We consider fourth order nonlinear problems which describe electrostatic actuation in MicroElectroMechanicalSystems (MEMS) both in the stationary case and in the evolution case; we prove existence, uniqueness and regularity theorems by exploiting the Near Operators Theory

A. Tarsia

D. Cassani

L. Fattorusso

CASSANI, DANIELE

Archivio istituzionale della ricerca - Università dell'Insubria

Nonlocal Singular Problems and Applications toMEMSD. Cassani, L. Fattorusso, A. TarsiaAbstract—We consider fourth order nonlinear problemswhich describe electrostatic actuation in MicroElectroMechan-icalSystems (MEMS) both in the stationary case and in theevolution case; we prove existence, uniqueness and regularitytheorems by exploiting the Near Operators Theory.Index Terms—Singular nonlinearities; Integro-differentialequations; Higher order elliptic and hyperbolic PDE; Regular-ity results; Steklov boundary conditions; Near operators theory;Implicit function theorem; MEMS and NEMS; Electrostaticactuation.I. INTRODUCTIONRecently a lot of attention has been devoted to thestudy of mathematical models which describe, with differentlevels of accuracy, the so-called electrostatic actuation inMicroElectroMechanicalSystems (MEMS), see e.g. [11] andreferences therein. These models are studied by consideringnonlinear problems involving nonlinearities which developsingularities.As an example we consider a plate set on a micro scale whichis suitably fixed at boundary of a region Ω ⊂ RN . Once thata drop voltage is applied between the deflecting plate and aground plate, the micro-plate leaves the steady state u = 0moving towards the ground plate set at height u = 1.The deformation profile u of the MEMS is then governed,in the stationary case, by the following model:α∆2u =(β∫Ω|∇u|2 dx+ γ)∆u+λf(x)(1− u)σ(1 + χ∫Ωdx(1− u)σ−1) , x ∈ Ωu = ∆u− duν = 0, x ∈ ∂Ω, d ≥ 00 ≤ u < 1, x ∈ Ω(1)Here:• Ω ⊂ RN is a smooth bounded domain;• u : Ω −→ R is the unknown profile of the deflectingMEMS plate;• f : Ω −→ R+ is a bounded function which carries di-electric properties of the material (permittivity profile);Daniele Cassani, Department of Science and High Tecnology, Universita`degli Studi dell’Insubria, Via Valleggio 11, 22100–Como (Italy), email:daniele.cassani@uninsubria.itLuisa Fattorusso, Department of Information Engineering, Infrastruc-ture and Sustainable Energy (DIIES), Universita` Mediterranea di Reg-gio Calabria, Via Graziella, 89122 Reggio Calabria (Italy), e-mail:luisa.fattorusso@unirc.it.Antonio Tarsia, Department of Mathematics, Universita` di Pisa, ViaF. Buonarroti 2, 56127–Pisa (Italy), email: tarsia@dm.unipi.it• λ ≥ 0 is the drop voltage between the ground plate andthe deflecting plate;• the positive parameters α, β, γ, χ which are respectivelyrelated to the thickness (rigidity) of the deflecting plate,material deformation (self-stretching), tangential tensionforces (stretching), and nonlocal dependence of theelectrostatic potential on the solution itself (non uniformelectric charge distribution) and for σ ≥ 2 which takesinto account more general potential than Coulomb’s.This is an extension of the nonlocal MEMS problem studiedby Cassani-do O´-Ghoussoub [8] where the case σ = 2(Coulomb potential) was considered, as well as α = 1,β = 0, γ = 0 and χ = 0, namely:∆2 u(x) =λ f(x)[1− u(x)]2 ;0 ≤ u(x) < 1, in Ω,u = ∆u− duν = 0, on ∂Ω, d ≥ 0(2)It is well known that the role of boundary conditions inhigher order problems is very delicate, as pointed out in [12].Here we deal with a rather general physical situation inwhich Steklov boundary conditions are considered and givenbyu = ∆u− duν = 0, x ∈ ∂Ω, d ≥ 0and from which we obtain Dirichlet (u = uν = 0) and Navier(u = ∆u = 0) boundary conditions by setting respectivelyd =∞ and d = 0.In the stationary case we consider: u(t, x) = u(x); λ(t) =λ ≥ 0; 0 ≤ f(x) ≤ 1(Sλ)∆2u = λf(x)(1− u)2 , in Ω ⊂ RN0 ≤ u < 1, in Ωu = ∆u = 0 (Navier)u = uν = 0 (Dirichlet) on ∂Ωu = ∆u− duν = 0 (Steklov),(3)where f is the permittivity profile of the material. Solutionshave to be understood in the following