Search

In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine mesh and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.

This paper deals with the pulsating waves and entire solutions for a spatially periodic nonlocal dispersal model with a quiescent stage. By the method of super- and subsolutions together with the comparison principle, we establish the existence of pulsating waves with any speed larger than the spreading speed. The construction of the super- and subsolutions depends on the principal eigenvalue theory for a periodic eigenvalue problem with partially degenerate nonlocal dispersal. The joint results of this paper and our recent work (Wang J B, Li W T, Sun J W. Global dynamics and spreading speeds for a partially degenerate system with non-local dispersal in periodic habitats. Proc Roy Soc Edinburgh Sect A, 2018, 148: 849–880) show that the spreading speed coincides with the minimal wave speed of pulsating waves for the considered system. Moreover, combining the rightward and leftward pulsating waves with different speeds and a spatially periodic solution, we prove the existence and qualitative properties of entire solutions other than pulsating waves, which provide some new spreading ways in a heterogeneous habitat.

Nonconforming grids with hanging nodes are frequently used in adaptive finite element computations. In all earlier works on such methods, proper constraints should be enforced on degrees of freedom on edges/faces with hanging nodes to keep continuity, which yield numerical computations much complicated. In 2014, Zhao et al. (2014) presented quadrilateral constraint-free finite element methods on quadrilateral grids with hanging nodes. This paper further develops a hexahedral constraint-free finite element method on hexahedral grids with hanging nodes, which is of greater challenge than the two-dimensional case. Residual-based a posteriori error reliability and efficiency are also established in this paper.

In this paper, we find all positive squarefree integers d such that the Pell equation X2-dY2 = +-1 has at least two positive integer solutions (X,Y) and (X',Y') such that both X and X' have Zeckendorf representations with at most two terms. This paper has been accepted for publication in SCIENCE CHINA Mathematics.

This paper is concerned with the time-step condition of linearized implicit finite difference method for solving the Gross-Pitaevskii equation with an angular momentum rotation term. Unlike the existing studies in the literature, where the cut-off function technique was used to establish the error estimates under some conditions of the time-step size, this paper introduces an induction argument and a ‘lifting’ technique as well as some useful inequalities to build the optimal maximum error estimate without any constraints on the time-step size. The analysis method can be directly extended to the general nonlinear Schrodinger-type equations in twoand three-dimensions and other linear implicit finite difference schemes. As a by-product, this paper defines a new type of energy functional of the grid functions by using a recursive relation to prove that the proposed scheme preserves well the total mass and energy in the discrete sense. Several numerical results are reported to verify the error estimates and conservation laws.

In this paper, we obtain the necessary and sufficient condition of the pre-compact sets in the variable exponent Lebesgue spaces, which is also called the Riesz-Kolmogorov theorem. The main novelty appearing in this approach is the constructive approximation which does not rely on the boundedness of the Hardy-Littlewood maximal operator in the considered spaces such that we do not need the log-Holder continuous conditions on the variable exponent. As applications, we establish the boundedness of Riemann-Liouville integral operators and prove the compactness of truncated Riemann-Liouville integral operators in the variable exponent Lebesgue spaces. Moreover, applying the Riesz-Kolmogorov theorem established in this paper, we obtain the existence and the uniqueness of solutions to a Cauchy type problem for fractional differential equations in variable exponent Lebesgue spaces.

Let A,K be positive integers and u=-2,-1,1 or 2. The main contribution of the paper is a proof that each of the D(u^2)-triples {K,A^2K+2uA,(A+1)^2K+2u(A+1)} has unique extension to a D(u^2)-quadruple., This paper has 20 pages and has been accepted for publication in SCIENCE CHINA Mathematics

This paper is devoted to the existence of the traveling waves of the equations describing a diffusive susceptible-exposed-infected-recovered (SEIR) model. The existence of traveling waves depends on the basic reproduction rate and the minimal wave speed. We obtain a more precise estimation of the minimal wave speed of the epidemic model, which is of great practical value in the control of serious epidemics. The approach in this paper is to use the Schauder fixed point theorem and the Laplace transform. We also give some numerical results on the minimal wave speed.

