , it is an open problem proposed by Buttazzo, Carlier and Laborde in the paper On the Wasserstein distance between mutually singular measures. We prove the existence of minimizers to this problem when the dimension d ⩾ 1 d\geqslant 1 , 1 p + 2 d > 1 \frac{1}{p}+\frac{2}{d}>1 , Ω = R d \Omega=\mathbb{R}^{d} and 𝜆 is sufficiently small.

scholarly journals On the Wasserstein distance between mutually singular measures

2020 ◽  
Vol 13 (2) ◽  
pp. 141-154
Author(s):  
Giuseppe Buttazzo ◽  
Guillaume Carlier ◽  
Maxime Laborde
Keyword(s):  
Characteristic Function ◽  
Numerical Simulations ◽  
Minimization Problem ◽  
Optimal Solution ◽  
Wasserstein Distance ◽  
Singular Measures ◽  
Minimization Problems ◽  
Regularity Properties ◽  
Two Measures ◽  
Fixed Constant

AbstractWe study the Wasserstein distance between two measures {\mu,\nu} which are mutually singular. In particular, we are interested in minimization problems of the formW(\mu,\mathcal{A})=\inf\{W(\mu,\nu):\nu\in\mathcal{A}\},where μ is a given probability and {\mathcal{A}} is contained in the class {\mu^{\perp}} of probabilities that are singular with respect to μ. Several cases for {\mathcal{A}} are considered; in particular, when {\mathcal{A}} consists of {L^{1}} densities bounded by a constant, the optimal solution is given by the characteristic function of a domain. Some regularity properties of these optimal domains are also studied. Some numerical simulations are included, as well as the double minimization problem\min\{P(B)+kW(A,B):|A\cap B|=0,\,|A|=|B|=1\},where {k>0} is a fixed constant, {P(A)} is the perimeter of A, and both sets {A,B} may vary.

Monotonicity and symmetry of positive solutions to fractional p-Laplacian equation

2021 ◽  
pp. 2150005
Author(s):  
Wei Dai ◽  
Zhao Liu ◽  
Pengyan Wang
Keyword(s):  
Dirichlet Problem ◽  
Nonlinear Equations ◽  
Positive Solutions ◽  
Unbounded Domain ◽  
Fractional Laplacian ◽  
Unbounded Domains ◽  
Laplacian Operator ◽  
Maximum Principles ◽  
Laplacian Equation ◽  
Laplacian Equations

In this paper, we are concerned with the following Dirichlet problem for nonlinear equations involving the fractional [Formula: see text]-Laplacian: [Formula: see text] where [Formula: see text] is a bounded or an unbounded domain which is convex in [Formula: see text]-direction, and [Formula: see text] is the fractional [Formula: see text]-Laplacian operator defined by [Formula: see text] Under some mild assumptions on the nonlinearity [Formula: see text], we establish the monotonicity and symmetry of positive solutions to the nonlinear equations involving the fractional [Formula: see text]-Laplacian in both bounded and unbounded domains. Our results are extensions of Chen and Li [Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math. 335 (2018) 735–758] and Cheng et al. [The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math. 19(6) (2017) 1750018].

Least energy solutions for elliptic equations in unbounded domains

1996 ◽  
Vol 126 (1) ◽  
pp. 195-208 ◽  
Author(s):  
Manuel A. del Pino ◽  
Patricio L. Felmer
Keyword(s):  
Critical Point ◽  
Unbounded Domain ◽  
Elliptic Equations ◽  
Unbounded Domains ◽  
Asymptotic Results ◽  
Semilinear Elliptic ◽  
Superlinear Growth ◽  
Image Position ◽  
Least Energy Solutions ◽  
Least Energy

In this paper we study the existence of least energy solutions to subcritical semilinear elliptic equations of the formwhere Ω is an unbounded domain in RN and f is a C1 function, with appropriate superlinear growth. We state general conditions on the domain Ω so that the associated functional has a nontrivial critical point, thus yielding a solution to the equation. Asymptotic results for domains stretched in one direction are also provided.

