Journal of Functional Analysis最新文献

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On positivity of the Q-curvatures of conformal metrics

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-04-18 DOI: 10.1016/j.jfa.2025.111011

Mingxiang Li , Xingwang Xu

We mainly show that for a conformal metric

g = u^{\frac{4}{n - 2 m}} | d x |^{2}

R^{n}

with

n \geq 2 m + 1

, if the

2 m -

order Q-curvature

Q_{g}^{(2 m)}

is positive and has slow decay barrier near infinity, the lower order Q-curvature

Q_{g}^{(2)}

and

Q_{g}^{(4)}

are both positive if m is at least two.

我们主要证明，对于 Rn 上 n≥2m+1 的共形度量 g=u4n-2m|dx|2，如果 2m 阶 Qcurvature Qg(2m) 为正且在无穷远附近有缓慢衰减障，则如果 m 至少为 2，则低阶 Qcurvature Qg(2) 和 Qg(4) 均为正。

引用次数: 0

Well-posedness of the obstacle problem for stochastic nonlinear diffusion equations: An entropy formulation

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-04-18 DOI: 10.1016/j.jfa.2025.111012

Kai Du , Ruoyang Liu

In this paper, we establish the existence, uniqueness and stability results for the obstacle problem associated with a degenerate nonlinear diffusion equation perturbed by conservative gradient noise. Our approach revolves round introducing a new entropy formulation for stochastic variational inequalities. As a consequence, we obtain a novel well-posedness result for the obstacle problem of deterministic porous medium equations with nonlinear reaction terms.

引用次数: 0

Lp-boundedness of wave operators for fourth order Schrödinger operators with zero resonances on R3

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-04-18 DOI: 10.1016/j.jfa.2025.111013

Haruya Mizutani , Zijun Wan , Xiaohua Yao

<div><div>Let <math><mi>H</mi><mo>=</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>V</mi></math> be the fourth-order Schrödinger operator on <math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math> with a real-valued fast-decaying potential V. If zero is neither a resonance nor an eigenvalue of H, then it was recently shown that the wave operators <math><msub><mrow><mi>W</mi></mrow><mrow><mo>±</mo></mrow></msub><mo>(</mo><mi>H</mi><mo>,</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math> are bounded on <math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math> for all <math><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mo>∞</mo></math> and unbounded at the endpoints <math><mi>p</mi><mo>=</mo><mn>1</mn></math> and <math><mi>p</mi><mo>=</mo><mo>∞</mo></math>.</div><div>This paper is to further establish the <math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math>-boundedness of <math><msub><mrow><mi>W</mi></mrow><mrow><mo>±</mo></mrow></msub><mo>(</mo><mi>H</mi><mo>,</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math> that exhibit all types of singularities at the zero energy threshold. We first prove that <math><msub><mrow><mi>W</mi></mrow><mrow><mo>±</mo></mrow></msub><mo>(</mo><mi>H</mi><mo>,</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math> are bounded on <math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math> for all <math><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mo>∞</mo></math> in the first kind resonance case, and then proceed to establish for the second kind resonance case that they are bounded on <math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math> for all <math><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>3</mn></math>, but not if <math><mn>3</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mo>∞</mo></math>. In the third kind resonance case, we also show that <math><msub><mrow><mi>W</mi></mrow><mrow><mo>±</mo></mrow></msub><mo>(</mo><mi>H</mi><mo>,</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math> are bounded on <math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math> for all <math><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>3</mn></math> and generically unbounded on <math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></

假设H=Δ2+V是R3上的四阶薛定谔算子，具有实值快衰减势V。如果零既不是共振也不是H的特征值，那么最近的研究表明，波算子W±(H,Δ2)在Lp(R3)上对于所有1<p<∞都是有界的，而在端点p=1和p=∞处是无界的。本文将进一步建立 W±(H,Δ2)的 Lp 有界性，它在零能量阈值处表现出所有类型的奇异性。我们首先证明，在第一种共振情况下，W±(H,Δ2) 在 Lp(R3) 上对所有 1<p<∞ 都是有界的；然后继续证明，在第二种共振情况下，它们在 Lp(R3) 上对所有 1<p<3 都是有界的，但如果 3≤p≤∞ 则不是。在第三种共振情况下，我们还证明了 W±(H,Δ2)在 Lp(R3) 上对所有 1<p<3 都是有界的，而在 Lp(R3) 上对任何 3≤p≤∞ 通常都是无界的。此外，还证明了在所有 3≤p<∞ 条件下，W±(H,Δ2) 在 Lp(R3) 上都是有界的，如果此外 H 具有零特征值，但没有 p 波零共振，并且对于所有 i,j,k=1,2,3，x=(x1,x2,x3)∈R3，所有零特征函数都与 L2(R3) 中的 xixjxkV 正交。这些结果精确地描述了 W±(H,Δ2)在 R3 中的 Lp 有界性对于零能量阈值处所有类型奇点的有效性，但对于端点情况 p=1,∞ 有些例外。作为应用，还推导出了具有零共振奇点的四阶薛定谔方程和梁方程的 Lp-Lq 衰变估计值。