This paper concerns with the existence of solutions for the following fractional Kirchhoff problem with critical nonlinearity: $${\left( {\int {\int {_{{\mathbb{R}^{2N}}}\frac{{{{\left| {u\left( x \right) - u\left( y \right)} \right|}^2}}}{{{{\left| {x - y} \right|}^{N + 2s}}}}dxdy} } } \right)^{\theta - 1}}{\left( { - \Delta } \right)^s}u = \lambda h\left( x \right){u^{p - 1}} + {u^{2_s^* - 1}} in {\mathbb{R}^N},$$ where (−Δ) s is the fractional Laplacian operator with 0 < s < 1, 2 s * = 2N/(N − 2s), N > 2s, p ∈ (1, 2 s *), θ ∈ [1, 2 s */2), h is a nonnegative function and λ a real positive parameter. Using the Ekeland variational principle and the mountain pass theorem, we obtain the existence and multiplicity of solutions for the above problem for suitable parameter λ > 0. Furthermore, under some appropriate assumptions, our result can be extended to the setting of a class of nonlocal integro-differential equations. The remarkable feature of this paper is the fact that the coefficient of fractional Laplace operator could be zero at zero, which implies that the above Kirchhoff problem is degenerate. Hence our results are new even in the Laplacian case.

In this paper, we consider a nonlinear integral eigenvalue problem, which is a reformulation of the transmission eigenvalue problem arising in the inverse scattering theory. The boundary element method is employed for discretization, which leads to a generalized matrix eigenvalue problem. We propose a novel method based on the spectral projection. The method probes a given region on the complex plane using contour integrals and decides if the region contains eigenvalue(s) or not. It is particularly suitable to test if zero is an eigenvalue of the generalized eigenvalue problem, which in turn implies that the associated wavenumber is a transmission eigenvalue. Effectiveness and efficiency of the new method are demonstrated by numerical examples., Comment: The paper has been accepted for publication in SCIENCE CHINA Mathematics

The classical critical Trudinger-Moser inequality in ℝ2 under the constraint $\int_{\mathbb{R}{^2}} {\left( {{{\left| {\nabla u} \right|}^2} + {{\left| u \right|}^2}} \right)dx\;} \leqslant \;1$ was established through the technique of blow-up analysis or the rearrangement-free argument: for any τ > 0, it holds that $$\mathop {\sup }\limits_{\matrix{{u \in {H^1}\left( {{\mathbb{R}^2}} \right)} \cr {\int_{{\mathbb{R}^2}} {\left( {\tau {{\left| u \right|}^2} + {{\left| {{\nabla u}} \right|}^2}} \right)dx} \leqslant 1} \cr } } \int_{\mathbb{R}{^2}} {\left( {{\rm{e}^{4\pi }}^{{{\left| u \right|}^2}} - 1} \right)dx} \;\leqslant \;C\left( \tau \right) < + \infty,$$ and 4π is sharp. However, if we consider the less restrictive constraint $\int_{\mathbb{R}{^2}} {\left( {{{\left| {\nabla u} \right|}^2} + V\left( x \right){u^2}} \right)dx\;} \leqslant \;1$ , where V(x) is nonnegative and vanishes on an open set in ℝ2, it is unknown whether the sharp constant of the Trudinger-Moser inequality is still 4π. The loss of a positive lower bound of the potential V(x) makes this problem become fairly nontrivial. The main purpose of this paper is two-fold. We will first establish the Trudinger-Moser inequality $$\mathop {\sup }\limits_{u \in {H^1}\left( {\mathbb{R}{^2}} \right),\;\int_{\mathbb{R}{^2}} {\left( {{{\left| {\nabla u} \right|}^2} + V\left( x \right){u^2}} \right)dx} \leqslant 1} \int_{\mathbb{R}{^2}} {\left( {{\rm{e}^{4\pi {u^2}}} - 1} \right)dx} \;\leqslant \;C\left( V \right) < \infty,$$ when V is nonnegative and vanishes on an open set in ℝ2. As an application, we also prove the existence of ground state solutions to the following Schrodinger equations with critical exponential growth (0.1) $$ - {\rm{\Delta }} u + V\left( x \right)u = f\left( u \right)\;\;\;\;{\rm{in}}\;\mathbb{R}{^2},$$ where V(x) ⩾ 0 and vanishes on an open set of ℝ2 and f has critical exponential growth. Having the positive constant lower bound for the potential V(x) (e.g., the Rabinowitz type potential) has been the standard assumption when one deals with the existence of solutions to the above Schroodinger equations when the nonlinear term has the exponential growth. Our existence result seems to be the first one without this standard assumption.

