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Discrete & Continuous Dynamical Systems - S

August 2017 , Volume 10 , Issue 4

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Preface
Hans-Christoph Kaiser, Dorothee Knees, Alexander Mielke, Joachim Rehberg, Elisabetta Rocca, Marita Thomas and Enrico Valdinoci
2017, 10(4): i-iv doi: 10.3934/dcdss.201704i +[Abstract](275) +[HTML](140) +[PDF](171.2KB)
Abstract:
Intrinsic geometry and De Giorgi classes for certain anisotropic problems
Paolo Baroni, Agnese Di Castro and Giampiero Palatucci
2017, 10(4): 647-659 doi: 10.3934/dcdss.2017032 +[Abstract](327) +[HTML](7) +[PDF](384.7KB)
Abstract:

We analyze a natural approach to the regularity of solutions of problems related to some anisotropic Laplacian operators, and a subsequent extension of the usual De Giorgi classes, by investigating the relation of the functions in such classes with the weak solutions to some anisotropic elliptic equations as well as with the quasi-minima of the corresponding functionals with anisotropic polynomial growth.

The Dirichlet-to-Neumann map for Schrödinger operators with complex potentials
Jussi Behrndt and A. F. M. ter Elst
2017, 10(4): 661-671 doi: 10.3934/dcdss.2017033 +[Abstract](323) +[HTML](2) +[PDF](373.5KB)
Abstract:

Let \begin{document}$\Omega \subset \mathbb{R}^d$\end{document} be a bounded open set with Lipschitz boundary and let \begin{document}$q \colon \Omega \to \mathbb{C}$\end{document} be a bounded complex potential. We study the Dirichlet-to-Neumann graph associated with the operator \begin{document}$- \Delta + q$\end{document} and we give an example in which it is not \begin{document}$m$\end{document}-sectorial.

Modeling and analysis of reactive multi-component two-phase flows with mass transfer and phase transition the isothermal incompressible case
Dieter Bothe and Jan Prüss
2017, 10(4): 673-696 doi: 10.3934/dcdss.2017034 +[Abstract](376) +[HTML](6) +[PDF](482.0KB)
Abstract:

Isothermal incompressible multi-component two-phase flows with mass transfer, chemical reactions, and phase transition are modeled based on first principles. It is shown that the resulting system is thermodynamically consistent in the sense that the available energy is a strict Lyapunov functional, and the equilibria are identified. It is proved that the problem is well-posed in an \begin{document}$L_p$\end{document}-setting, and generates a local semiflow in the proper state manifold. It is further shown that each non-degenerate equilibrium is dynamically stable in the natural state manifold. Finally, it is proved that a solution, which does not develop singularities, exists globally and converges to an equilibrium in the state manifold.

Thermistor systems of p(x)-Laplace-type with discontinuous exponents via entropy solutions
Miroslav Bulíček, Annegret Glitzky and Matthias Liero
2017, 10(4): 697-713 doi: 10.3934/dcdss.2017035 +[Abstract](655) +[HTML](5) +[PDF](437.2KB)
Abstract:

We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two difficulties appear: (ⅰ) the elliptic term in the current-flow equation is of p(x)-Laplacian-type with discontinuous exponent p, which limits the use of standard methods, and (ⅱ) in the heat equation, we have to deal with an a priori L1 term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder's fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.

Volume constrained minimizers of the fractional perimeter with a potential energy
Annalisa Cesaroni and Matteo Novaga
2017, 10(4): 715-727 doi: 10.3934/dcdss.2017036 +[Abstract](314) +[HTML](4) +[PDF](363.1KB)
Abstract:

We consider volume-constrained minimizers of the fractional perimeter with the addition of a potential energy in the form of a volume integral. Such minimizers are solutions of the prescribed fractional curvature problem. We prove existence and regularity of minimizers under suitable assumptions on the potential energy, which cover the periodic case. In the small volume regime we show that minimizers are close to balls, with a quantitative estimate.

Characterizations of Sobolev functions that vanish on a part of the boundary
Moritz Egert and Patrick Tolksdorf
2017, 10(4): 729-743 doi: 10.3934/dcdss.2017037 +[Abstract](494) +[HTML](4) +[PDF](415.2KB)
Abstract:

Let \begin{document}$\Omega$\end{document} be a bounded domain in \begin{document}$\mathbb{R}^n$\end{document} with a Sobolev extension property around the complement of a closed part \begin{document}$D$\end{document} of its boundary. We prove that a function \begin{document}$u \in {\rm{W}}^{1,p}(\Omega)$\end{document} vanishes on \begin{document}$D$\end{document} in the sense of an interior trace if and only if it can be approximated within \begin{document}${\rm{W}}^{1,p}(\Omega)$\end{document} by smooth functions with support away from \begin{document}$D$\end{document}. We also review several other equivalent characterizations, so to draw a rather complete picture of these Sobolev functions vanishing on a part of the boundary.

