Nonlinear Analysis-Theory Methods & Applications最新文献

英文中文

Boundary value problems for Choquard equations

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-13 DOI: 10.1016/j.na.2024.113745

Chiara Bernardini , Annalisa Cesaroni

We consider the following nonlinear Choquard equation

- Δ u + V u = (I_{α} * {| u |}^{p}) {| u |}^{p - 2} u in Ω \subset R^{N},

where

N \geq 2

p \in (1, + \infty)

V (x)

is a continuous radial function such that

{inf}_{x \in Ω} V > 0

and

I_{α} (x)

is the Riesz potential of order

α \in (0, N)

. Assuming Neumann or Dirichlet boundary conditions, we prove existence of a positive radial solution to the corresponding boundary value problem when

Ω

is an annulus, or an exterior domain of the form

R^{N} ∖ \bar{B_{r} (0)}

. We also provide a nonexistence result: if

p \geq \frac{N + α}{N - 2}

the corresponding Dirichlet problem has no nontrivial regular solution in strictly star-shaped domains. Finally, when considering annular domains, letting

α \to 0^{+}

we recover existence results for the corresponding local problem with power-type nonlinearity.

引用次数: 0

Mild regularity of weak solutions to the Navier – Stokes equations

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-09 DOI: 10.1016/j.na.2024.113744

Ira Herbst

We use a well known integral equation to derive some regularity properties of Leray–Hopf weak solutions of the Navier–Stokes equations in

L^{p} (R^{3}), p \in [1, 2)

引用次数: 0

Convergence of the area functional on spaces with lower Ricci bounds and applications

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-06 DOI: 10.1016/j.na.2024.113738

Alessandro Cucinotta

We show that the heat flow provides good approximation properties for the area functional on proper

RCD (K, \infty)

spaces, implying that in this setting the area formula for functions of bounded variation holds and that the area functional coincides with its relaxation. We then obtain partial regularity and uniqueness results for functions whose hypographs are perimeter minimizing. Finally, we consider sequences of

RCD (K, N)

spaces and we show that, thanks to the previously obtained properties, Sobolev minimizers of the area functional in a limit space can be approximated with minimizers along the converging sequence of spaces. Using this last result, we obtain applications on Ricci-limit spaces.

引用次数: 0

Composition of rough singular integral operators on rearrangement invariant Banach type spaces

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-06 DOI: 10.1016/j.na.2024.113742

Jiawei Tan, Qingying Xue

Let

Ω

be a homogeneous function of degree zero that enjoys the vanishing condition on the unit sphere

S^{n - 1} (n \geq 2)

. Let

T_{Ω}

be the convolution singular integral operator with kernel

Ω (x) {| x |}^{- n}

. In this paper, when

Ω \in L^{\infty} (S^{n - 1})

, we consider quantitative weighted bounds of composite operators of

T_{Ω}

on rearrangement invariant Banach function spaces. These spaces contain classical Lorentz spaces and Orlicz spaces as special examples. Weighted boundedness of the composite operators on rearrangement invariant quasi-Banach spaces are also given.

引用次数: 0

A partially overdetermined problem for the p-Laplace equation in a convex cone

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-01 DOI: 10.1016/j.na.2024.113743

Hui Ma , Mingxuan Yang , Jiabin Yin

We consider a partially overdetermined problem for the

p

-Laplace equation in a convex cone

C

intersected with the exterior of a smooth bounded domain

\bar{Ω}

R^{n}

(

n \geq 2

). First, we establish the existence, regularity, and asymptotic behavior of a capacitary potential. Then, based on these properties of the potential, we obtain a rigidity result under the assumption of orthogonal intersection, by using

P

-function, the isoperimetric inequality, and a Heintze–Karcher type inequality in a convex cone.

引用次数: 0

Local second order regularity of solutions to elliptic Orlicz–Laplace equation

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-01 DOI: 10.1016/j.na.2024.113737

Arttu Karppinen , Saara Sarsa

We consider Orlicz–Laplace equation

- div (\frac{φ^{'} (| \nabla u |)}{| \nabla u |} \nabla u) = f

where

φ

is an Orlicz function and either

f = 0

f \in L^{\infty}

. We prove local second order regularity results for the weak solutions

u

of the Orlicz–Laplace equation. More precisely, we show that if

ψ

is another Orlicz function that is close to

φ

in a suitable sense, then

\frac{ψ^{'} (| \nabla u |)}{| \nabla u |} \nabla u \in W_{loc}^{1, 2}

. This work contributes to the building up of quantitative second order Sobolev regularity for solutions of nonlinear equations.