scholarly journals Exponential decay for the damped wave equation in unbounded domains

2016 ◽  
Vol 18 (06) ◽  
pp. 1650012 ◽  
Author(s):  
Nicolas Burq ◽  
Romain Joly
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Exponential Decay ◽  
Unbounded Domain ◽  
Unbounded Domains ◽  
Geometric Control ◽  
Damped Wave Equation ◽  
Logarithmic Decay ◽  
And Control

We study the decay of the semigroup generated by the damped wave equation in an unbounded domain. We first prove under the natural geometric control condition the exponential decay of the semigroup. Then we prove under a weaker condition the logarithmic decay of the solutions (assuming that the initial data are smoother). As corollaries, we obtain several extensions of previous results of stabilization and control.

Rearrangements of functions on unbounded domains

1994 ◽  
Vol 124 (4) ◽  
pp. 621-644 ◽  
Author(s):  
R. J. Douglas
Keyword(s):  
Variational Problem ◽  
Unbounded Domain ◽  
Extreme Points ◽  
Unbounded Domains ◽  
Compact Set ◽  
Weakly Compact ◽  
Weak Closure ◽  
Weakly Compact Set ◽  
Steady Vortices

A characterisation is provided for the weak closure of the set of rearrangements of a function on an unbounded domain. The extreme points of this convex, weakly compact set are classified. This result is used to study the maximising sequences of a variational problem for steady vortices.

scholarly journals Use of Finite Point Method for Wave Propagation in Nonhomogeneous Unbounded Domains

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
S. Moazam ◽  
Z. Nalbantoglu ◽  
M. Celikag
Keyword(s):  
Wave Propagation ◽  
Numerical Methods ◽  
Unbounded Domain ◽  
Unbounded Domains ◽  
Point Method ◽  
Finite Point ◽  
New Approach ◽  
Boundary Points ◽  
Finite Point Method

Wave propagation in an unbounded domain surrounding the stimulation resource is one of the important issues for engineers. Past literature is mainly concentrated on the modelling and estimation of the wave propagation in partially layered, homogeneous, and unbounded domains with harmonic properties. In this study, a new approach based on the Finite Point Method (FPM) has been introduced to analyze and solve the problems of wave propagation in any nonhomogeneous unbounded domain. The proposed method has the ability to use the domain properties by coordinate as an input. Therefore, there is no restriction in the form of the domain properties, such as being periodical as in the case of existing similar numerical methods. The proposed method can model the boundary points between phases with trace of errors and the results of this method satisfy both conditions of decay and radiation.

scholarly journals Query-Specific Deep Embedding of Content-Rich Network

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Yue Li ◽  
Hongqi Wang ◽  
Liqun Yu ◽  
Sarah Yvonne Cooper ◽  
Jing-Yan Wang
Keyword(s):  
Minimization Problem ◽  
Upper Bound ◽  
Wasserstein Distance ◽  
Model Complexity ◽  
Innovation Networks ◽  
Convolutional Network ◽  
Gaussian Distributions ◽  
Deep Embedding ◽  
Query Node ◽  
Wasserstein Distances

In this paper, we propose to embed a content-rich network for the purpose of similarity searching for a query node. In this network, besides the information of the nodes and edges, we also have the content of each node. We use the convolutional neural network (CNN) to represent the content of each node and then use the graph convolutional network (GCN) to further represent the node by merging the representations of its neighboring nodes. The GCN output is further fed to a deep encoder-decoder model to convert each node to a Gaussian distribution and then convert back to its node identity. The dissimilarity between the two nodes is measured by the Wasserstein distance between their Gaussian distributions. We define the nodes of the network to be positives if they are relevant to the query node and negative if they are irrelevant. The labeling of the positives/negatives is based on an upper bound and a lower bound of the Wasserstein distances between the candidate nodes and the query nodes. We learn the parameters of CNN, GCN, encoder-decoder model, Gaussian distributions, and the upper bound and lower bounds jointly. The learning problem is modeled as a minimization problem to minimize the losses of node identification, network structure preservation, positive/negative query-specific relevance-guild distance, and model complexity. An iterative algorithm is developed to solve the minimization problem. We conducted experiments over benchmark networks, especially innovation networks, to verify the effectiveness of the proposed method and showed its advantage over the state-of-the-art methods.

