arXiv subjects

Xifeng Su

Publications and source records attributed to Xifeng Su.

At least 19 recordsLinked to original sources

arXiv2026

Integrability for conformally symplectic systems

The goal of this paper is to study the dynamics of conformally symplectic Hamiltonian flows under the light of integrability. As the dynamics of conformally symplectic Hamiltonian flows are dissipative and differ fundamentally from their conservative counterpart we start by proposing several notions of integrability that are better suited to the problem. We will propose two notions of integrability: $C^1$-integrability and Hopf integrability, that depend on the existence of a global attractor and on its shape. Then our main theorem focuses on Tonelli Hamiltonians whose conformally symplectic flows do not have conjugate points. We prove that such flows are automatically Hopf integrable. The proof is geometric and studies the long time evolution of vertical subspaces under the flow. It also makes use of (discounted) weak KAM theory. We also establish several results about the asymptotic Maslov index for integrable conformally symplectic Hamiltonian flows. Finally, we describe some examples to illustrate differences between symplectic and conformally symplectic Hamiltonian flows and to illustrate the pertinence of our definitions of integrability.

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arXiv2026

Phonon gap analysis for equilibria of perturbed almost-periodic Frenkel-Kontorova models

For generalized Frenkel-Kontorova models subjected to almost-periodic media, employing both the KAM method and the approach of `anti-integrable' limits, two different types of equilibria are obtained in \cite{an2024kamtheoryalmostperiodicequilibria} and \cite{du2024anti} respectively. We study the phonon gap around these equilibria and we find that the KAM equilibria do not have a phonon gap but the equilibria obtained by anti-integrable limits do have.

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arXiv2025

Qualitative properties of positive solutions to mixed local and nonlocal critical problems in $\mathbb{R}^n$

We consider the following mixed local and non-local critical elliptic equation: \begin{equation*}\label{0.1} \left\{ \begin{array}{lll} -\Delta u+(-\Delta)^su=\lambda h u^{p}+u^{2^*-1}, &\text{in}\,\, \mathbb{R}^n, u>0, &\text {in} \,\, \mathbb{R}^n, \lim\limits_{|x|\to\infty} u(x) = 0, \end{array} \right. \end{equation*} where $n\geqslant4, \,\, p\in (0,2^*-1),\,\, 2^*:=\frac{2n}{n-2}$ and $h$ is a positive function. We first show the existence and regularity results of viscosity solutions to the above critical elliptic equation. More precisely, from \cite{Su-Xu} weak solutions are obtained and we prove they are indeed viscosity solutions and their regularity is: $ u \in C^{\alpha}(\mathbb{R}^n) $ for $p\in(0,1);$ $ u \in C^{2,\beta}(\mathbb{R}^n) $ for $p\in [1, 2^*-1).$ Moreover, for $p\in [1, 2^*-1)$, these viscosity solutions are indeed classical ones and we then prove the existence of positive solutions with the qualitative properties such as the decay estimates and the radial symmetry.

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arXiv2025

Gevrey KAM equilibria for quasi-periodic long-range Frenkel-Kontorova models

We consider models of one-dimensional chains of non-nearest neighbor and many-body interacting particles subjected to quasi-periodic media. We extend the results in \cite{12Su&delaLlavelongrange} from analytic to Gevrey regularity potentials. More precisely, we establish an a posteriori KAM theorem showing that in the Gevrey topology, given an approximate solution of equilibrium equation, which satisfies some appropriate non-degeneracy conditions and decay property, then there is a true solution nearby and the solution preserves both the quasi-periodicity and Gevrey regularity. The method of proof is based on a combination of quasi-Newton methods and delicate estimates in spaces of Gevrey functions.