In this paper, we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schrodinger equation $$\left\{ {\matrix{ { - {\rm{\Delta }}u + V\left( x \right)u = {\mu _q}u + a{{\left| u \right|}^q}u\,\,\,{\rm{in}}\,{\mathbb{R}^2},} \hfill \cr {\int_{{\mathbb{R}^2}} {{{\left| u \right|}^2}dx = 1,} } \hfill \cr } } \right.$$ where μq is the Lagrange multiplier. We show that for q > 2 close to 2, the problem admits two solutions: one is the local minimal solution uq and the other one is the mountain pass solution υq. Furthermore, we study the limiting behavior of uq and υq when q → 2+. Particularly, we describe precisely the blow-up formation of the excited state υq.

This paper continues to derive the globally hyperbolic moment model of arbitrary order for the three-dimensional special relativistic Boltzmann equation with the Anderson-Witting collision. The method is the model reduction by the operator projection. Finding an orthogonal basis of the weighted polynomial space is crucial and built on infinite families of the complicate relativistic Grad type orthogonal polynomials depending on a parameter and the real spherical harmonics instead of the irreducible tensors. We study the properties of those functions carefully, including their recurrence relations, their derivatives with respect to the independent variable and parameter, and the zeros of the orthogonal polynomials. Our moment model is proved to be globally hyperbolic and linearly stable. Moreover, the Lorentz-covariance and the quasi-one-dimensional case, the non-relativistic and ultra-relativistic limits are also studied.

Unique expansions in non-integer bases have been investigated in many papers during the last thirty years. They are often conveniently generated by labeled directed graphs. We give a precise description of the set of sequences generated by these graphs. This provides a geometric explanation of many former abstract results in this domain. Our results are illustrated by many examples.

In this paper, a topological classification of piecewise monotone functions (abbreviated as PM functions) with nonmonotonicity height equal to 1 which are strictly increasing on their characteristic intervals and have finitely many fixed points is presented.

In this paper, we apply local discontinuous Galerkin methods to the pattern formation dynamical model in polymerizing action flocks. Optimal error estimates for the density and filament polarization in different norms are established. We use a semi-implicit spectral deferred correction time method for time discretization, which allows a relative large time step and avoids computation of a Jacobian matrix. Numerical experiments are presented to verify the theoretical analysis and to show the capability for simulations of action wave formation.

This paper is concerned with the well-posedness and large-time behavior of a two-dimensional PDE-ODE hybrid chemotaxis system describing the initiation of tumor angiogenesis. We first transform the system via a Cole-Hopf type transformation into a parabolic-hyperbolic system and then show that the solution of the transformed system converges to a constant equilibrium state as time tends to infinity. Finally we reverse the Cole-Hopf transformation and obtain the relevant results for the pre-transformed PDE-ODE hybrid system. In contrast to the existing related results, where continuous initial data is imposed, we are able to prove the asymptotic stability for discontinuous initial data with large oscillations. The key ingredient in our proof is the use of the so-called “effective viscous flux”, which enables us to obtain the desired energy estimates and regularity. The technique of the “effective viscous flux” turns out to be a very useful tool to study chemotaxis systems with initial data having low regularity and was rarely (if not) used in the literature for chemotaxis models.

