Journal of Functional Analysis最新文献

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Propagation of regularity for a class of systems of real vector fields on torus

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-01-06 DOI: 10.1016/j.jfa.2024.110807

I.A. Ferra, L.A. Carvalho dos Santos

We characterize the global hypoellipticity, almost hypoellipticity and solvability for a class of systems of real vector fields on the

(n + 1)

-dimensional torus as well as the same properties about the sum of squares associated to the system. The key result is a theorem about propagation of regularity for solutions of the system.

引用次数: 0

One-bubble nodal blow-up for asymptotically critical stationary Schrödinger-type equations

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-01-06 DOI: 10.1016/j.jfa.2024.110808

Bruno Premoselli , Frédéric Robert

We investigate in this work families

{(u_{ε})}_{ε > 0}

of sign-changing blowing-up solutions of asymptotically critical stationary nonlinear Schrödinger equations of the following type:

Δ_{g} u_{ε} + h_{ε} u_{ε} = | u_{ε} |^{p_{ε} - 2} u_{ε}

in a closed Riemannian manifold

(M, g)

, where

h_{ε}

converges to h in

C^{1} (M)

and

p_{ε} \to 2_{-}^{⋆} : = \frac{2 n}{n - 2}

ε \to 0

. Assuming that

{(u_{ε})}_{ε > 0}

blows-up as a single sign-changing bubble, we obtain necessary conditions for blow-up that constrain the localisation of blow-up points and exhibit a strong interaction between h, the geometry of

(M, g)

and the bubble itself. These conditions are new and are a consequence of the sign-changing nature of

u_{ε}

引用次数: 0

Quantitative spectral stability for the Neumann Laplacian in domains with small holes

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2025-01-06 DOI: 10.1016/j.jfa.2024.110817

Veronica Felli , Lorenzo Liverani , Roberto Ognibene

The aim of the present paper is to investigate the behavior of the spectrum of the Neumann Laplacian in domains with little holes excised from the interior. More precisely, we consider the eigenvalues of the Laplacian with homogeneous Neumann boundary conditions on a bounded, Lipschitz domain. Then, we singularly perturb the domain by removing Lipschitz sets which are “small” in a suitable sense and satisfy a uniform extension property. In this context, we provide an asymptotic expansion for all the eigenvalues of the perturbed problem which are converging to simple eigenvalues of the limit one. The eigenvalue variation turns out to depend on a geometric quantity resembling the notion of (boundary) torsional rigidity: understanding this fact is one of the main contributions of the present paper. In the particular case of a hole shrinking to a point, through a fine blow-up analysis, we identify the exact vanishing order of such a quantity and we establish some connections between the location of the hole and the sign of the eigenvalue variation.

引用次数: 0

Sobolev differentiability properties of logarithmic modulus of real analytic functions

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-12-30 DOI: 10.1016/j.jfa.2024.110804

Ziming Shi , Ruixiang Zhang

Let f be the germ of a real analytic function at the origin in

R^{n}

for

n \geq 2

, and suppose the codimension of the zero set of f at 0 is at least 2. We show that

\log | f |

W_{loc}^{1, 1}

near 0. In particular, this implies that the differential inequality

| \nabla f | \leq V | f |

holds with

V \in L_{loc}^{1}

引用次数: 0

Strongly convergent unitary representations of limit groups

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-12-30 DOI: 10.1016/j.jfa.2024.110803

Larsen Louder , Michael Magee , Will Hide

We prove that all finitely generated fully residually free groups (limit groups) have a sequence of finite dimensional unitary representations that ‘strongly converge’ to the regular representation of the group. The corresponding statement for finitely generated free groups was proved by Haagerup and Thorbjørnsen in 2005. In fact, we can take the unitary representations to arise from representations of the group by permutation matrices, as was proved for free groups by Bordenave and Collins.

As for Haagerup and Thorbjørnsen, the existence of such representations implies that for any non-abelian limit group, the Ext-invariant of the reduced

C^{⁎}

-algebra is not a group (has non-invertible elements).

An important special case of our main theorem is in application to the fundamental groups of closed orientable surfaces of genus at least two. In this case, our results can be used as an input to the methods previously developed by the authors of the appendix. The output is a variation of our previous proof of Buser's 1984 conjecture that there exist a sequence of closed hyperbolic surfaces with genera tending to infinity and first eigenvalue of the Laplacian tending to

\frac{1}{4}

. In this variation of the proof, the systoles of the surfaces are bounded away from zero and the surfaces can be taken to be arithmetic.