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arXiv2025

Existence and regularity of weak solutions for mixed local and nonlocal semilinear elliptic equations

We study the existence, multiplicity and regularity results of weak solutions for the Dirichlet problem of a semi-linear elliptic equation driven by the mixture of the usual Laplacian and fractional Laplacian \begin{equation*} \left\{% \begin{array}{ll} -\Delta u + (-\Delta)^{s} u+ a(x)\ u =f(x,u) & \hbox{in $\Omega$,} u=0 & \hbox{in $\mathbb{R}^n\backslash\Omega$} \end{array}% \right. \end{equation*} where $s \in (0,1)$, $\Omega \subset \mathbb{R}^{n}$ is a bounded domain, the coefficient $a$ is a function of $x$ and the subcritical nonlinearity $f(x,u)$ has superlinear growth at zero and infinity. We show the existence of a non-trivial weak solution by Linking Theorem and Mountain Pass Theorem respectively for $\lambda_{1} \leqslant 0$ and $\lambda_{1} > 0$, where $\lambda_{1}$ denotes the first eigenvalue of $-\Delta + (-\Delta)^{s} +a(x)$. In particular, adding a symmetric condition to $f$, we obtain infinitely many solutions via Fountain Theorem. Moreover, for the regularity part, we first prove the $L^{\infty}$-boundedness of weak solutions and then establish up to $C^{2, \alpha}$-regularity up to boundary.

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arXiv2025

Multiple solutions for elliptic equations driven by higher order fractional Laplacian

We consider an elliptic partial differential equation driven by higher order fractional Laplacian $(-\Delta)^{s}$, $s \in (1,2)$ with homogeneous Dirichlet boundary condition \begin{equation*} \left\{% \begin{array}{ll} (-\Delta)^{s} u=f(x,u) & \text{ in }\Omega, u=0 & \text{ in } \mathbb{R}^n \setminus \Omega. \end{array}% \right. \end{equation*} The above equation has a variational nature, and we investigate the existence and multiplicity results for its weak solutions under various conditions on the nonlinear term $f$: superlinear growth, concave-convex and symmetric conditions and their combinations. The existence of two different non-trivial weak solutions is established by Mountain Pass Theorem and Ekeland's variational principle, respectively. Furthermore, due to Fountain Theorem and its dual form, both infinitely many weak solutions with positive energy and infinitely many weak solutions with negative energy are obtained.

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arXiv2025

Concentration phenomena for a mixed local/nonlocal Schr\"{o}dinger equation with Dirichlet datum

We consider the mixed local/nonlocal semilinear equation \begin{equation*} -\epsilon^{2}\Delta u +\epsilon^{2s}(-\Delta)^s u +u=u^p\qquad \text{in } \Omega \end{equation*} with zero Dirichlet datum, where $\epsilon>0$ is a small parameter, $s\in(0,1)$, $p\in(1,\frac{n+2}{n-2})$ and $\Omega$ is a smooth, bounded domain. We construct a family of solutions that concentrate, as $\epsilon\rightarrow 0$, at an interior point of $\Omega$ having uniform distance to $\partial\Omega$ (this point can also be characterized as a local minimum of a nonlocal functional). In spite of the presence of the Laplace operator, the leading order of the relevant reduced energy functional in the Lyapunov-Schmidt procedure is polynomial rather than exponential in the distance to the boundary, in light of the nonlocal effect at infinity. A delicate analysis is required to establish some uniform estimates with respect to $\epsilon$, due to the difficulty caused by the different scales coming from the mixed operator.

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arXiv2025

Anti-integrable limits for generalized Frenkel-Kontorova models on almost-periodic media

We study the equilibrium configurations for generalized Frenkel-Kontorova models subjected to almost-periodic media. By contrast with the spirit of the KAM theory, our approach consists in establishing the other perturbation theory for fully chaotic systems far away from the integrable, which is called "anti-integrable" limits. More precisely, we show that for large enough potentials, there exists a locally unique equilibrium with any prescribed rotation number/vector/plane, which is hyperbolic. The assumptions are general enough to satisfy both short-range and long-range Frenkel-Kontorova models and their multidimensional analogues.

