Related papers: Touchdown solutions in general MEMS models
A new supersymmetric model for electrons with generalized hopping terms and Hubbard interaction on a one-dimensional lattice is solved by means of the Bethe Ansatz. We investigate the phase diagram of this model by studying the ground state…
In this paper we study, both numerically and analytically, the asymptotic behavior of the principal eigenfunction of \eqref{1.1}, normalized by \eqref{1.2}, as $s\uparrow +\infty$. Based on the numerical computations of this paper, we can…
We are interested in the differential equation $\ddot u(t) = -A u(t) - c A \dot u(t) + \lambda u(t) + F(t,u(t))$, where $c > 0$ is a damping factor, $A$ is a sectorial operator and $F$ is a continuous map. We consider the situation where…
We discuss the general dynamical behaviors of quintessence field, in particular, the general conditions for tracking and thawing solutions are discussed. We explain what the tracking solutions mean and in what sense the results depend on…
In this paper we study the existence and summability of the solutions to the following parabolic-elliptic system of partial differential equations with discontinuous coefficients: \begin{equation*} \begin{cases} u_t -…
For $1<p<\infty$, we consider the following problem $$ -\Delta_p u=f(u),\quad u>0\text{ in }\Omega,\quad\partial_\nu u=0\text{ on }\partial\Omega, $$ where $\Omega\subset\mathbb R^N$ is either a ball or an annulus. The nonlinearity $f$ is…
The method of fundamental solutions (MFS) is known to be effective for solving 3D Laplace and Stokes Dirichlet boundary value problems in the exterior of a large collection of simple smooth objects. Here we present new scalable MFS…
We investigate a perturbed Gelfand problem involving a mixed local-nonlocal $p$-Laplacian operator with singular nonlinearity: \begin{equation*} \begin{aligned} -\Delta_p u + (-\Delta_p)^s u = \lambda \frac{f(u)}{u^{\beta}}\ \text{in} \…
Our principal object of study is the modulus of continuity of a periodic or uniformly vanishing function \( u: \mathbb{R} ^{n} \rightarrow \mathbb{R} \) which satisfies a degenerate elliptic equation \( F(x, u, \nabla u, D^{2} u) = 0 \) in…
Optical metasurfaces (subwavelength-patterned surfaces typically described by variable effective surface impedances) are typically modeled by an approximation akin to ray optics: the reflection or transmission of an incident wave at each…
In this work we study the following nonlocal problem \begin{equation*} \left\{ \begin{aligned} M(\|u\|^2_X)(-\Delta)^s u&= \lambda {f(x)}|u|^{\gamma-2}u+{g(x)}|u|^{p-2}u &&\mbox{in}\ \ \Omega, u&=0 &&\mbox{on}\ \ \mathbb R^N\setminus…
In this paper, we investigate the asymptotic behavior, as $\beta \to 0$, of positive solutions to the semilinear elliptic Robin problem \begin{equation*} \begin{cases} -\Delta u = u^p, & \text{in } \Omega,\\ u > 0, & \text{in } \Omega,\\…
In the current work we study a nonlocal parabolic problem with Robin boundary conditions. The problem arises from the study of an idealized electrically actuated MEMS (Micro-Electro-Mechanical System) device. Initially we study the…
We consider the reaction-diffusion problem $-\Delta_g u = f(u)$ in $\mathcal{B}_R$ with zero Dirichlet boundary condition, posed in a geodesic ball $\mathcal{B}_R$ with radius $R$ of a Riemannian model $(M,g)$. This class of Riemannian…
Micro-Electro-Mechanical Systems (MEMS) normally have fixed or moving structures (plates or array of thin beams) with cross-sections of the order of microns and lengths of the order of tens or hundreds of microns. Electrostatic forces play…
We present a generalization of Stampacchia Lemma and give applications to regularity property of weak and entropy solutions of degenerate elliptic equations of the form $$ \left\{ \begin{array}{llll} -\mbox{div} (a(x,u(x)) Du (x)) =f(x), &…
We study a one dimensional metastable dynamics of internal interfaces for the initial boundary value problem for the following convection-reaction-diffusion equation \begin{equation*} \partial_t u = \varepsilon \partial_x^2 u -\partial_x…
We study a stationary scattering problem related to the nonlinear Helmholtz equation $-\Delta u - k^2 u = f(x,u) \ \ \text{in $\mathbb{R}^N$,}$ where $N \ge 3$ and $k>0$. For a given incident free wave $\varphi \in L^\infty(\mathbb{R}^N)$,…
In this paper, we consider the asymptotic behavior of the nonlocal parabolic problem \[ u_{t}=\Delta u+\displaystyle\frac{\lambda f(u)}{\big(\int_{\Omega}f(u)dx\big)^{p}}, x\in \Omega, t>0, \] with homogeneous Dirichlet boundary condition,…
Despite considerable developments in the literature of the past decades, a standing open problem in the analysis of continuum mechanics appears to consist of determining how far the prototypical model for small-strain thermoviscoelastic…