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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.

Global existence and uniqueness conditions for a dimensionless fourth-order integro-differential model for an electrostatic-elastic MEMS device with parallel plates and fringing field contribution were recently achieved by the Authors. Moving from this work, once the dielectric profile of the deformable plate according with experimental setups has been assigned, some technical conditions of applicability for the intended use of the device as well as the mechanical tension of the deformable plate are presented and discussed. Then, highlighting the link between the fringing field and the electrostatic force, finite differences were exploited for recovering the deformable plate profile according both global existence and uniqueness conditions. Moreover, the influence of the electro-mechanical properties of the deformable plate on both the numerical approach and on the intended uses of the device is discussed, comparing the results with experimental setups regarding pull-in voltage and electrostatic pressure.

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.

The membrane MEMSs represent a good design solution for the industry requirements about the construction of micro-dimensional devices, because easily constructible and extremely versatile. In this domain, the experience of the authors in the modeling of membrane MEMS devices has matured. In this chapter, they present a formalization of stationary 1D-membrane MEMS in which the electric field magnitude, |E|, is proportional to the curvature of the membrane, C, obtaining a semilinear elliptic model. Next, techniques based on fixed point Theorems provide results of existence, while an approach based on the joint use of Poincare’s inequality and Gronwall’s Lemma establish conditions of uniqueness. Finally, some numerical tests complete the work.

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 .

In the framework of 2D circular membrane Micro-Electric-Mechanical-Systems (MEMS), a new non-linear second-order differential model with singularity in the steady-state case is presented in this paper. In particular, starting from the fact that the electric field magnitude is locally proportional to the curvature of the membrane, the problem is formalized in terms of the mean curvature. Then, a result of the existence of at least one solution is achieved. Finally, two different approaches prove that the uniqueness of the solutions is not ensured.

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 $A_x.$

In this work we prove that the solutions $$u\in L^{q}(-T, 0,H^{1,q}(\Omega,\mathbb{R}^{N})) \cap C^{0,\lambda}(\overline{Q},\mathbb{R}^{N})$$ (1 1) to the second order nonlinear parabolic system of variational type $$-\sum^{n}\limits_{j=1}D_{j}a^{j}(X, u,Du) + \frac{\partial u}{\partial t} = B^{0}(X, u,Du),\quad X= (x, t)\in Q,$$ belong to the space \({L^{q}(-a, 0,H^{2,q}_{\rm loc} (\Omega,\mathbb{R}^{N})) \cap H^{1,q}(-a, 0, L^{q}_{\rm loc}(\Omega,\mathbb{R}^{N}))}\) , for all \({a\in ]0, T[}\) , provided suitable hypotheses of q-nonlinearity (1 < q < 2) on the coefficients are adopted.

Let Ω be a bounded open subset of ℝ n , n > 2. In Ω we deduce the global differentiability result $$ u \in H^2 (\Omega ,\mathbb{R}^N ) $$ for the solutions u ∈ H 1 (Ω, ℝ n ) of the Dirichlet problem $$ u - g \in H_0^1 (\Omega ,\mathbb{R}^N ), - \sum\limits_i {D_i a^i (x,u,Du)} = B_0 (x,u,Du) $$ with controlled growth and nonlinearity q = 2. The result was obtained by first extending the interior differentiability result near the boundary and then proving the global differentiability result making use of a covering procedure.

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.

We study the regularity of solutions of Von Karman equations with variables coefficients.We study the regularity of solutions of Von Karman equations with variables coefficients.

We study the existence and uniqueness of solutions (v,w)∈H4,p∩H02,p(Ω)×H4,p∩H02,p(Ω), 1

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.

We show a new theorem in Campanato’s near operators theory and give an example of one of its applications to a classical problem in the nonlinear elasticity theory: Von Karman equations.

In this paper we extend the results of [4] on the local differentiability of the weak solutions of the nonlinear parabolic systems with quadratic growth $$ - \sum\limits_{j = 1}^n {D_j a^j (X, u, Du) + \frac{{\partial u}}{{\partial t}} = B^0 (X, u, Du )} $$ to the case in which the coefficientsa j (X,u,p) are uniformly monotone inP.

Let Ω be a bounded open subset of ℝn, let X = (x,t) be a point of ℝn × ℝN. In the cylinder Q = Ω × (−T,0), T>0, we deduce the local differentiability result $$ u \in L^2 \left( { - a,0,H^2 \left( {B\left( \sigma \right),\mathbb{R}^N } \right)} \right) \cap H^l \left( { - a,0,L^2 \left( {B\left( \sigma \right),\mathbb{R}^N } \right)} \right) $$ for the solutions u of the class \( L^q \left( { - T,0,H^{l,q} \left( {\Omega ,\mathbb{R}^N } \right)} \right) \cap C^{0,\lambda } \left( {\bar Q,\mathbb{R}^N } \right) \) (0 < λ < 1, N integer ≥ 1) of the non linear parabolic system $$ - \sum\limits_{i = 1}^n {D_i a^i \left( {X,u,Du} \right) + \partial u\partial t = B^0 \left( {X,u,Du} \right)} $$ with quadratic growth and non linearity q ≥ 2. This result had been obtained making use of the interpolation theory and an imbedding theorem of Gagliardo-Nirenberg type for functions u belonging to W l,q ⋂ C 0,λ .

In this paper we give some existence and uniqueness theorems for weak solutions of boundary value problems for a particular class of parabolic systems of linear partial differential equations of the second order with real coefficients. In particular some uniqueness theorems for equations with complex coefficients are deduced from the previous results.