A survey on second order free boundary value problems modelling MEMS with general permittivity profile
Joachim Escher and Christina Lienstromberg
2017, 10(4): 745-771 doi: 10.3934/dcdss.2017038 +[Abstract](512) +[HTML](8) +[PDF](537.8KB)
Abstract:

In this survey we review some recent results on microelectromechanical systems with general permittivity profile. Different systems of differential equations are derived by taking various physical modelling aspects into account, according to the particular application. In any case an either semi-or quasilinear hyperbolic or parabolic evolution problem for the displacement of an elastic membrane is coupled with an elliptic moving boundary problem that determines the electrostatic potential in the region occupied by the elastic membrane and a rigid ground plate. Of particular interest in all models is the influence of different classes of permittivity profiles.

The subsequent analytical investigations are restricted to a dissipation dominated regime for the membrane's displacement. For the resulting parabolic evolution problems local well-posedness, global existence, the occurrence of finite-time singularities, and convergence of solutions to those of the so-called small-aspect ratio model, respectively, are investigated. Furthermore, a topic is addressed that is of note not till non-constant permittivity profiles are taken into account -the direction of the membrane's deflection or, in mathematical parlance, the sign of the solution to the evolution problem. The survey is completed by a presentation of some numerical results that in particular justify the consideration of the coupled problem by revealing substantial qualitative differences of the solutions to the widely-used small-aspect ratio model and the coupled problem.

Derivation of effective transmission conditions for domains separated by a membrane for different scaling of membrane diffusivity
Markus Gahn, Maria Neuss-Radu and Peter Knabner
2017, 10(4): 773-797 doi: 10.3934/dcdss.2017039 +[Abstract](314) +[HTML](7) +[PDF](566.5KB)
Abstract:

We consider a system of non-linear reaction-diffusion equations in a domain consisting of two bulk regions separated by a thin layer with periodic structure. The thickness of the layer is of order \begin{document}$\epsilon$\end{document}, and the equations inside the layer depend on the parameter \begin{document}$\epsilon$\end{document} and an additional parameter \begin{document}$\gamma \in [-1,1)$\end{document}, which describes the size of the diffusion in the layer. We derive effective models for the limit \begin{document}$\epsilon \to 0 $\end{document}, when the layer reduces to an interface \begin{document}$\Sigma$\end{document} between the two bulk domains. The effective solution is continuous across \begin{document}$\Sigma$\end{document} for all \begin{document}$\gamma \in [-1,1)$\end{document}. For \begin{document}$\gamma \in (-1,1)$\end{document}, the jump in the normal flux is given by a non-linear ordinary differential equation on \begin{document}$\Sigma$\end{document}. In the critical case \begin{document}$\gamma = -1$\end{document}, a dynamic transmission condition of Wentzell-type arises at the interface \begin{document}$\Sigma$\end{document}.

On the geometry of the p-Laplacian operator
Bernd Kawohl and Jiří Horák
2017, 10(4): 799-813 doi: 10.3934/dcdss.2017040 +[Abstract](666) +[HTML](282) +[PDF](1487.0KB)
Abstract:

The \begin{document}$p$\end{document}-Laplacian operator \begin{document}$\Delta_pu={\rm div }\left(|\nabla u|^{p-2}\nabla u\right)$\end{document} is not uniformly elliptic for any \begin{document}$p\in(1,2)\cup(2,\infty)$\end{document} and degenerates even more when \begin{document}$p\to \infty$\end{document} or \begin{document}$p\to 1$\end{document}. In those two cases the Dirichlet and eigenvalue problems associated with the \begin{document}$p$\end{document}-Laplacian lead to intriguing geometric questions, because their limits for \begin{document}$p\to\infty$\end{document} or \begin{document}$p\to 1$\end{document} can be characterized by the geometry of \begin{document}$\Omega$\end{document}. In this little survey we recall some well-known results on eigenfunctions of the classical 2-Laplacian and elaborate on their extensions to general \begin{document}$p\in[1,\infty]$\end{document}. We report also on results concerning the normalized or game-theoretic \begin{document}$p$\end{document}-Laplacian

and its parabolic counterpart \begin{document}$u_t-\Delta_p^N u=0$\end{document}. These equations are homogeneous of degree 1 and \begin{document}$\Delta_p^N$\end{document} is uniformly elliptic for any \begin{document}$p\in (1,\infty)$\end{document}. In this respect it is more benign than the \begin{document}$p$\end{document}-Laplacian, but it is not of divergence type.