引用次数: 0

The direct moving sphere for fractional Laplace equation

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-04-17 DOI: 10.1016/j.jfa.2025.111010

Congming Li , Meiqing Xu , Hui Yang , Ran Zhuo

This paper works on the direct method of moving spheres and establishes a Liouville-type theorem for the fractional elliptic equation

{(- Δ)}^{α / 2} u = f (u) in R^{n}

with general non-linearity. One of the key improvements over the previous work is that we do not require the usual Lipschitz condition. In fact, we only assume the structural condition that

f (t) t^{- \frac{n + α}{n - α}}

is monotonically decreasing. This differs from the usual approach such as Chen-Li-Li (Adv. Math. 2017), which needs the Lipschitz condition on f, or Chen-Li-Zhang (J. Funct. Anal. 2017), which relies on both the structural condition and the monotonicity of f. We also use the direct moving spheres method to give an alternative proof for the Liouville-type theorem of the fractional Lane-Emden equation in a half space. Similarly, our proof does not depend on the integral representation of solutions compared to existing ones. The methods developed here should also apply to problems involving more general non-local operators, especially if no equivalent integral equations exist.

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引用次数: 0

Nonlocal Liouville theorems with gradient nonlinearity

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-04-17 DOI: 10.1016/j.jfa.2025.111008

Anup Biswas , Alexander Quaas , Erwin Topp

In this article we consider a large family of nonlinear nonlocal equations involving gradient nonlinearity and provide a unified approach, based on the Ishii-Lions type technique, to establish Liouville properties of the solutions. We also answer an open problem raised by Cirant and Goffi [24]. Some applications to regularity issues are also studied.

引用次数: 0

Phase space analysis of finite and infinite dimensional Fresnel integrals

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-04-17 DOI: 10.1016/j.jfa.2025.111009

Sonia Mazzucchi , Fabio Nicola , S. Ivan Trapasso

The full characterization of the class of Fresnel integrable functions is an open problem in functional analysis, with significant applications to mathematical physics (Feynman path integrals) and the analysis of the Schrödinger equation. In finite dimension, we prove the Fresnel integrability of functions in the Sjöstrand class

M^{\infty, 1}

— a family of continuous and bounded functions, locally enjoying the mild regularity of the Fourier transform of an integrable function. This result broadly extends the current knowledge on the Fresnel integrability of Fourier transforms of finite complex measures, and relies upon ideas and techniques of Gabor wave packet analysis. We also discuss infinite-dimensional extensions of this result. In this connection, we extend and make more concrete the general framework of projective functional extensions introduced by Albeverio and Mazzucchi. In particular, we obtain a concrete example of a continuous linear functional on an infinite-dimensional space beyond the class of Fresnel integrable functions. As an interesting byproduct, we obtain a sharp

M^{\infty, 1} \to L^{\infty}

operator norm bound for the free Schrödinger evolution operator.

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引用次数: 0

Interior W2,δ type estimates for degenerate fully nonlinear elliptic equations with Ln data

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-04-17 DOI: 10.1016/j.jfa.2025.111007

Sun-Sig Byun , Hongsoo Kim , Jehan Oh

We establish interior

W^{2, δ}

type estimates for a class of degenerate fully nonlinear elliptic equations with

L^{n}

data. The main idea of our approach is to slide

C^{1, α}

cones, instead of paraboloids, vertically to touch the solution, and estimate the contact set in terms of the measure of the vertex set. This shows that the solution has tangent

C^{1, α}

cones almost everywhere, which leads to the desired Hessian estimates. Accordingly, we are able to develop a kind of counterpart to the estimates for divergent structure quasilinear elliptic problems, as discussed in [6], [16].

