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Discrete & Continuous Dynamical Systems - A

April 2019 , Volume 39 , Issue 4

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On the global well-posedness of 3-D Boussinesq system with partial viscosity and axisymmetric data
Saoussen Sokrani
2019, 39(4): 1613-1650 doi: 10.3934/dcds.2019072 +[Abstract](763) +[HTML](108) +[PDF](584.38KB)
Abstract:

In this paper we prove the global well-posedness for the Three dimensional Boussinesq system with axisymmetric initial data. This system couples the Navier-Stokes equation with vanishing the horizontal viscosity with a transport-diffusion equation governing the temperature.

Asymptotic limit and decay estimates for a class of dissipative linear hyperbolic systems in several dimensions
Thinh Tien Nguyen
2019, 39(4): 1651-1684 doi: 10.3934/dcds.2019073 +[Abstract](607) +[HTML](69) +[PDF](607.49KB)
Abstract:

In this paper, we study the large-time behavior of solutions to a class of partially dissipative linear hyperbolic systems with applications in, for instance, the velocity-jump processes in several dimensions. Given integers \begin{document}$ n,d\ge 1 $\end{document}, let \begin{document}$ \mathbf A: = (A^1,\dots,A^d)\in (\mathbb R^{n\times n})^d $\end{document} be a matrix-vector and let \begin{document}$ B\in \mathbb R^{n\times n} $\end{document} be not necessarily symmetric but have one single eigenvalue zero, we consider the Cauchy problem for \begin{document}$ n\times n $\end{document} linear systems having the form

Under appropriate assumptions, we show that the solution \begin{document}$ u $\end{document} is decomposed into \begin{document}$ u = u^{(1)}+u^{(2)} $\end{document} such that the asymptotic profile of \begin{document}$ u^{(1)} $\end{document} denoted by \begin{document}$ U $\end{document} is a solution to a parabolic equation, \begin{document}$ u^{(1)}-U $\end{document} decays at the rate \begin{document}$ t^{-\frac d2(\frac 1q-\frac 1p)-\frac 12} $\end{document} as \begin{document}$ t\to +\infty $\end{document} in any \begin{document}$ L^p $\end{document}-norm and \begin{document}$ u^{(2)} $\end{document} decays exponentially in \begin{document}$ L^2 $\end{document}-norm, provided \begin{document}$ u(\cdot,0)\in L^q(\mathbb R^d)\cap L^2(\mathbb R^d) $\end{document} for \begin{document}$ 1\le q\le p\le \infty $\end{document}. Moreover, \begin{document}$ u^{(1)}-U $\end{document} decays at the optimal rate \begin{document}$ t^{-\frac d2(\frac 1q-\frac 1p)-1} $\end{document} as \begin{document}$ t\to +\infty $\end{document} if the original system satisfies a symmetry property. The main proofs are based on the asymptotic expansions of the fundamental solution in the frequency space and the Fourier analysis.

Riccati equations for linear Hamiltonian systems without controllability condition
Peter Šepitka
2019, 39(4): 1685-1730 doi: 10.3934/dcds.2019074 +[Abstract](590) +[HTML](85) +[PDF](830.28KB)
Abstract:

In this paper we develop new theory of Riccati matrix differential equations for linear Hamiltonian systems, which do not require any controllability assumption. When the system is nonoscillatory, it is known from our previous work that conjoined bases of the system with eventually the same image form a special structure called a genus. We show that for every such a genus there is an associated Riccati equation. We study the properties of symmetric solutions of these Riccati equations and their connection with conjoined bases of the system. For a given genus, we pay a special attention to distinguished solutions at infinity of the associated Riccati equation and their relationship with the principal solutions at infinity of the system in the considered genus. We show the uniqueness of the distinguished solution at infinity of the Riccati equation corresponding to the minimal genus. This study essentially extends and completes the work of W. T. Reid (1964, 1972), W. A. Coppel (1971), P. Hartman (1964), W. Kratz (1995), and other authors who considered the Riccati equation and its distinguished solution at infinity for invertible conjoined bases, i.e., for the maximal genus in our setting.