A general analysis framework is presented in this paper for many different types of finite element methods (including various discontinuous Galerkin methods). For the second-order elliptic equation −div(α∇u) = f, this framework employs four different discretization variables, uh, ph, ŭh and $${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over p} _h}$$ , where uh and ph are for approximation of u and p = −α∇u inside each element, and ŭh and $${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over p} _h}$$ are for approximation of residual of u and p · n on the boundary of each element. The resulting 4-field discretization is proved to satisfy two types of inf-sup conditions that are uniform with respect to all discretization and penalization parameters. As a result, many existing finite element and discontinuous Galerkin methods can be analyzed using this general framework by making appropriate choices of discretization spaces and penalization parameters.

In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials. Our argument is to establish some Hardy-Rellich type inequalities with boundary terms and introduce an Almgren’s type frequency function to show some doubling conditions for the solutions to the above-mentioned equations.

The present paper is concerned with a class of quasi-linear elliptic degenerate equations. The degenerate operator arises from analysis of manifolds with singularities. The variational methods are applied here to verify the existence of infinitely many solutions for the problem.

Fractional operators are widely used in mathematical models describing abnormal and nonlocal phenomena. Although there are extensive numerical methods for solving the corresponding model problems, theoretical analysis such as the regularity result, or the relationship between the left-side and right-side fractional operators is seldom mentioned. Instead of considering the fractional derivative spaces, this paper starts from discussing the image spaces of Riemann-Liouville fractional integrals of LP(Ω) functions, since the fractional derivative operators that are often used are all pseudo-differential. Then the high regularity situation—the image spaces of Riemann-Liouville fractional integral operators on the Wm,p(Ω) space is considered. Equivalent characterizations of the defined spaces, as well as those of the intersection of the left-side and right-side spaces are given. The behavior of the functions in the defined spaces at both the nearby boundary point/points and the points in the domain is demonstrated in a clear way. Besides, tempered fractional operators are shown to be reciprocal to the corresponding Riemann-Liouville fractional operators, which is expected to contribute some theoretical support for relevant numerical methods. Last, we also provide some instructions on how to take advantage of the introduced spaces when numerically solving fractional equations.

This paper deals with numerical solutions of nonlinear stiff stochastic differential equations with jump-diffusion and piecewise continuous arguments. By combining compensated split-step methods and balanced methods, a class of compensated split-step balanced (CSSB) methods are suggested for solving the equations. Based on the one-sided Lipschitz condition and local Lipschitz condition, a strong convergence criterion of CSSB methods is derived. It is proved under some suitable conditions that the numerical solutions produced by CSSB methods can preserve the mean-square exponential stability of the corresponding analytical solutions. Several numerical examples are presented to illustrate the obtained theoretical results and the effectiveness of CSSB methods. Moreover, in order to show the computational advantage of CSSB methods, we also give a numerical comparison with the adapted split-step backward Euler methods with or without compensation and tamed explicit methods.

This article is concerned with the well-posedness as well as long-term dynamics of a wide class of non-autonomous, non-local, fractional, stochastic FitzHugh-Nagumo systems driven by nonlinear noise defined on the entire spaceℝRN. The well-posedness is proved for the systems with polynomial drift terms of arbitrary order as well as locally Lipschitz nonlinear diffusion terms by utilizing the pathwise and mean square uniform estimates. The mean random dynamical system generated by the solution operators is proved to possess a unique weak pullback mean random attractor in a Bochner space. The existence of invariant measures is also established for the autonomous systems with globally Lipschitz continuous diffusion terms. The idea of uniform tail-estimates of the solutions in the appropriate spaces is employed to derive the tightness of a family of probability distributions of the solutions in order to overcome the non-compactness of the standard Sobolev embeddings on ℝN as well as the lack of smoothing effect on one component of the solutions. The results of this paper are new even when the fractional Laplacian is replaced by the standard Laplacian.

In this paper, we investigate the smoothness of invariant manifolds and foliations for random dynamical systems with nonuniform pseudo-hyperbolicity in Hilbert spaces. We discuss on the effect of temperedness and the spectral gaps in the nonuniform pseudo-hyperbolicity so as to prove the existence of invariant manifolds and invariant foliations, which preserve the CN,τ(ω) HOlder smoothness of the random system in the space variable and the measurability of the random system in the sample point. Moreover, we also prove that the stable foliation is CN−1,τ(ω) in the base point.