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arXiv2024

On some regularity properties of mixed local and nonlocal elliptic equations

This article is concerned with ``up to $C^{2, \alpha}$-regularity results'' about a mixed local-nonlocal nonlinear elliptic equation which is driven by the superposition of Laplacian and fractional Laplacian operators. First of all, an estimate on the $L^\infty$ norm of weak solutions is established for more general cases than the ones present in the literature, including here critical nonlinearities. We then prove the interior $C^{1,\alpha}$-regularity and the $C^{1,\alpha}$-regularity up to the boundary of weak solutions, which extends previous results by the authors [X. Su, E. Valdinoci, Y. Wei and J. Zhang, Math. Z. (2022)], where the nonlinearities considered were of subcritical type. In addition, we establish the interior $C^{2,\alpha}$-regularity of solutions for all $s\in(0,1)$ and the $C^{2,\alpha}$-regularity up to the boundary for all $s\in(0,\frac{1}{2})$, with sharp regularity exponents. For further perusal, we also include a strong maximum principle and some properties about the principal eigenvalue.

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arXiv2024

Qualitative properties of positive solutions of a mixed order nonlinear Schr\"{o}dinger equation

In this paper, we deal with the following mixed local/nonlocal Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{ll} - \Delta u + (-\Delta)^s u+u = u^p \quad \hbox{in $\mathbb{R}^n$,} u>0 \quad \hbox{in $\mathbb{R}^n$,} \lim\limits_{|x|\to+\infty}u(x)=0, \end{array} \right. \end{equation*} where $n\geqslant2$, $s\in (0,1)$ and $p\in\left(1,\frac{n+2}{n-2}\right)$. The existence of positive solutions for the above problem is proved, relying on some new regularity results. In addition, we study the power-type decay and the radial symmetry properties of such solutions. The methods make use also of some basic properties of the heat kernel and the Bessel kernel associated with the operator $- \Delta + (-\Delta)^s$: in this context, we provide self-contained proofs of these results based on Fourier analysis techniques.

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arXiv2024

KAM Theory for almost-periodic equilibria in one dimensional almost-periodic media

We consider one dimensional chains of interacting particles subjected to one dimensional almost-periodic media. We formulate and prove two KAM type theorems corresponding to both short-range and long-range interactions respectively. Both theorems presented have an a posteriori format and establish the existence of almost-periodic equilibria. The new part here is that the potential function is given by some almost-periodic function with infinitely many incommensurate frequencies. In both cases, we do not need to assume that the system is close to integrable. We will show that if there exists an approximate solution for the functional equations, which satisfies some appropriate non-degeneracy conditions, then a true solution nearby is obtained. This procedure may be used to validate efficient numerical computations. Moreover, to well understand the role of almost-periodic media which can be approximated by quasi-periodic ones, we present a different approach -- the step by step increase of complexity method -- to the study of the above results of the almost-periodic models.

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arXiv2024

Uniqueness and Nondegeneracy of ground states of $ -\Delta u + (-\Delta)^s u+u = u^{p+1} \quad \hbox{in $\mathbb{R}^n$}$ when $s$ is close to $0$ and $1$

We are concerned with the mixed local/nonlocal Schr\"{o}dinger equation \begin{equation} - \Delta u + (-\Delta)^s u+u = u^{p+1} \quad \hbox{in $\mathbb{R}^n$,} \end{equation} for arbitrary space dimension $n\geqslant1$, $s\in(0,1)$, and $p\in(0,2^*-2)$ with $2^*$ the critical Sobolev exponent. We provide the existence and several fundamental properties of nonnegative solutions for the above equation. And then, we prove that, if $s$ is close to $0$ and $1$, respectively, such equation then possesses a unique (up to translations) ground state, which is nondegenerate.

