American Institute of Mathematical Sciences

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.

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.

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.

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.

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.

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.

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.

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}.

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

\begin{document}$\Delta_p^Nu:=\tfrac{1}{p}|\nabla u|^{2-p}\Delta_pu=\tfrac{1}{p}\Delta_1^Nu+\tfrac{p-1}{p}\Delta_\infty^Nu$\end{document}

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.

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.

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

\begin{document}$\text{(1)}\quad \nabla\cdot \boldsymbol{J}=0,\qquad \text{(2)}\quad \frac{\partial u}{\partial t}+\nabla\cdot\boldsymbol{q}=f(x,t,u,\nabla\varphi)\;\text{ in }\; \Omega\times\,]\,0,T\,[\,,$\end{document}

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}.

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.

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.

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

\begin{document}$P_0(u):=u_t-\Delta u=a(t,x)u-b(t,x)|u|^{p-1}u\ \ \ \text{ in } [0,T)\times\Omega,$\end{document}

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:

\begin{document}$a_1(t)\geqslant c_0\exp(-\omega(T-t)(T-t)^{-1})\ \ \ \forall t<T,\ c_0=\textrm{const}>0a_1(t)\geqslant c_0\exp(-\omega(T-t)(T-t)^{-1})\ \ \ \forall t<T,\ c_0=\textrm{const}>0$\end{document}

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

\begin{document}$\omega(\tau)\rightarrow\omega_0=\textrm{const}>0\ \ \ \text{ as }\tau\rightarrow0$\end{document}

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}.

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

0.1\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.

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.