Entropy rigidity and Hilbert volume
Ilesanmi Adeboye, Harrison Bray and David Constantine
2019, 39(4): 1731-1744 doi: 10.3934/dcds.2019075 +[Abstract](545) +[HTML](56) +[PDF](406.32KB)
Abstract:

For a closed, strictly convex projective manifold that admits a hyperbolic structure, we show that the ratio of Hilbert volume to hyperbolic volume is bounded below by a constant that depends only on dimension. We also show that for such spaces, if topological entropy of the geodesic flow goes to zero, the volume must go to infinity. These results follow from adapting Besson–Courtois–Gallot's entropy rigidity result to Hilbert geometries.

Vortex structures for some geometric flows from pseudo-Euclidean spaces
Ruiqi Jiang, Youde Wang and Jun Yang
2019, 39(4): 1745-1777 doi: 10.3934/dcds.2019076 +[Abstract](386) +[HTML](55) +[PDF](615.66KB)
Abstract:

For some geometric flows (such as wave map equations, Schrödinger flows) from pseudo-Euclidean spaces to a unit sphere contained in a three-dimensional Euclidean space, we construct solutions with various vortex structures (vortex pairs, elliptic/hyperbolic vortex circles, and also elliptic vortex helices). The approaches base on the transformations associated with the symmetries of the nonlinear problems, which will lead to two-dimensional elliptic problems with resolution theory given by the finite-dimensional Lyapunov-Schmidt reduction method in nonlinear analysis.

Spectra of expanding maps on Besov spaces
Yushi Nakano and Shota Sakamoto
2019, 39(4): 1779-1797 doi: 10.3934/dcds.2019077 +[Abstract](479) +[HTML](43) +[PDF](516.25KB)
Abstract:

A typical approach to analysing statistical properties of expanding maps is to show spectral gaps of associated transfer operators in adapted function spaces. The classical function spaces for this purpose are Hölder spaces and Sobolev spaces. Natural generalisations of these spaces are Besov spaces, on which we show a spectral gap of transfer operators.

Linear response for Dirac observables of Anosov diffeomorphisms
Matthieu Porte
2019, 39(4): 1799-1819 doi: 10.3934/dcds.2019078 +[Abstract](435) +[HTML](41) +[PDF](573.5KB)
Abstract:

We consider a \begin{document}$ \mathcal{C}^3 $\end{document} family \begin{document}$ t\mapsto f_t $\end{document} of \begin{document}$ \mathcal{C}^4 $\end{document} Anosov diffeomorphisms on a compact Riemannian manifold \begin{document}$ M $\end{document}. Denoting by \begin{document}$ \rho_t $\end{document} the SRB measure of \begin{document}$ f_t $\end{document}, we prove that the map \begin{document}$ t\mapsto\int \theta d\rho_t $\end{document} is differentiable if \begin{document}$ \theta $\end{document} is of the form \begin{document}$ \theta(x) = h(x)\delta(g(x)-a) $\end{document}, with \begin{document}$ \delta $\end{document} the Dirac distribution, \begin{document}$ g:M\rightarrow \mathbb{R} $\end{document} a \begin{document}$ \mathcal{C}^4 $\end{document} function, \begin{document}$ h:M\rightarrow\mathbb{C} $\end{document} a \begin{document}$ \mathcal{C}^3 $\end{document} function and \begin{document}$ a $\end{document} a regular value of \begin{document}$ g $\end{document}. We also require a transversality condition, namely that the intersection of the support of \begin{document}$ h $\end{document} with the level set \begin{document}$ \{g(x) = a\} $\end{document} is foliated by 'admissible stable leaves'.