Let ( $${\cal X}$$ , d,μ) be a non-homogeneous metric measure space satisfying the so-called upper doubling and geometrically doubling conditions, which includes the space of homogeneous type and the Euclidean space with the non-doubling measure as special cases. Let T be a multilinear Calderon-Zygmund operator and $$\vec b: = ({b_1}, \ldots ,{b_m})$$ be a finite family of $$\widetilde{{\rm{RBMO}}}(\mu)$$ functions. In this paper, some weak-type multiple weighted estimates for the iterated commutator $${{\rm{T}}_{\Pi \vec b}}$$ generated by T and $$\vec b$$ are obtained.

In this paper we are concerned with the global regularity of solutions to the Dirichlet problem for a class of Monge-Ampere type equations. By employing the concept of (a, η) type domain, we emphasize that the boundary regularity depends on the convexity of the domain in nature. The key idea of our proof is to provide more effective global Holder estimates of convex solutions to the problem based on carefully choosing auxiliary functions and constructing sub-solutions.

In this paper, we obtain a necessary and sufficient condition for a $U(n)$-invariant complex Finsler metric $F$ on domains in $\mathbb{C}^n$ to be strongly convex, which also makes it possible to investigate relationship between real and complex Finsler geometry via concrete and computable examples. We prove a rigid theorem which states that a $U(n)$-invariant strongly convex complex Finsler metric $F$ is a real Berwald metric if and only if $F$ comes from a $U(n)$-invariant Hermitian metric. We give a characterization of $U(n)$-invariant weakly complex Berwald metrics with vanishing holomorphic sectional curvature and obtain an explicit formula for holomorphic curvature of $U(n)$-invariant strongly pseudoconvex complex Finsler metric. Finally, we prove that the real geodesics of some $U(n)$-invariant complex Finsler metric restricted on the unit sphere $\pmb{S}^{2n-1}\subset\mathbb{C}^n$ share a specific property as that of the complex Wrona metric on $\mathbb{C}^n$.cc, 22 pages, has been accepted for publication in SCIENCE CHINA Mathematics

In this paper we study the nonhomogeneous semilinear fractional Schrodinger equation with critical growth $$\left\{ {\begin{array}{*{20}{c}} {{{( - \Delta )}^s}u + u = {u^{2_s^* - 1}} + \lambda (f(x,u)) + h(x)),}&{x \in {\mathbb{R}^N},} \\ {u \in {H^s}({\mathbb{R}^N}),\;\;\;u(x) > 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&{x \in {\mathbb{R}^N},} \end{array}} \right.$$ where s ∈ (0, 1), N > 4s, and λ > 0 is a parameter, $$2_s^* = {{2N} \over {N - 2s}}$$ is the fractional critical Sobolev exponent, f and h are some given functions. We show that there exists 0 λ*, a unique solution (λ*,uλ*) if λ = λ*, which shows that (λ*,uλ*) is a turning point in HS(ℝN) for the problem. Our proofs are based on the variational methods and the principle of concentration-compactness.

In this paper we put forward a viral propagation model with a nonlinear infection rate and free boundaries and investigate the dynamical properties. This model is composed of two ordinary differential equations and one partial differential equation, in which the spatial range of the first equation is the whole space ℝ, and the last two equations have free boundaries. As a new mathematical model, we prove the existence, uniqueness and uniform estimates of the global solution, and provide the criteria for spreading and vanishing, and the long time behavior of the solution components u, v and w. Comparing this model with the corresponding ordinary differential systems, the basic reproduction number $${{\cal R}_0}$$ plays a different role. We find that when $${{\cal R}_0} \le 1$$, the virus cannot spread successfully; when $${{\cal R}_0} > 1$$, the successful spread of virus depends on the initial value and varying parameters.