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引用次数: 0

Deformed single ring theorems

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-12-12 DOI: 10.1016/j.jfa.2024.110797

Ching-Wei Ho , Ping Zhong

Given a sequence of deterministic matrices

A = A_{N}

and a sequence of deterministic nonnegative matrices

Σ = Σ_{N}

such that

A \to a

and

Σ \to σ

in ⁎-distribution for some operators a and σ in a finite von Neumann algebra

A

. Let

U = U_{N}

and

V = V_{N}

be independent Haar-distributed unitary matrices. We use free probability techniques to prove that, under mild assumptions, the empirical eigenvalue distribution of

U Σ V^{⁎} + A

converges to the Brown measure of

T + a

, where

T \in A

is an R-diagonal operator freely independent from a and

| T |

has the same distribution as σ. The assumptions can be removed if A is Hermitian or unitary. By putting

A = 0

, our result removes a regularity assumption in the single ring theorem by Guionnet, Krishnapur and Zeitouni. We also prove a local convergence on optimal scale, extending the local single ring theorem of Bao, Erdős and Schnelli.

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引用次数: 0

Stability threshold of Couette flow for the 3D MHD equations

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-12-10 DOI: 10.1016/j.jfa.2024.110796

Yulin Rao , Zhifei Zhang , Ruizhao Zi

In this paper, we consider the stability of 3D Couette flow

{(y, 0, 0)}^{⊤}

in a uniform background magnetic field

α {(σ, 0, 1)}^{⊤}

. In particular, the MHD equations on

T \times R \times T

that we are concerned with are of different viscosity coefficient ν and magnetic diffusion coefficient μ. It is shown that if the background magnetic field

α {(σ, 0, 1)}^{⊤}

with

σ \in R ﹨ Q

satisfying a generic Diophantine condition is so strong that

| α | ≫ \frac{ν + μ}{\sqrt{ν μ}}

, and the initial perturbations

u_{in}

and

b_{in}

satisfy

{‖ (u_{in}, b_{in}) ‖}_{H^{N + 2}} ≪ \min {ν, μ}

for sufficiently large N, then the resulting solution remains close to the steady state in

L^{2}

at the same order for all time. Compared with the result of Liss [Comm. Math. Phys., 377(2020), 859–908], we use a more general energy method to address the physically relevant case

ν \neq μ

based on some new observations.

引用次数: 0

Second order Sobolev regularity results for the generalized p-parabolic equation

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-12-10 DOI: 10.1016/j.jfa.2024.110799

Yawen Feng , Mikko Parviainen , Saara Sarsa

We study a general class of parabolic equations

u_{t} - {| D u |}^{γ} (Δ u + (p - 2) Δ_{\infty}^{N} u) = 0,

which can be highly degenerate or singular. This class contains as special cases the standard parabolic p-Laplace equation and the normalized version that arises from stochastic game theory. Utilizing the systematic approach developed in our previous work we establish second order Sobolev regularity together with a priori estimates and improved range of parameters. In addition we derive second order Sobolev estimate for a nonlinear quantity. This quantity contains many useful special cases. As a corollary we also obtain that a viscosity solution has locally

L^{2}

-integrable Sobolev time derivative.

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引用次数: 0

Interior Hölder and Calderón-Zygmund estimates for fully nonlinear equations with natural gradient growth

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-12-10 DOI: 10.1016/j.jfa.2024.110800

Alessandro Goffi

We establish local Hölder estimates for viscosity solutions of fully nonlinear second order equations with quadratic growth in the gradient and unbounded right-hand side in

L^{q}

spaces, for an integrability threshold q guaranteeing the validity of the maximum principle. This is done through a nonlinear Harnack inequality for nonhomogeneous equations driven by a uniformly elliptic Isaacs operator and perturbed by a Hamiltonian term with natural growth in the gradient. As a byproduct, we derive a new Liouville property for entire

L^{p}

viscosity solutions of fully nonlinear equations as well as a nonlinear Calderón-Zygmund estimate for strong solutions of such equations.

引用次数: 0

Uncertainty qualification of Vlasov-Poisson-Boltzmann equations in the diffusive scaling

IF 1.7 2区数学 Q1 MATHEMATICS

Journal of Functional Analysis

Pub Date : 2024-12-10 DOI: 10.1016/j.jfa.2024.110794

Ning Jiang , Xu Zhang

For the Vlasov-Poisson-Boltzmann equations with random uncertainties from the initial data or collision kernels, we proved the sensitivity analysis and energy estimates uniformly with respect to both the Knudsen number and the random variables in the diffusive scaling using hypocoercivity method developed in [6], [7], [14]. As a consequence, we also justified the incompressible Navier-Stokes-Poisson limit with random inputs. In particular, for the first time, we obtain the precise convergence rate without employing any results based on Hilbert expansion (in other words, we don't need any information from the limiting fluid equations a priori). We not only generalized the previous deterministic Navier-Stokes-Fourier-Poisson limits to random initial data case, but also improve the previous uncertainty quantification results to the case where the initial data include both kinetic and fluid parts. This is the first uncertainty qualification (UQ) result for spatially high dimension kinetic equations in diffusive limits containing Navier-Stokes dynamics, and generalizes the previous UQ results which does not contain fluid equations (for example, [34]).

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引用次数: 0

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