A non-local problem for the Fokker-Planck equation related to the Becker-Döring model
Joseph G. Conlon and André Schlichting
2019, 39(4): 1821-1889 doi: 10.3934/dcds.2019079 +[Abstract](458) +[HTML](44) +[PDF](1138.71KB)
Abstract:

This paper concerns a Fokker-Planck equation on the positive real line modeling nucleation and growth of clusters. The main feature of the equation is the dependence of the driving vector field and boundary condition on a non-local order parameter related to the excess mass of the system.

The first main result concerns the well-posedness and regularity of the Cauchy problem. The well-posedness is based on a fixed point argument, and the regularity on Schauder estimates. The first a priori estimates yield Hölder regularity of the non-local order parameter, which is improved by an iteration argument.

The asymptotic behavior of solutions depends on some order parameter \begin{document}$ \rho $\end{document} depending on the initial data. The system shows different behavior depending on a value \begin{document}$ \rho_s>0 $\end{document}, determined from the potentials and diffusion coefficient. For \begin{document}$ \rho \leq \rho_s $\end{document}, there exists an equilibrium solution \begin{document}$ c^ {{ \rm{eq}}} _{(\rho)} $\end{document}. If \begin{document}$ \rho\le\rho_s $\end{document} the solution converges strongly to \begin{document}$ c^ {{ \rm{eq}}} _{(\rho)} $\end{document}, while if \begin{document}$ \rho > \rho_s $\end{document} the solution converges weakly to \begin{document}$ c^ {{ \rm{eq}}} _{(\rho_s)} $\end{document}. The excess \begin{document}$ \rho - \rho_s $\end{document} gets lost due to the formation of larger and larger clusters. In this regard, the model behaves similarly to the classical Becker-Döring equation.

The system possesses a free energy, strictly decreasing along the evolution, which establishes the long time behavior. In the subcritical case \begin{document}$ \rho<\rho_s $\end{document} the entropy method, based on suitable weighted logarithmic Sobolev inequalities and interpolation estimates, is used to obtain explicit convergence rates to the equilibrium solution.

The close connection of the presented model and the Becker-Döring model is outlined by a family of discrete Fokker-Planck type equations interpolating between both of them. This family of models possesses a gradient flow structure, emphasizing their commonality.

Asymptotic expansion of the mean-field approximation
Thierry Paul and Mario Pulvirenti
2019, 39(4): 1891-1921 doi: 10.3934/dcds.2019080 +[Abstract](425) +[HTML](45) +[PDF](703.6KB)
Abstract:

We consider the \begin{document}$ N $\end{document}-body quantum evolution of a particle system in the mean-field approximation. We show that the \begin{document}$ j $\end{document}th order marginals \begin{document}$ F^N_j(t) $\end{document}, for factorized initial data \begin{document}$ F(0)^{\otimes N} $\end{document}, are explicitly expressed, modulo \begin{document}$ N^{-\infty} $\end{document}, out of the solution \begin{document}$ F(t) $\end{document} of the corresponding non-linear mean-field equation and the solution of its linearization around \begin{document}$ F(t) $\end{document}. The result is valid for all times \begin{document}$ t $\end{document}, uniformly in \begin{document}$ j = O(N^{\frac12-\alpha}) $\end{document} for any \begin{document}$ \alpha>0 $\end{document}. We establish and estimate the full asymptotic expansion in integer powers of \begin{document}$ \frac1N $\end{document} of \begin{document}$ F^N_j(t) $\end{document}, \begin{document}$ j = O(\sqrt N) $\end{document}, whose computation at order \begin{document}$ n $\end{document} involves a finite number of operations depending on \begin{document}$ j $\end{document} and \begin{document}$ n $\end{document} but not on \begin{document}$ N $\end{document}. Our results are also valid for more general models including Kac models. As a by-product we get that the rate of convergence to the mean-field limit in \begin{document}$ \frac1N $\end{document} is optimal in the sense that the first correction to the mean-field limit does not vanish.