In this paper, numerical analysis is carried out for a class of history-dependent variational-hemivariational inequalities arising in contact problems. Three different numerical treatments for temporal discretization are proposed to approximate the continuous model. Fixed-point iteration algorithms are employed to implement the implicit scheme and the convergence is proved with a convergence rate independent of the time step-size and mesh grid-size. A special temporal discretization is introduced for the history-dependent operator, leading to numerical schemes for which the unique solvability and error bounds for the temporally discrete systems can be proved without any restriction on the time step-size. As for spatial approximation, the finite element method is applied and an optimal order error estimate for the linear element solutions is provided under appropriate regularity assumptions. Numerical examples are presented to illustrate the theoretical results., 33 pages,6 figures

In this paper we consider iteration of single-plateau functions, an important class of continuous functions with infinitely many forts, and investigate changes of number and length of plateaux under iteration. We use the indices flatness, plateau limit and limit length to formulate those changes. Furthermore, we compute flatness, plateau limit and limit length for all nine types of single-plateau functions.

This paper studies limit behaviors of stationary measures for stochastic ordinary differential equations with nondegenerate noise and presents a criterion to guarantee that a repeller with zero Lebesgue measure is a null set of any limit measure. Using this criterion, we first provide a series of nontrivial concrete examples to show that their repelling limit cycles or quasi-periodic orbits are null sets for all limit measures, which deduces that all their limit measures are concentrated on stable equilibria and stable limit cycles or quasi-periodic orbits, and saddles. Interesting open questions on exact supports of limit measures are proposed.

In this paper we study some extremal problems for the family $S_g^0(\mathbb{B}_X)$ of normalized univalent mappings with $g$-parametric representation on the unit ball $\mathbb{B}_X$ of an $n$-dimensional JB$^*$-triple $X$ with $r\geq 2$, where $r$ is the rank of $X$ and $g$ is a convex (univalent) function on the unit disc $\mathbb{U}$, which satisfies some natural assumptions. We obtain sharp coefficient bounds for the family $S_g^0(\mathbb{B}_X)$, and examples of bounded support points for various subsets of $S_g^0(\mathbb{B}_X)$. Our results are generalizations to bounded symmetric domains of known recent results related to support points for families of univalent mappings on the Euclidean unit ball $\mathbb{B}^n$ and the unit polydisc $\mathbb{U}^n$ in $\mathbb{C}^n$. Certain questions will be also mentioned. Finally, we point out sharp coefficient bounds and bounded support points for the family $S_g^0(\mathbb{B}^n)$ and for special compact subsets of $S_g^0(\mathbb{B}^n)$, in the case $n\geq 2$., SCIENCE CHINA Mathematics

In this paper, new Levin methods are presented for calculating oscillatory integrals with algebraic and/or logarithmic singularities. To avoid singularity, the technique of singularity separation is applied and then the singular ODE occurring in classic Levin methods is converted into two kinds of non-singular ODEs. The solutions of one can be obtained explicitly, while those of the other can be solved efficiently by collocation methods. The proposed methods can attach arbitrarily high asymptotic orders and also enjoy superalgebraic convergence with respect to the number of collocation points. Several numerical experiments are presented to validate the efficiency of the proposed methods., Accepted for publication in SCIENCE CHINA Mathematics

In this paper, we consider the {\it tensor absolute value equations} (TAVEs), which is a newly introduced problem in the context of multilinear systems. Although the system of TAVEs is an interesting generalization of matrix {\it absolute value equations} (AVEs), the well-developed theory and algorithms for AVEs are not directly applicable to TAVEs due to the nonlinearity (or multilinearity) of the problem under consideration. Therefore, we first study the solutions existence of some classes of TAVEs with the help of degree theory, in addition to showing, by fixed point theory, that the system of TAVEs has at least one solution under some checkable conditions. Then, we give a bound of solutions of TAVEs for some special cases. To find a solution to TAVEs, we employ the generalized Newton method and report some preliminary results., Comment: 26 pages and 4 tables

In this paper, we first present a Gearhardt-Pr\"uss type theorem with a sharp bound for m-accretive operators. Then we give two applications: (1) give a simple proof of the result proved by Constantin et al. on relaxation enhancement induced by incompressible flows; (2) show that shear flows with a class of Weierstrass functions obey logarithmically fast dissipation time-scales., Comment: 12 pages