scholarly journals A Second Order Finite-Difference Ghost-Point Method for Elasticity Problems on Unbounded Domains with Applications to Volcanology

2014 ◽  
Vol 16 (4) ◽  
pp. 983-1009 ◽  
Author(s):  
Armando Coco ◽  
Gilda Currenti ◽  
Ciro Del Negro ◽  
Giovanni Russo
Keyword(s):  
Linear System ◽  
Finite Difference ◽  
Unbounded Domain ◽  
Unbounded Domains ◽  
Numerical Tests ◽  
Arbitrary Geometries ◽  
Elasticity Problems ◽  
Grid Nodes ◽  
Interior Points ◽  
Point Stencil

AbstractWe propose a finite-difference ghost-point approach for the numerical solution of Cauchy-Navier equations in linear elasticity problems on arbitrary unbounded domains. The technique is based on a smooth coordinate transformation, which maps an unbounded domain into a unit square. Arbitrary geometries are defined by suitable level-set functions. The equations are discretized by classical nine-point stencil on interior points, while boundary conditions and high order reconstructions are used to define the field variables at ghost-points, which are grid nodes external to the domain with a neighbor inside the domain. The linear system arising from such discretization is solved by a multigrid strategy. The approach is then applied to solve elasticity problems in volcanology for computing the displacement caused by pressure sources. The method is suitable to treat problems in which the geometry of the source often changes (explore the effects of different scenarios, or solve inverse problems in which the geometry itself is part of the unknown), since it does not require complex re-meshing when the geometry is modified. Several numerical tests are successfully performed, which asses the effectiveness of the present approach.

Stability of Elastic Half-Spaces by an Energy Method

1999 ◽  
Author(s):  
Yi-chao Chen ◽  
Lewis T. Wheeler
Keyword(s):  
Half Space ◽  
Unbounded Domain ◽  
Minimization Problem ◽  
Energy Method ◽  
Stability Criterion ◽  
Elastic Half Space ◽  
Stability Conditions ◽  
Second Variation ◽  
The Stability ◽  
First And Second Variation

Abstract An energy stability criterion is used to study the stability of deformations of a compressible elastic half-space. A minimization problem is formulated in an unbounded domain, and the first and second variation conditions are derived for this problem. Algebraic stability conditions are derived for general compressible isotropic materials, as well as for neo-Hookean class of Hadamard materials.

scholarly journals Extended weak maximum principles for parabolic partial differential inequalities on unbounded domains

2014 ◽  
Vol 470 (2167) ◽  
pp. 20140079 ◽  
Author(s):  
J. C. Meyer ◽  
D. J. Needham
Keyword(s):  
Maximum Principle ◽  
Unbounded Domain ◽  
Differential Inequality ◽  
Unbounded Domains ◽  
Comparison Theorems ◽  
Weight Functions ◽  
Maximum Principles ◽  
Differential Inequalities ◽  
Partial Differential ◽  
Conditional Maximum

In this paper, we establish extended maximum principles for solutions to linear parabolic partial differential inequalities on unbounded domains, where the solutions satisfy a variety of growth/decay conditions on the unbounded domain. We establish a conditional maximum principle, which states that a solution u to a linear parabolic partial differential inequality satisfies a maximum principle whenever a suitable weight function can be exhibited. Our extended maximum principles are then established by exhibiting suitable weight functions and applying the conditional maximum principle. In addition, we include several specific examples, to highlight the importance of certain generic conditions, which are required in the statements of maximum principles of this type. Furthermore, we demonstrate how to obtain associated comparison theorems from our extended maximum principles.

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