引用次数: 0

Stability of a class of exact solutions of the incompressible Euler equation in a disk

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-04-15 DOI: 10.1016/j.jfa.2025.110998

Guodong Wang

We prove a sharp orbital stability result for a class of exact steady solutions, expressed in terms of Bessel functions of the first kind, of the two-dimensional incompressible Euler equation in a disk. A special case of these solutions is the truncated Lamb dipole, whose stream function corresponds to the second eigenfunction of the Dirichlet Laplacian. The proof is achieved by establishing a suitable variational characterization for these solutions via conserved quantities of the Euler equation and employing a compactness argument.

引用次数: 0

Global wellposedness of general nonlinear evolution equations for distributions on the Fourier half space

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-04-15 DOI: 10.1016/j.jfa.2025.111004

Kenji Nakanishi , Baoxiang Wang

The Cauchy problem is studied for very general systems of evolution equations, where the time derivative of solution is written by Fourier multipliers in space and analytic nonlinearity, with no other structural requirement. We construct a function space for the Fourier transform embedded in the space of distributions, and establish the global wellposedness with no size restriction. The major restriction on the initial data is that the Fourier transform is supported on the half space, decaying at the boundary in the sense of measure. We also require uniform integrability for the orthogonal directions in the distribution sense, but no other condition. In particular, the initial data may be much more rough than the tempered distributions, and may grow polynomially at the spatial infinity. A simpler argument is also presented for the solutions locally integrable in the frequency. When the Fourier support is slightly more restricted to a conical region, the generality of equations is extremely wide, including those that are even locally illposed in the standard function spaces, such as the backward heat equations, as well as those with infinite derivatives and beyond the natural boundary of the analytic nonlinearity. As more classical examples, our results may be applied to the incompressible and compressible Navier-Stokes and Euler equations, the nonlinear diffusion and wave equations, and so on. In particular, the wellposedness includes uniqueness of very weak solution for those equations, under the Fourier support condition, but with no restriction on regularity or size of solutions. The major drawback of the Fourier support restriction is that the solutions cannot be real valued.

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引用次数: 0

A sharp higher order Sobolev inequality on Riemannian manifolds

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-04-14 DOI: 10.1016/j.jfa.2025.111001

Samuel Zeitler

<div><div>Let <math><mi>m</mi><mo>,</mo><mi>n</mi></math> be integers such that <math><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>></mo><mi>m</mi><mo>≥</mo><mn>1</mn></math> and let <math><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></math> be a closed n-dimensional Riemannian manifold. We prove there exists some <math><mi>B</mi><mo>∈</mo><mi>R</mi></math> depending only on <math><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></math>, m, and n such that for all <math><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>(</mo><mi>M</mi><mo>)</mo></math>,<math><msubsup><mrow><mo>‖</mo><mi>u</mi><mo>‖</mo></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>#</mi></mrow></msup></mrow><mrow><mn>2</mn></mrow></msubsup><mo>≤</mo><mi>K</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo><munder><mo>∫</mo><mrow><mi>M</mi></mrow></munder><msup><mrow><mo>(</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mfrac><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mi>u</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><msub><mrow><mi>v</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>+</mo><mi>B</mi><msubsup><mrow><mo>‖</mo><mi>u</mi><mo>‖</mo></mrow><mrow><msubsup><mrow><mi>H</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>(</mo><mi>M</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msubsup></math> where <math><msup><mrow><mn>2</mn></mrow><mrow><mi>#</mi></mrow></msup><mo>=</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn><mi>m</mi></mrow></mfrac></math>, <math><mi>K</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math> is the square of the best constant for the embedding <math><msup><mrow><mi>W</mi></mrow><mrow><mi>m</mi><mo>,</mo><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>⊂</mo><msup><mrow><mi>L</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>#</mi></mrow></msup></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math>, <math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>(</mo><mi>M</mi><mo>)</mo></math> is the Sobolev space consisting of functions on M with m weak derivatives in <math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo></math>, and <math><msup><mrow><mi>Δ</mi></mrow><mrow><mfrac><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>=</mo><mi>∇</mi><msup><mrow><mi>Δ</mi></mrow><mrow><mfrac><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math> if m is odd. This

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