The Schnakenberg model with precursors
Weiwei Ao and Chao Liu
2019, 39(4): 1923-1955 doi: 10.3934/dcds.2019081 +[Abstract](497) +[HTML](62) +[PDF](599.75KB)
Abstract:

In this paper, we mainly consider the following Schnakenberg model with a precursor \begin{document}$ \mu(x) $\end{document} on the interval \begin{document}$ (-1,1) $\end{document}:

where \begin{document}$ D_{1}>0 $\end{document}, \begin{document}$ D_{2}>0 $\end{document}, \begin{document}$ B>0 $\end{document}.

We establish the existence and stability of \begin{document}$ N- $\end{document}peaked steady-states in terms of the precursor \begin{document}$ \mu(x) $\end{document} and the diffusion coefficients \begin{document}$ D_{1} $\end{document} and \begin{document}$ D_{2} $\end{document}. It is shown that \begin{document}$ \mu(x) $\end{document} plays an essential role for both existence and stability of the above pattern. Similar result has been obtained for the Gierer-Meinhardt system by Wei and Winter [21].

Binary differential equations with symmetries
Miriam Manoel and Patrícia Tempesta
2019, 39(4): 1957-1974 doi: 10.3934/dcds.2019082 +[Abstract](511) +[HTML](54) +[PDF](908.0KB)
Abstract:

This paper introduces the study of occurrence of symmetries in binary differential equations (BDEs). These are implicit differential equations given by the zeros of a quadratic 1-form, \begin{document}$ a(x,y)dy^2 + b(x,y)dxdy + c(x,y)dx^2 = 0, $\end{document} for \begin{document}$ a, b, c $\end{document} smooth real functions defined on an open set of \begin{document}$ \mathbb{R}^2 $\end{document}. Generically, solutions of a BDE are given as leaves of a pair of foliations, and the action of a symmetry must depend not only whether it preserves or inverts the plane orientation, but also whether it preserves or interchanges the foliations. The first main result reveals this dependence, which is given algebraically by a formula relating three group homomorphisms defined on the symmetry group of the BDE. The second main result adapts methods from invariant theory of compact Lie groups to obtain an algorithm to compute general expressions of equivariant quadratic 1-forms under each compact subgroup of the orthogonal group \begin{document}$ {{\bf{O}}(2)} $\end{document}.

Effect of quantified irreducibility on the computability of subshift entropy
Silvère Gangloff and Benjamin Hellouin de Menibus
2019, 39(4): 1975-2000 doi: 10.3934/dcds.2019083 +[Abstract](511) +[HTML](44) +[PDF](471.35KB)
Abstract:

We study the algorithmic computability of topological entropy of subshifts subjected to a quantified version of a strong condition of mixing, called irreducibility. For subshifts of finite type, it is known that this problem goes from uncomputable to computable as the rate of irreducibility decreases. Furthermore, the set of possible values for the entropy goes from all right-recursively computable numbers to some subset of the computable numbers. However, the exact nature of the transition is not understood.

In this text, we characterize a computability threshold for subshifts with decidable language (in any dimension), expressed as a summability condition on the rate function. This class includes subshifts of finite type under the threshold, and offers more flexibility for the constructions involved in the proof of uncomputability above the threshold. These constructions involve bounded density subshifts that control the density of particular symbols in all subwords.

Entire solutions and traveling wave solutions of the Allen-Cahn-Nagumo equation
Hirokazu Ninomiya
2019, 39(4): 2001-2019 doi: 10.3934/dcds.2019084 +[Abstract](608) +[HTML](69) +[PDF](766.22KB)
Abstract:

In this paper, the propagation phenomena in the Allen-Cahn-Nagumo equation are considered. Especially, the relation between traveling wave solutions and entire solutions is discussed. Indeed, several types of one-dimensional entire solutions are constructed by composing one-dimensional traveling wave solutions. Combining planar traveling wave solutions provides several types of multi-dimensional traveling wave solutions. The relation between multi-dimensional traveling wave solutions and entire solutions suggests the existence of new traveling wave solutions and new entire solutions.

