Search

We consider the Dirichlet problem for a class of quasilinear elliptic systems in domain with irregular boundary. The principal part satisfies componentwise coercivity condition and the nonlinear terms are Carath\'eodory maps having Morrey regularity in $x$ and verifying controlled growth conditions with respect to the other variables. We have obtained boundedness of the weak solution to the problem that permits to apply an iteration procedure in order to find optimal Morrey regularity of its gradient., Comment: 15 pages

We deal with nonlinear systems of parabolic type satisfying component-wise structural conditions. The nonlinear terms are Carath\'eodory maps having controlled growth with respect to the solution and the gradient and the data are in anisotropic Lebesgue spaces. Under these assumptions we obtain essential boundedness of the weak solutions., Comment: 16 pages

The recovery of the membrane profile of an electrostatic micro-electro-mechanical system (MEMS) device is an important issue because, when applying an external voltage, the membrane deforms with the consequent risk of touching the upper plate of the device (a condition that should be avoided). Then, during the deformation of the membrane, it is useful to know if this movement admits stable equilibrium configurations. In such a context, our present work analyze the behavior of an electrostatic 1D membrane MEMS device when an external electric voltage is applied. In particular, starting from a well-known second-order elliptical semi-linear di erential model, obtained considering the electrostatic field inside the device proportional to the curvature of the membrane, the only possible equilibrium position is obtained, and its stability is analyzed. Moreover, considering that the membrane has an inertia in moving and taking into account that it must not touch the upper plate of the device, the range of possible values of the applied external voltage is obtained, which accounted for these two particular operating conditions. Finally, some calculations about the variation of potential energy have identified optimal control conditions.

In this paper, the authors present a new condition of the uniqueness of the solution for a previous 1 D semi-linear elliptic boundary value problem of membrane MEMS devices, where the amplitude of the electric field is considered proportional to the curvature of the membrane. The existence of the solution (membrane deflection) depends on the material of the membrane, which is obtained by Schauder–Tychonoff’s fixed point approach. Thus, in this paper, the result of uniqueness has been completely reformulated to obtain a condition depending on the material of the membrane achieving a new result of existence and uniqueness, depending on both the material of the membrane and the geometrical characteristics of the device. Then, by shooting numerical method, more realistic conditions for detecting eventual ghost solutions and new ranges of both operational parameters and mechanical tension of the membrane ensuring convergence have been achieved confirming the useful information on the industrial applicability of the model under study.

In a framework of 1D membrane MEMS theory, we consider the MEMS boundary semi-linear elliptic problem with fringing field $$\begin{aligned} u''=-\frac{\lambda ^2(1+\delta |u'|^2)}{(1-u)^2}\;\;\;\text {in}\;\;\varOmega , \;\;u=0\;\;\text {on}\;\;\partial \varOmega , \end{aligned}$$ where $$\lambda ^2$$ and $$\delta $$ are positive parameters, $$\varOmega =[-L,L] \subset {\mathbb {R}}$$ , and u is the deflection of the membrane. In this model, since the electric field $$ {\mathbf {E}} $$ on the membrane is locally orthogonal to the straight line tangent to the membrane at the same point, $$ | {\mathbf {E}} | $$ , proportional to $$\lambda ^2/(1-u)^2$$ , is considered locally proportional to the curvature of the membrane. Thus, we achieve interesting results of existence writing it into its equivalent integral formulation by means of a suitable Green function and applying on it the Schauder–Tychonoff fixed point theory. Therefore, the uniqueness of the solution is proved exploiting both Poincare inequality and Gronwall Lemma. Then once the instability of the only obtained equilibrium position is verified, an interesting limitation for the potential energy dependent on the fringing field capacitance is obtained and studied.

In this paper we present, in a framework of 1D-membrane Micro-Electro-Mechanical- Systems (MEMS) theory, a formalization of the problem of existence and uniqueness of a solution related to the membrane deformation u for electrostatic actuation in the steady- state case. In particular, we propose a new model in which the electric field magnitude E is proportional to the curvature of the membrane and, for it, we obtain results of existence by Schauder-Tychono's fixed point application and subsequently we establish conditions of uniqueness. Finally, some numerical tests have been carried out to further support the analytical results.

We show existence and uniqueness theorems of global strong solutions of Cauchy–Dirichlet problem, for second order fully nonlinear parabolic systems, that satisfy the Campanato's condition of ellipticity. We use the Campanato's near operators theory .

Let Ω be a bounded convex open set of ℝ n , n ≥ 2, ∂Ω of class C 2. We consider the following Cauchy–Dirichlet problem where f ∈ L 2,λ(0, T; L 2(Ω, ℝ N )), 0

We prove the existence of solutions for a nonlocal equation arising from the mathematical modeling of MicroElectroMechanicalSystems (MEMS). The existence result, obtained within a suitable Implicit Function Theorem framework, is established under rather general boundary conditions and for bounded domains whose diameter is fairly small.

Let $\Omega$ be a bounded convex open set of $\mathbb{R}^n,$ $n\geq 2,$ $\partial \Omega $ of class $C^{2,1}.$ We consider the following Dirichlet problem \begin{equation} \left\{\begin{array}{l} u\in H^2\cap H^1_0(\Omega,\mathbb{R}^N) \\ F(x,D^2 u(x))= f(x), \quad \text{a.e. in} \,\,\,\Omega, \end{array} \right. \end{equation} where $f\in {\mathcal L}^{2,\lambda}(\Omega,\mathbb{R}^N),$ $n \leq$ $\lambda< n+2$, $F$ satisfies Campanato's Condition $A_x$ and is Holder continuous in $\Omega$ with exponent $b.$ We show that there exist $\varepsilon, \overline{\varepsilon}\in (0,1),$ ($\varepsilon,\overline{\varepsilon}$ depend on $\gamma$ and $\delta$), such that for any $\zeta \in (0,\overline{\varepsilon}\, n) ,$ and $ \mu \in( 0,\lambda],$ with $ \mu< (2b+\zeta)\wedge [\epsilon\,(n+2)],$ we have $D^2 u \in {\mathcal L}^{2,\mu}(\Omega,\mathbb{R}^{n^2N}),$ where $\varepsilon$ and $\overline{\varepsilon}$ depend on the constants appearing in Condition $A_x.$

We establish a Caccioppoli-type inequality for a second-order degenerate elliptic systems of p -Laplacian type. As a direct consequence, exploiting classical Campanato’s approach and the hole-filling technique due to Widman, we are able to prove a global regularity result on Morrey’s and L q spaces, with q > p .

We establish existence and regularity results for a time dependent fourth-order integro-differential equation with a possibly singular nonlinearity which has applications in designing MicroElectroMechanicalSystems. The key ingredient in our approach, besides basic theory of hyperbolic equations in Hilbert spaces, exploits the Near Operators Theory introduced by Campanato.

We show existence theorems of global strong solutions of Dirichlet problem for second-order fully nonlinear systems that satisfy the Campanato's condition of ellipticity. We use the Campanato's near operators theory.

We show the existence of solution in L p spaces for a generalized form of the classical Von Karman equations where the coefficients of nonlinear terms are variable. We use Campanato's near operators theory.

Let Ω be a bounded convex open set of ℝ n , n ≥ 2, of class C 2,1. We consider the following Dirichlet problem where f ∈ L 2,λ€(Ω, ℝ N ), 0 < λ < n, and F satisfying the condition A x . We show that there exists ϵ ∈ (0, 1) such that D 2 u ∈ L 2,μ(Ω, ℝ N ), where μ ∈ (0, λ) and μ < nϵ.