Effective acoustic properties of a meta-material consisting of small Helmholtz resonators
Agnes Lamacz and Ben Schweizer
2017, 10(4): 815-835 doi: 10.3934/dcdss.2017041 +[Abstract](339) +[HTML](3) +[PDF](482.5KB)
Abstract:

We investigate the acoustic properties of meta-materials that are inspired by sound-absorbing structures. We show that it is possible to construct meta-materials with frequency-dependent effective properties, with large and/or negative permittivities. Mathematically, we investigate solutions \begin{document}$u^\varepsilon : \Omega_\varepsilon \to \mathbb{R}$\end{document} to a Helmholtz equation in the limit \begin{document}$\varepsilon \to 0$\end{document} with the help of two-scale convergence. The domain \begin{document}$\Omega_\varepsilon $\end{document} is obtained by removing from an open set \begin{document}$\Omega\subset \mathbb{R}^n$\end{document} in a periodic fashion a large number (order \begin{document}$\varepsilon ^{-n}$\end{document}) of small resonators (order \begin{document}$\varepsilon $\end{document}). The special properties of the meta-material are obtained through sub-scale structures in the perforations.

On the existence of weak solutions of an unsteady p-Laplace thermistor system with strictly monotone electrical conductivities
Joachim Naumann
2017, 10(4): 837-852 doi: 10.3934/dcdss.2017042 +[Abstract](314) +[HTML](2) +[PDF](462.3KB)
Abstract:

Let \begin{document}$\Omega\subset\mathbb{R}^n$\end{document} (\begin{document}$n=2$\end{document} or \begin{document}$n=3$\end{document}) be a bounded domain. We consider the thermistor system

where (1) is a \begin{document}$p$\end{document}-Laplace type equation for \begin{document}$\varphi$\end{document} (\begin{document}$u=$\end{document} temperature, \begin{document}$\varphi=$\end{document} electrostatic potential). We prove the existence of a weak solution \begin{document}$(\varphi,u)$\end{document} of (1)–(2) under mixed boundary conditions for \begin{document}$\varphi$\end{document}, and a Robin boundary condition and an initial condition for \begin{document}$u$\end{document}.

A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class
Fabio Paronetto
2017, 10(4): 853-866 doi: 10.3934/dcdss.2017043 +[Abstract](298) +[HTML](6) +[PDF](415.3KB)
Abstract:

We define a homogeneous parabolic De Giorgi class of order 2 which suits a mixed type class of evolution equations whose simplest example is \begin{document}$\mu (x) \frac{\partial u}{\partial t} - \Delta u = 0$\end{document} where \begin{document}$\mu$\end{document} can be positive, null and negative. The functions belonging to this class are local bounded and satisfy a Harnack type inequality. Interesting by-products are Hölder-continuity, at least in the "evolutionary" part of \begin{document}$\Omega$\end{document} and in particular in the interface \begin{document}$I$\end{document} where \begin{document}$\mu$\end{document} change sign, and an interesting maximum principle.

An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat
Tomáš Roubíček
2017, 10(4): 867-893 doi: 10.3934/dcdss.2017044 +[Abstract](464) +[HTML](9) +[PDF](646.3KB)
Abstract:

The model of brittle cracks in elastic solids at small strains is approximated by the Ambrosio-Tortorelli functional and then extended into evolution situation to an evolutionary system, involving viscoelasticity, inertia, heat transfer, and coupling with Cahn-Hilliard-type diffusion of a fluid due to Fick's or Darcy's laws. Damage resulting from the approximated crack model is considered rate independent. The fractional-step Crank-Nicolson-type time discretisation is devised to decouple the system in a way so that the energy is conserved even in the discrete scheme. The numerical stability of such a scheme is shown, and also convergence towards suitably defined weak solutions. Various generalizations involving plasticity, healing in damage, or phase transformation are mentioned, too.