Global existence and optimal decay estimates of the compressible viscoelastic flows in $ L^p $ critical spaces
Xinghong Pan and Jiang Xu
2019, 39(4): 2021-2057 doi: 10.3934/dcds.2019085 +[Abstract](723) +[HTML](60) +[PDF](653.07KB)
Abstract:

In this paper, we are concerned with the compressible viscoelastic flows in whole space \begin{document}$ \mathbb{R}^n $\end{document} with \begin{document}$ n\geq2 $\end{document}. We aim at extending the global existence in energy spaces (see [18] by Hu & Wang and [30] by Qian & Zhang) such that it holds in more general \begin{document}$ L^p $\end{document} critical spaces, which allows to the case of large highly oscillating initial velocity. Precisely, We define "two effective velocities" which are used to eliminate the coupling between the density, velocity and deformation tensor. Consequently, the global existence in the \begin{document}$ L^p $\end{document} critical framework is constructed by elementary energy approaches. In addition, the optimal time-decay estimates of strong solutions are firstly shown in the \begin{document}$ L^p $\end{document} framework, which improve recent decay efforts for compressible viscoelastic flows.

Q-entropy for general topological dynamical systems
Yun Zhao, Wen-Chiao Cheng and Chih-Chang Ho
2019, 39(4): 2059-2075 doi: 10.3934/dcds.2019086 +[Abstract](686) +[HTML](66) +[PDF](432.18KB)
Abstract:

The aim of this paper is to extend the \begin{document}$ q $\end{document}-entropy from symbolic systems to a general topological dynamical system. Using a (weak) Gibbs measure as the reference measure, this paper defines \begin{document}$ q $\end{document}-topological entropy and \begin{document}$ q $\end{document}-metric entropy, then studies basic properties of these entropies. In particular, this paper describes the relations between \begin{document}$ q $\end{document}-topological entropy and topological pressure for almost additive potentials, and the relations between \begin{document}$ q $\end{document}-metric entropy and local metric entropy. Although these relations are quite similar to that described in [19], the methods used here need more techniques from the theory of thermodynamic formalism with almost additive potentials.

Pattern formation in the doubly-nonlocal Fisher-KPP equation
Christian Kuehn and Pasha Tkachov
2019, 39(4): 2077-2100 doi: 10.3934/dcds.2019087 +[Abstract](538) +[HTML](44) +[PDF](1858.33KB)
Abstract:

We study the existence, bifurcations, and stability of stationary solutions for the doubly-nonlocal Fisher-KPP equation. We prove using Lyapunov-Schmidt reduction that under suitable conditions on the parameters, a bifurcation from the non-trivial homogeneous state can occur. The kernel of the linearized operator at the bifurcation is two-dimensional and periodic stationary patterns are generated. Then we prove that these patterns are, again under suitable conditions, locally asymptotically stable. We also compare our results to previous work on the nonlocal Fisher-KPP equation containing a local diffusion term and a nonlocal reaction term. If the diffusion is approximated by a nonlocal kernel, we show that our results are consistent and reduce to the local ones in the local singular diffusion limit. Furthermore, we prove that there are parameter regimes, where no bifurcations can occur for the doubly-nonlocal Fisher-KPP equation. The results demonstrate that intricate different parameter regimes are possible. In summary, our results provide a very detailed classification of the multi-parameter dependence of the stationary solutions for the doubly-nonlocal Fisher-KPP equation.

Global existence and decay to equilibrium for some crystal surface models
Rafael Granero-Belinchón and Martina Magliocca
2019, 39(4): 2101-2131 doi: 10.3934/dcds.2019088 +[Abstract](516) +[HTML](60) +[PDF](241.05KB)
Abstract:

In this paper we study the large time behavior of the solutions to the following nonlinear fourth-order equations

These two PDE were proposed as models of the evolution of crystal surfaces by J. Krug, H.T. Dobbs, and S. Majaniemi (Z. Phys. B, 97,281-291, 1995) and H. Al Hajj Shehadeh, R. V. Kohn, and J. Weare (Phys. D, 240, 1771-1784, 2011), respectively. In particular, we find explicitly computable conditions on the size of the initial data (measured in terms of the norm in a critical space) guaranteeing the global existence and exponential decay to equilibrium in the Wiener algebra and in Sobolev spaces.