In this paper, we establish a relationship between the Morse index at rest points in the saddle point reduction and the brake-orbit-type Maslov index at corresponding brake orbits. As an application, we give a criterion to find brake orbits which are contractible and start at $$\left\{0 \right\} \times \mathbb{T}{^n} \subset {\mathbb{T}^{2n}}$$ for even Hamiltonian on $${\mathbb{T}^{2n}}$$ by the methods of the Maslov-index theory and a critical point theorem formulated by Bartsch and Wang (1997). Explicitly, if all trivial solutions of a Hamiltonian are nondegenerate in the brake orbit boundary case, there are at least max{iL0 (z0)} pairs of nontrivial 1-periodic brake orbits if iL0 (z0) > 0 or at least max{−iL0 (z0) − n} pairs of nontrivial 1-periodic brake orbits if iL0 (z0) < −n. In the end, we give an example to find brake orbits for certain Hamiltonian via this criterion.

In this paper, we study the integrals of the Ricci curvature over metric balls in a Finsler manifold, which can be viewed as an Lq-norm of the Ricci curvature. By bounding such integrals from above, we obtain several Myers type theorems.

In this paper, we study locally strongly convex affine hyperspheres in the unimodular affine space ℝn+1 which, as Riemannian manifolds, are locally isometric to the Riemannian product of two Riemannian manifolds both possessing constant sectional curvature. As the main result, a complete classification of such affine hyperspheres is established. Moreover, as direct consequences, 3- and 4-dimensional affine hyperspheres with parallel Ricci tensor are also classified.

In this paper, we consider the following fractional Schrodinger-Poisson problem, $$\begin{cases}\epsilon^{2s}(-\Delta)^{s}u+V(x)u+\phi{u}=\mid{u}\mid^{p-1}u, & x\in\mathbb{R}^N,\\(-\Delta)^t\phi=u^2, & x\in\mathbb{R}^N,\end{cases}$$ where e > 0 is a small parameter, N ⩾ 3 and V(x) is a potential function. We construct non-radial sign-changing solutions, whose components may have spikes clustering at the local minimum point of V(x).

In this paper, we consider an initial-boundary value problem for the 2D incompressible magneto-micropolar fluid equations with zero magnetic diffusion and zero spin viscosity in the horizontally infinite flat layer with Navier-type boundary conditions. We establish the global well-posedness of strong solutions around the equilibrium $(0,e_1,0)$.

In this paper, we prove the existence of quasi-periodic solutions and boundeness of all solutions of the general semilinear quasi-periodic differential equation x″ + ax+ − bx− = Gx(x, t) + f(t), where x+ = max{x, 0}, x− = max{ − x, 0}, a and b are two different positive constants, f(t) is $${{\cal C}^{39}}$$ smooth in t, G(x, t) is $${{\cal C}^{35}}$$ smooth in x and t, f (t) and G(x, t) are quasi-periodic in t with the Diophantine frequency ω = (ω1, ω2), and $$D_x^iD_t^jG\left({x,\;t} \right)$$ is bounded for 0 ⩽ i + j ⩽ 35.

In this paper, we prove the local well-posedness of the viscous surface wave equation in low regularity Sobolev spaces. The key points are to establish several new Stokes estimates depending only on the optimal boundary regularity and to construct a new iteration scheme on a known moving domain. Our method could be applied to some other fluid models with free boundaries.

For equations of order two with the Dirichlet boundary condition, as the Laplace problem, the Stokes and the Navier-Stokes systems, perforated domains were only studied when the distance between the holes $d_{\varepsilon}$ is equal or much larger than the size of the holes $\varepsilon$. Such a diluted porous medium is interesting because it contains some cases where we have a non-negligible effect on the solution when $(\varepsilon,d_{\varepsilon})\to (0,0)$. Smaller distance was avoided for mathematical reasons and for theses large distances, the geometry of the holes does not affect -- or few -- the asymptotic result. Very recently, it was shown for the 2D-Euler equations that a porous medium is non-negligible only for inter-holes distance much smaller than the size of the holes. For this result, the regularity of holes boundary plays a crucial role, and the permeability criterium depends on the geometry of the lateral boundary. In this paper, we relax slightly the regularity condition, allowing a corner, and we note that a line of irregular obstacles cannot slow down a perfect fluid in any regime such that $\varepsilon \ln d_{\varepsilon} \to 0$., Dedicated to Professor Jean-Yves Chemin on the Occasion of his 60th Birthday. Accepted for publication in SCIENCE CHINA Mathematics

In this paper, we prove Poincaré and Sobolev inequalities for differential forms in L1(ℝn). The singular integral estimates that it is possible to use for Lp, p > 1, are replaced here with inequalities which go back to Bourgain and Brezis (2007).