引用次数: 0

Fourier integral operators on Hardy spaces with amplitudes in forbidden Hörmander classes

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-01-01 DOI: 10.1016/j.na.2024.113741

Xiaofeng Ye , Chunjie Zhang , Xiangrong Zhu

<div><div>In this note, we consider a Fourier integral operator defined by <math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>ϕ</mi><mo>,</mo><mi>a</mi></mrow></msub><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mo>∫</mo></mrow><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>ϕ</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>ξ</mi><mo>)</mo></mrow></mrow></msup><mi>a</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>ξ</mi><mo>)</mo></mrow><mover><mrow><mi>f</mi></mrow><mrow><mo>̂</mo></mrow></mover><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mi>d</mi><mi>ξ</mi><mo>,</mo></mrow></math>where <math><mi>a</mi></math> is the amplitude, and <math><mi>ϕ</mi></math> is the phase.</div><div>Let <math><mrow><mn>0</mn><mo>≤</mo><mi>ρ</mi><mo>≤</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></math> or <math><mrow><mn>0</mn><mo>≤</mo><mi>ρ</mi><mo><</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>=</mo><mn>1</mn></mrow></math> and <math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>ρ</mi><mo>−</mo><mi>n</mi></mrow><mrow><mi>p</mi></mrow></mfrac><mo>+</mo><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>min</mo><mrow><mo>{</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mi>ρ</mi><mo>}</mo></mrow><mo>.</mo></mrow></math>If <math><mi>a</mi></math> belongs to the forbidden Hörmander class <math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>ρ</mi><mo>,</mo><mn>1</mn></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msubsup></math> and <math><mrow><mi>ϕ</mi><mo>∈</mo><msup><mrow><mi>Φ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math> satisfies the strong non-degeneracy condition, then for any <math><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo><</mo><mi>p</mi><mo>≤</mo><mn>1</mn></mrow></math>, we can show that the Fourier integral operator <math><msub><mrow><mi>T</mi></mrow><mrow><mi>ϕ</mi><mo>,</mo><mi>a</mi></mrow></msub></math> is bounded from the local Hardy space <math><msup><mrow><mi>h</mi></mrow><mrow><mi>p</mi></mrow></msup></math> to <math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math>. Furthermore, if <math><mi>a</mi></math> has compact support in variable <math><mi>x</mi></math>, then we can extend this result to <math><mrow><mn>0</mn><mo><</mo><mi>p</mi><mo>≤</mo><mn>1</mn></mrow></math>. As <math><mrow><msubsup><mrow><mi>S</mi></mrow><mrow><mi>ρ</mi><mo>,</mo><mi>δ</mi></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msubsup><mo>⊂</mo><msubsup><mrow><mi>S</mi></m

引用次数: 0

Microscopic derivation of non-local models with anomalous diffusions from stochastic particle systems

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-12-28 DOI: 10.1016/j.na.2024.113736

Marielle Simon , Christian Olivera

This paper considers a large class of nonlinear integro-differential scalar equations which involve an anomalous diffusion (e.g. driven by a fractional Laplacian) and a non-local singular convolution kernel. Each of those singular equations is obtained as the macroscopic limit of an interacting particle system modeled as

N

coupled stochastic differential equations driven by Lévy processes. In particular we derive quantitative estimates between the microscopic empirical measure of the particle system and the solution to the limit equation in some non-homogeneous Sobolev space. Our result only requires very weak regularity on the interaction kernel, therefore it includes numerous applications, e.g.: the

2 d

turbulence model (including the quasi-geostrophic equation) in sub-critical regime, the

2 d

generalized Navier–Stokes equation, the fractional Keller–Segel equation in any dimension, and the fractal Burgers equation.

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引用次数: 0

On local energy decay for solutions to the b-family of peakon equations

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-12-27 DOI: 10.1016/j.na.2024.113739

Christian Hong

This work is concerned with the long time behavior of solutions to the

b

-family of peakon equations. We prove local energy decay of global solutions under suitable hypotheses. Assuming the global bound of the

H^{1} (R)

norm, we show local energy decay along sequences of time in an expanding region around the origin. If we assume nonnegativity for an initial momentum,

m_{0} (x) = (u - \partial_{x}^{2} u) (0, x)

, we show strict decay on a similar region. Moreover, we show decay in an exterior region given the same nonnegativity condition.

引用次数: 0

Hyperbolic circle packings and total geodesic curvatures on surfaces with boundary

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-12-21 DOI: 10.1016/j.na.2024.113735

Guangming Hu , Yi Qi , Yu Sun , Puchun Zhou

This paper investigates a generalized hyperbolic circle packing (including circles, horocycles or hypercycles) with respect to the total geodesic curvatures on the surface with boundary. We mainly focus on the existence and rigidity of circle packing whose contact graph is the 1-skeleton of a finite polygonal cellular decomposition, which is analogous to the construction of Bobenko and Springborn (2004). Motivated by Colin de Verdiere’s method (Colin de Verdiere’s, 1991), we introduce the variational principle for generalized hyperbolic circle packings on polygons. By analyzing limit behaviors of generalized circle packings on polygons, we obtain an existence and rigidity for the generalized hyperbolic circle packing with conical singularities regarding the total geodesic curvature on each vertex of the contact graph. As a consequence, we introduce the combinatoral Ricci flow to find a desired circle packing with a prescribed total geodesic curvature on each vertex of the contact graph.

引用次数: 0

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