Nonlinear Analysis-Theory Methods & Applications最新文献

英文中文

Hypergraph p-Laplacian regularization on point clouds for data interpolation

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-04-03 DOI: 10.1016/j.na.2025.113807

Kehan Shi , Martin Burger

As a generalization of graphs, hypergraphs are widely used to model higher-order relations in data. This paper explores the benefit of the hypergraph structure for the interpolation of point cloud data that contain no explicit structural information. We define the

ɛ_{n}

-ball hypergraph and the

k_{n}

-nearest neighbor hypergraph on a point cloud and study the

p

-Laplacian regularization on the hypergraphs. We prove the variational consistency between the hypergraph

p

-Laplacian regularization and the continuum

p

-Laplacian regularization in a semisupervised setting when the number of points

n

goes to infinity while the number of labeled points remains fixed. A key improvement compared to the graph case is that the results rely on weaker assumptions on the upper bound of

ɛ_{n}

and

k_{n}

. To solve the convex but non-differentiable large-scale optimization problem, we utilize the stochastic primal–dual hybrid gradient algorithm. Numerical experiments on data interpolation verify that the hypergraph

p

-Laplacian regularization outperforms the graph

p

-Laplacian regularization in preventing the development of spikes at the labeled points.

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引用次数: 0

Beurling–Deny formula for Sobolev–Bregman forms

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-04-01 DOI: 10.1016/j.na.2025.113808

Michał Gutowski, Mateusz Kwaśnicki

For an arbitrary regular Dirichlet form

E

and the associated symmetric Markovian semigroup

T_{t}

, we consider the corresponding Sobolev–Bregman form

E_{p} (u) = - \frac{1}{p} \frac{d}{d t} |_{t = 0} {‖ T_{t} u ‖}_{p}^{p}

, where

p \in (1, \infty)

. We prove a variant of the Beurling–Deny formula for

E_{p}

. As an application, we prove the corresponding Hardy–Stein identity. Our results extend previous works in this area, which either required that

E

is translation-invariant, or that

u

is sufficiently regular.

引用次数: 0

The horospherical p-Christoffel–Minkowski problem in hyperbolic space

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-04-01 DOI: 10.1016/j.na.2025.113799

Tianci Luo, Yong Wei

The horospherical

p

-Christoffel–Minkowski problem, introduced by Li and Xu (2022), involves prescribing the

k

-th horospherical

p

-surface area measure for

h

-convex domains in hyperbolic space

H^{n + 1}

. This problem generalizes the classical

L^{p}

Christoffel–Minkowski problem in Euclidean space

R^{n + 1}

. In this paper, we study a fully nonlinear equation associated with this problem and establish the existence of a uniformly

h

-convex solution under suitable assumptions on the prescribed function. The proof relies on a full rank theorem, which we demonstrate using a viscosity approach inspired by the work of Bryan et al. (2023).

When

p = 0

, the horospherical

p

-Christoffel–Minkowski problem in

H^{n + 1}

reduces to a Nirenberg-type problem on

S^{n}

in conformal geometry. As a consequence, our result also provides the existence of solutions to this Nirenberg-type problem.

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引用次数: 0

Long time asymptotic behavior of a self-similar fragmentation equation

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-03-28 DOI: 10.1016/j.na.2025.113805

Gaetano Agazzotti , Madalina Deaconu , Antoine Lejay

Using the Mellin transform, we study self-similar fragmentation equations whose breakage rate follows the power law distribution, and a particle is split into a fixed number of smaller particles. First, we show how to extend the solution of such equations to measure-valued initial conditions, by a closure argument on the Mellin space. Second, we use appropriate series representations to give a rigorous proof to the asymptotic behavior of the moments, completing some results known through heuristic derivations.

引用次数: 0

Carleson conditions for weights: The quantitative small constant case

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-03-27 DOI: 10.1016/j.na.2025.113802

Simon Bortz , Moritz Egert , Olli Saari

We investigate the small constant case of a characterization of

A_{\infty}

weights due to Fefferman, Kenig and Pipher. In their work, Fefferman, Kenig and Pipher bound the logarithm of the

A_{\infty}

constant by the Carleson norm of a measure built out of the heat extension, up to a multiplicative and additive constant (as well as the converse). We prove, qualitatively, that when one of these quantities is small, then so is the other. In fact, we show that these quantities are bounded by a constant times the square root of the other, provided at least one of them is sufficiently small.