Memory equations as reduced Markov processes
Artur Stephan and Holger Stephan
2019, 39(4): 2133-2155 doi: 10.3934/dcds.2019089 +[Abstract](495) +[HTML](41) +[PDF](509.62KB)
Abstract:

A large class of linear memory differential equations in one dimension, where the evolution depends on the whole history, can be equivalently described as a projection of a Markov process living in a higher dimensional space. Starting with such a memory equation, we propose an explicit construction of the corresponding Markov process. From a physical point of view the Markov process can be understood as a change of the type of some quasiparticles along one-way loops. Typically, the arising Markov process does not have the detailed balance property. The method leads to a more realistic modeling of memory equations. Moreover, it carries over the large number of investigation tools for Markov processes to memory equations like the calculation of the equilibrium state. The method can be used for an approximative solution of some degenerate memory equations like delay differential equations.

Quasiregular semigroups with examples
Alastair Fletcher
2019, 39(4): 2157-2172 doi: 10.3934/dcds.2019090 +[Abstract](331) +[HTML](36) +[PDF](389.26KB)
Abstract:

Rational semigroups were introduced by Hinkkanen and Martin as a generalization of the iteration of a single rational map. There has subsequently been much interest in the study of rational semigroups. Quasiregular semigroups were introduced shortly after rational semigroups as analogues in higher real dimensions, but have received far less attention. Each map in a quasiregular semigroup must necessarily be a uniformly quasiregular map. While there is a completely viable theory for the iteration of uniformly quasiregular maps, it is a highly non-trivial matter to construct them. In this paper, we study properties of the Julia and Fatou sets of quasiregular semigroups and, equally as importantly, give several families of examples illustrating some of the behaviours that can arise.

Exponential stabilization of the stochastic Burgers equation by boundary proportional feedback
Ionuţ Munteanu
2019, 39(4): 2173-2185 doi: 10.3934/dcds.2019091 +[Abstract](600) +[HTML](45) +[PDF](425.0KB)
Abstract:

In the present paper it is designed a simple, finite-dimensional, linear deterministic stabilizing boundary feedback law for the stochastic Burgers equation with unbounded time-dependent coefficients. The stability of the system is guaranteed no matter how large the level of the noise is.

Existence of the normalized solutions to the nonlocal elliptic system with partial confinement
Jun Wang, Qiuping Geng and Maochun Zhu
2019, 39(4): 2187-2201 doi: 10.3934/dcds.2019092 +[Abstract](527) +[HTML](50) +[PDF](459.0KB)
Abstract:

In present paper we study the existence and orbital stability of the standing waves to the nonlocal elliptic system with partial confinement. This type equations arises from the basic quantum chemistry model of small number of electrons interacting with static nucleii. On the one hand, we prove the existence of global minimizer of the associate energy functional subject to the \begin{document}$ L^2 $\end{document}-constraint. On the other hand, we discuss the orbital stability of the global minimizers. Comparing to the local case, we need establish the new inequality related to the Steiner rearrangement.

NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph
Roy H. Goodman
2019, 39(4): 2203-2232 doi: 10.3934/dcds.2019093 +[Abstract](537) +[HTML](45) +[PDF](1237.5KB)
Abstract:

We consider the bifurcations of standing wave solutions to the nonlinear Schrödinger equation (NLS) posed on a quantum graph consisting of two loops connected by a single edge, the so-called dumbbell, recently studied in [27]. The authors of that study found the ground state undergoes two bifurcations, first a symmetry-breaking, and the second which they call a symmetry-preserving bifurcation. We clarify the type of the symmetry-preserving bifurcation, showing it to be transcritical. We then reduce the question, and show that the phenomena described in that paper can be reproduced in a simple discrete self-trapping equation on a combinatorial graph of bowtie shape. This allows for complete analysis by parameterizing the full solution space. We then expand the question, and describe the bifurcations of all the standing waves of this system, which can be classified into three families, and of which there exists a countably infinite set.