In this paper, we study the effect of lower orderperturbations in the existence of positive solutions to thefractional Kirchhoff equation with critical growth where $a,b>0$ areconstants, $\\mu>0$ is a parameter, $s\\in~(\\frac{3}{4},1)$,$1

This paper is concerned withexact boundary behavior of large solutions to semilinear elliptic equations$\Delta~u~=b(x)f(u)(C_0+|\nabla~u|^q)$, $x\in~\xO$, where$\Omega$ is a bounded domain with a smooth boundaryin $\mathbb~R^N$,$C_0\geq~0$, $q\in~[0,~2)$, $b~\in~C^{\xa}_{\rm~loc}({\xO})$ ispositive in $\Omega$, but may be vanishing or appropriatelysingular on the boundary, $f\in~C[0,~\infty)$, $f(0)=0$, and $f$ isstrictly increasing on $[0,~\infty)$ (or $f\in~C(\mathbb~R)$,$f(s)>0$, $\forall~\,~s\in~\mathbb~R,$ $f$ is strictly increasing on$\mathbb~R$). We show unified boundary behavior of such solutions to the problem under a new structure condition on $f$.

The combinatorial Mandelbrot set is a continuum in the plane, whose boundary can be defined, up to a homeomorphism, as the quotient space of the unit circle by an explicit equivalence relation. This equivalence relation was described by Douady and, in different terms, by Thurston. Thurston used quadratic invariant laminations as a major tool. As has been previously shown by the authors, the combinatorial Mandelbrot set can be interpreted as a quotient of the space of all limit quadratic invariant laminations. The topology in the space of laminations is defined by the Hausdorff distance. In this paper, we describe two similar quotients. In the first case, the identifications are the same but the space is smaller than that taken for the Mandelbrot set. The result (the quotient space) is obtained from the Mandelbrot set by "unpinching" the transitions between adjacent hyperbolic components. In the second case, we do not identify non-renormalizable laminations while identifying renormalizable laminations according to which hyperbolic lamination they can be "unrenormalised" to., Comment: 29 pages, 4 figures

This paper is dedicated to studying the following elliptic system of Hamiltonian type: $$\begin{cases}-\varepsilon^2\Delta{u}+u+V(x)v=Q(x)F_v{(u,v)}, & x \in \mathbb{R}^N,\\-\varepsilon^2\Delta{v}+v+V(x)u=Q(x)F_u{(u,v)}, & x \in \mathbb{R}^N,\\|u(x)|+|v(x)|\rightarrow0, & as |x|\rightarrow \infty\end{cases}$$ where N ⩾ 3, V, $$Q\in\mathcal{C}(\mathbb{R}^N,\mathbb{R})$$, V (x) is allowed to be sign-changing and inf Q > 0, and $$F\in\mathcal{C}^1(\mathbb{R}^2,\mathbb{R})$$ is superquadratic at both 0 and infinity but subcritical. Instead of the reduction approach used in Ding et al. (2014), we develop a more direct approach—non-Nehari manifold approach to obtain stronger conclusions but under weaker assumptions than those in Ding et al. (2014). We can find an e0 > 0 which is determined by terms of N, V, Q and F, and then we prove the existence of a ground state solution of Nehari-Pankov type to the coupled system for all e ∈ (0, e0].

In this paper we develop the theory of variable exponent Hardy spaces associated with discrete Laplacians on infinite graphs. Our Hardy spaces are defined by square integrals, atomic and molecular decompositions. Also we study boundedness properties of Littlewood-Paley functions, Riesz transforms, and spectral multipliers for discrete Laplacians on variable exponent Hardy spaces.