Large solutions of parabolic logistic equation with spatial and temporal degeneracies
Andrey Shishkov
2017, 10(4): 895-907 doi: 10.3934/dcdss.2017045 +[Abstract](314) +[HTML](6) +[PDF](419.7KB)
Abstract:

There is studied asymptotic behavior as \begin{document}$t\rightarrow T$\end{document} of arbitrary solution of equation

where \begin{document}$\Omega$\end{document} is smooth bounded domain in \begin{document}$\mathbb{R}^N$\end{document}, \begin{document}$0 < T < \infty$\end{document}, \begin{document}$p>1$\end{document}, \begin{document}$a(\cdot)$\end{document} is continuous, \begin{document}$b(\cdot)$\end{document} is continuous nonnegative function, satisfying condition: \begin{document}$b(t, x)\geqslant a_1(t)g_1(d(x))$\end{document}, \begin{document}$d(x):=\textrm{dist}(x, \partial\Omega)$\end{document}. Here \begin{document}$g_1(s)$\end{document} is arbitrary nondecreasing positive for all \begin{document}$s>0$\end{document} function and \begin{document}$a_1(t)$\end{document} satisfies:

with some continuous nondecreasing function \begin{document}$\omega(\tau)\geqslant0$\end{document} \begin{document}$\forall\tau>0$\end{document}. Under additional condition:

it is proved that there exist constant \begin{document}$k:0 < k < \infty$\end{document}, such that all solutions of mentioned equation (particularly, solutions, satisfying initial-boundary condition \begin{document}$u|_\Gamma=\infty$\end{document}, where \begin{document}$\Gamma=(0, T)\times\partial\Omega\cup\{0\}\times\Omega$\end{document}) stay uniformly bounded in \begin{document}$\Omega_0:=\{x\in\Omega:d(x)>k\omega_0^{\frac12}\}$\end{document} as \begin{document}$t\rightarrow T$\end{document}. Method of investigation is based on local energy estimates and is applicable for wide class of equations. So in the paper there are obtained similar sufficient conditions of localization of singularity set of solutions near to the boundary of domain for equation with main part \begin{document}$P_0(u)=(|u|^{\lambda-1}u)_t-\sum_{i=1}^N(|\nabla_xu|^{q-1}u_{x_i})_{x_i}$\end{document} if \begin{document}$0 < \lambda\leqslant q < p$\end{document}.

On the variational representation of monotone operators
Augusto VisintiN
2017, 10(4): 909-918 doi: 10.3934/dcdss.2017046 +[Abstract](334) +[HTML](9) +[PDF](359.3KB)
Abstract:

Let \begin{document}$V$\end{document} be a Banach space, \begin{document}$z'\in V'$\end{document}, and \begin{document}$\alpha: V\to {\mathcal P}(V')$\end{document} be a maximal monotone operator. A large number of phenomena can be modelled by inclusions of the form \begin{document}$\alpha(u) \ni z'$\end{document}, or by the associated flow \begin{document}$D_tu + \alpha(u) \ni z'$\end{document}. Fitzpatrick proved that there exists a lower semicontinuous, convex representative function\begin{document}$f_\alpha: V \!\times\! V'\to \mathbb{R}\cup \{+\infty\}$\end{document} such that

\begin{document}$f_\alpha(v,v') \ge \langle v',v\rangle\quad\;\forall (v,v'), \qquad\quadf_\alpha(v,v') = \langle v',v\rangle\;\;\Leftrightarrow\;\;\; v'\in \alpha(v).$\end{document}

This provides a variational formulation for the above inclusions. Here we use this approach to prove two results of existence of a solution, without using the classical theory of maximal monotone operators. This is based on a minimax theorem, and on the duality theory of convex optimization.

The gradient flow of a generalized Fisher information functional with respect to modified Wasserstein distances
Jonathan Zinsl
2017, 10(4): 919-933 doi: 10.3934/dcdss.2017047 +[Abstract](396) +[HTML](1) +[PDF](425.0KB)
Abstract:

This article is concerned with the existence of nonnegative weak solutions to a particular fourth-order partial differential equation: it is a formal gradient flow with respect to a generalized Wasserstein transportation distance with nonlinear mobility. The corresponding free energy functional is referred to as generalized Fisher information functional since it is obtained by autodissipation of another energy functional which generates the heat flow as its gradient flow with respect to the aforementioned distance. Our main results are twofold: For mobility functions satisfying a certain regularity condition, we show the existence of weak solutions by construction with the well-known minimizing movement scheme for gradient flows. Furthermore, we extend these results to a more general class of mobility functions: a weak solution can be obtained by approximation with weak solutions of the problem with regularized mobility.

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