Construction solutions for Neumann problem with Hénon term in $ \mathbb{R}^2 $
Shengbing Deng
2019, 39(4): 2233-2253 doi: 10.3934/dcds.2019094 +[Abstract](501) +[HTML](52) +[PDF](526.15KB)
Abstract:

In this paper, we are interested in the following boundary value problem

where \begin{document}$ \Omega $\end{document} is a bounded domain in \begin{document}$ \mathbb{R}^2 $\end{document} with smooth boundary, points \begin{document}$ q_1,\ldots,q_n\in \Omega $\end{document}, \begin{document}$ \alpha_1,\cdots,\alpha_n\in(0,\infty)\backslash\mathbb{N} $\end{document}, \begin{document}$ \lambda>0 $\end{document} is a small parameter, \begin{document}$ 0< p <2 $\end{document}, and \begin{document}$ \nu $\end{document} denotes the outer normal vector to \begin{document}$ \partial\Omega $\end{document}. We construct solutions of this problem with \begin{document}$ k $\end{document} interior bubbling points and \begin{document}$ l $\end{document} boundary bubbling points, for any \begin{document}$ k\geq 1 $\end{document} and \begin{document}$ l\geq 1 $\end{document}.

On smoothness of solutions to projected differential equations
David Salas, Lionel Thibault and Emilio Vilches
2019, 39(4): 2255-2283 doi: 10.3934/dcds.2019095 +[Abstract](503) +[HTML](57) +[PDF](670.88KB)
Abstract:

Projected differential equations are known as fundamental mathematical models in economics, for electric circuits, etc. The present paper studies the (higher order) derivability as well as a generalized type of derivability of solutions of such equations when the set involved for projections is prox-regular with smooth boundary.

Prescribing the $ Q' $-curvature in three dimension
Pak Tung Ho
2019, 39(4): 2285-2294 doi: 10.3934/dcds.2019096 +[Abstract](668) +[HTML](59) +[PDF](368.08KB)
Abstract:

In this note, we consider the problem of prescribing \begin{document}$ \overline{Q}' $\end{document}-curvature on a three-dimensional pseudohermitian manifold. Given a positive CR pluriharmonic function \begin{document}$ f $\end{document}, we construct a contact form on the three-dimensional pseudo-Einstein manifold with \begin{document}$ \overline{Q}' $\end{document}-curvature being equal to \begin{document}$ f $\end{document}, under some natural positivity conditions. On the other hand, we prove a Kazdan-Warner type identity for the problem of prescribing \begin{document}$ \overline{Q}' $\end{document}-curvature on the standard CR three sphere.

Periodic homogenization of elliptic systems with stratified structure
Yao Xu and Weisheng Niu
2019, 39(4): 2295-2323 doi: 10.3934/dcds.2019097 +[Abstract](548) +[HTML](43) +[PDF](586.07KB)
Abstract:

This paper concerns with the quantitative homogenization of second-order elliptic systems with periodic stratified structure in Lipschitz domains. Under the symmetry assumption on coefficient matrix, the sharp \begin{document}$ O(\varepsilon) $\end{document}-convergence rate in \begin{document}$ L^{p_0}(\Omega) $\end{document} with \begin{document}$ p_0 = \frac{2d}{d-1} $\end{document} is obtained based on detailed discussions on stratified functions. Without the symmetry assumption, an \begin{document}$ O(\varepsilon^\sigma) $\end{document}-convergence rate is also derived for some \begin{document}$ \sigma<1 $\end{document} by the Meyers estimate. Based on this convergence rate, we establish the uniform interior Lipschitz estimate. The uniform interior \begin{document}$ W^{1, p} $\end{document} and Hölder estimates are also obtained by the real variable method.

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