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arXiv2024

Singular dynamics for discrete weak K.A.M. solutions of exact twist maps

For any exact twist map $f$ and any cohomology class $c\in\mathbb{R}$, let $u_c$ be any associated discrete weak K.A.M. solution, and we introduce an inherent Lipschitz dynamics $\Sigma_+$ given by the discrete forward Lax-Oleinik semigroup. We investigate several properties of $\Sigma_+$ and show that the non-differentiable points of $u_c$ are globally propagated and forward invariant by $\Sigma_+$. In particular, such propagating dynamics possesses the same rotation number $\alpha'(c)$ as the associated Aubry-Mather set at cohomology class $c$. As applications, we provide via $\Sigma_+$ {a discrete analogue of Bernard's regularization theorem \cite{Ber07} and} a detailed exposition of Arnaud's observation \cite{Arnaud_2011}. Furthermore, we construct and analyze the corresponding dynamics on the full pseudo-graphs of discrete weak K.A.M. solutions.

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arXiv2022

Regularity results for solutions of mixed local and nonlocal elliptic equations

We consider the mixed local-nonlocal semi-linear elliptic equations driven by the superposition of Brownian and Lévy processes \begin{equation*} \left\{ \begin{array}{ll} - Δu + (-Δ)^s u = g(x,u) & \hbox{in $Ω$,} u=0 & \hbox{in $\mathbb{R}^n\backslashΩ$.} \\ \end{array} \right. \end{equation*} Under mild assumptions on the nonlinear term $g$, we show the $L^\infty$ boundedness of any weak solution (either not changing sign or sign-changing) by the Moser iteration method. Moreover, when $s\in (0, \frac{1}{2}]$, we obtain that the solution is unique and actually belongs to $C^{1,α}(\overlineΩ)$ for any $α\in (0,1)$.

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arXiv2022

Lipschitz sub-actions for locally maximal hyperbolic sets of a $C^1$ flow

Liv\v{s}ic theorem for flows asserts that a Lipschitz observable that has zero mean average along every periodic orbit is necessarily a coboundary, that is the Lie derivative of a Lipschitz function smooth along the flow direction. The positive Liv\v{s}ic theorem bounds from below the observable by such a coboundary as soon as the mean average along every periodic orbit is non negative. Previous proofs give a H\"older coboundary. Assuming that the dynamics is given by a locally maximal hyperbolic flow, we show that the coboundary can be Lipschitz. We introduce a new tool: the Lax-Oleinik semigroup, inspired by Fathi's weak KAM theory.

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arXiv2022

Point-assigned distance-like functions on non-compact geodesic spaces

On a complete, connected, locally compact, non-compact geodesic space $(X,d)$, we assign each compact set a distance-like function. With the help of these functions, we obtain a pseudo-metric on the space of (non-empty) compact subsets of $X$ which is less than the Hausdorff distance. The quotient metric space is closely related to the large scale geometry of the ambient metric space. In particular, we study both extreme cases --the pseudo-metric either vanishes or equals the original distance. The compactness of the level sets as well as stability under the Gromov-Hausdorff topology of such dl-functions are also investigated. As an application, we also give a representation formula of any distance-like function in terms of the singleton-assigned distance-like functions defined here.

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arXiv2021

Lipschitz sub-actions for locally maximal hyperbolic sets of a $C^1$ maps

Livšic theorem asserts that, for Anosov diffeomorphisms/flows, a Lipschitz observable is a coboundary if all its Birkhoff sums on every periodic orbits are equal to zero. The transfer function is then Lipschitz. We prove a positive Livšic theorem which asserts that a Lipschitz observable is bounded from below by a coboundary if and only if all its Birkhoff sums on periodic orbits are non negative. The new result is that the coboundary can be chosen Lipschitz. The map is only assumed to be $C^1$ and hyperbolic, but not necessarily bijective nor transitive. We actually prove our main result in the setting of locally maximal hyperbolic sets for not general $C^1$ map. The construction of the coboundary uses a new notion of the Lax-Oleinik operator that is a standard tool in the discrete Aubry-Mather theory.

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arXiv2021

Essential forward weak KAM solution for the convex Hamilton-Jacobi equation

For a convex, coercive continuous Hamiltonian on a compact closed Riemannian manifold $M$, we construct a unique forward weak KAM solution of \[ H(x, d_x u)=c(H) \] by a vanishing discount approach, where $c(H)$ is the Mañé critical value. We also discuss the dynamical significance of such a special solution.

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