We also give an application of our result to the study of elliptic measures associated to elliptic operators with coefficients satisfying the “Dahlberg–Kenig–Pipher” condition.

引用次数: 0

Upper bounds on the dimension of the global attractor of the 2D Navier-Stokes equations on the β-plane

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-03-24 DOI: 10.1016/j.na.2025.113794

Aseel Farhat , Anuj Kumar , Vincent R. Martinez

This article establishes estimates on the dimension of the global attractor of the two-dimensional rotating Navier–Stokes equation for viscous, incompressible fluids on the

β

-plane. Previous results in this setting by Al-Jaboori and Wirosoetisno (2011) had proved that the global attractor collapses to a single point that depends only the latitudinal coordinate, i.e., zonal flow, when the rotation is sufficiently fast. However, an explicit quantification of the complexity of the global attractor in terms of

β

had remained open. In this paper, such estimates are established which are valid across a wide regime of rotation rates and are consistent with the dynamically degenerate regime previously identified. Additionally, a decomposition of solutions is established detailing the asymptotic behavior of the solutions in the limit of large rotation.

引用次数: 0

The L1-relaxed area of the graph of the vortex map: Optimal lower bound

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-03-22 DOI: 10.1016/j.na.2025.113803

Giovanni Bellettini , Alaa Elshorbagy , Riccardo Scala

We prove a lower bound for the value of the

L^{1}

-relaxed area of the graph of the map

u : B_{l} (0) ∖ {0} \subset R^{2} \to R^{2}

u (x) : = x / | x |

x \neq 0

, for all values of the radius

l > 0

. In the computation of the singular part of the relaxed area, for

l

in a certain range, in particular

l

not too large, a nonparametric Plateau-type problem with partial free boundary has to be solved. Our lower bound turns out to be optimal, in view of an upper bound proven in a companion paper.

引用次数: 0

Bifurcation of double eigenvalues for Aharonov–Bohm operators with a moving pole

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-03-19 DOI: 10.1016/j.na.2025.113798

Laura Abatangelo , Veronica Felli

We study double eigenvalues of Aharonov–Bohm operators with Dirichlet boundary conditions in planar domains containing the origin. We focus on the behavior of double eigenvalues when the potential’s circulation is a fixed half-integer number and the operator’s pole is moving on straight lines in a neighborhood of the origin. We prove that bifurcation occurs if the pole is moving along straight lines in a certain number of cones with positive measure. More precise information is given for symmetric domains; in particular, in the special case of the disk, any eigenvalue is double if the pole is located at the center, but there exists a whole neighborhood where it bifurcates into two distinct branches.

引用次数: 0

Some qualitative and quantitative properties of weak solutions to mixed anisotropic and nonlocal quasilinear elliptic and doubly nonlinear parabolic equations

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-03-13 DOI: 10.1016/j.na.2025.113796

Prashanta Garain

This article is divided into two parts. In the first part, we examine the Brezis–Oswald problem involving a mixed anisotropic and nonlocal

p

-Laplace operator. We establish results on existence, uniqueness, boundedness, and the strong maximum principle. Additionally, for certain mixed anisotropic and nonlocal

p

-Laplace equations, we prove a Sturmian comparison theorem, establish comparison and nonexistence results, derive a weighted Hardy-type inequality, and analyze a system of singular mixed anisotropic and nonlocal

p

-Laplace equations. A key component of our approach is the use of the Picone identity, which we adapt from the local and nonlocal cases. In the second part of the article, we focus on regularity estimates. In the elliptic setting, we establish a weak Harnack inequality and semicontinuity results. We also consider a class of doubly nonlinear mixed anisotropic and nonlocal parabolic equations, proving semicontinuity results and analyzing the pointwise behavior of solutions. These results rely on appropriate energy estimates, De Giorgi-type lemmas, and positivity expansions. Finally, we derive various energy estimates, which may be of independent interest.

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引用次数: 0

On the existence of a second positive solution to mixed local-nonlocal concave–convex critical problems

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-03-13 DOI: 10.1016/j.na.2025.113795

Stefano Biagi , Eugenio Vecchi

We prove the existence of a second positive weak solution for mixed local-nonlocal critical semilinear elliptic problems with a sublinear perturbation in the spirit of Ambrosetti et al. (1994).

引用次数: 0

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