sense:Weak solutions: uλ, 1/(1−uλ)2 ∈ L1(Ω), 0 ≤ uλ ≤ 1 suchthat ∫Ωuλ∆2ϕdx = λ∫Ωh(x)(1− uλ)2ϕdx, ϕ-testEnergy solutions: weak-solutions such that uλ ∈ H, theSobolev space H20 or H2 ∩ H10 , accordingly to boundaryconditions.The first result in the case of Dirichlet boundary conditionswas obtained in [8] and can be summarized as follows:Proceedings of the World Congress on Engineering 2013 Vol II, WCE 2013, July 3 - 5, 2013, London, U.K.ISBN: 978-988-19252-8-2 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)WCE 2013• There exists a minimal (pointwise) classical (smooth)solution uλ for 0 < λ < λ∗(Ω, f,N);• λ∗ = λ∗ =: sup{λ | there exists a weak solution};• u∗(x) = limλ↗λ∗ uλ(x) is an energy solution and it isunique,which was further developed in [10, Cowan-Esposito-Ghoussoub-Moradifam]. Navier boundary conditions wereconsidered in [16, Lin-Yang’07], [15, Guo-Wei’08] and [9,Cowan-Esposito-Ghoussoub’10] whereas Steklov boundaryconditions in [2, Berchio-Cassani-Gazzola’10].II. MAIN RESULTSThe following result was proved in [6, Cassani-Fattorusso-Tarsia ’11].Theorem 1: Let the dimension N < 8, Ω ⊂ RN be abounded domain, f ∈ L∞(Ω) and α, β, γ, χ > 0. Then,there exist λ∗, d0 ∈ (0,∞) such that for λ ∈ (0, λ∗) problem(1) possesses a solution u ∈ H4(Ω) provided one of thefollowing holds:(SN) 0 ≤ d < d0or(D) d =∞ and Ω is a balland the diameter of Ω is sufficiently small.A. Remarks1) It is worth to emphasize that the solution provided byTheorem 1, u = uλ for a fixed 0 < λ < λ∗, is suchthat ‖uλ‖∞ ≤ C < 1; as a consequence, by ellipticregularity uλ turns out to be smooth.2) Let us mention that in applications, the domain Ω rep-resents the region occupied by the undeflected MEMSplate which is set on a micro-scale basis; therefore, inthis respect, restrictions from above on the diameter ofthe domain do not seem too stringent.3) The restriction in Theorem 1 on the dimension N < 8is somehow expected as a consequence of [10], [2]as for N ≥ 9 the solution in the semilinear case andavoiding nonlocal effects, is singular approaching λ∗,in the sense that for λ = λ∗ one has ‖u0‖∞ = 1; thisclearly prevents any existence result to (1).B. The abstract setting: a version of the Implicit FunctionTheoremThe key-ingredient in the study of the stationary case isprovided by the following version of the implicit functiontheorem proved in [20, Tarsia ’98] and which remarkablyextends the Near Operator Theory introduced in [4, Cam-panato ’94].Theorem 2: Let X be a topological space, Y a set, Z aBanach space and the following mappings F : X×Y → Z,B : Y → Z. Suppose that:(i) there exists (x0,y0) ∈ X × Y such that F (x0,y0) = 0;(ii) the map x→ F (x0,y0) is continuous at x0;(iii) there exist k1 > 0, k2 ∈ (0, 1) and a neighborhood of x0,U(x0) ⊂ X, such that, for all y1, y2 ∈ Y and for all x ∈U(x0), we have‖B(y1)−B(y2)− k1[F (x,y1)− F (x,y2)]‖Z≤ k2‖B(y1)−B(y2)‖Z(iv) B is injective;(v) B(Y ) is a neighborhood of z0 = B(y0).Then, there exists a ball S(z0, r) ⊂ B(Y ) and a neigh-borhood of x0, V (x0) ⊂ U(x0), such that the followingproblem: F (x,y(x)) = 0, ∀x ∈ V (x0),y(x0) = y0(4)posseses an unique solution y : V (x0) → B−1(S(z0, r)).Moreover, if condition (iii) holds for all x ∈ X, then thesolution y = y(x) turns out to be defined in the whole X .The second main tool which is actually the starting pointto set up the strategy outlined in Theorem 2, can be obtainedby joining some results of [8] and [2] which we recall in thefollowingTheorem 3: Let β = γ = χ = 0 in (1). Then thereexists λ∗ ∈ (0,∞) such that for all λ ∈ (0, λ∗) problem (1)possesses a classical solution provided one of the followingholds:(S) 0 ≤ d < d0, where d0 is the first simple boundaryeigenvalue of the biharmonic operator ∆2 under Steklovboundary conditions;(D) Ω is a ball and d = ∞, which corresponds to theDirichlet boundary conditions u = uν = 0 on ∂Ω.C. Sketch of the proof of Theorem 1First we apply implicit function theorem to show that theproblem is well posed and for this purpose we assumex = (α, β, γ, χ, f, λ), y = y(x);F (x,y) = F (α, β, γ, χ, f, λ, y)= α∆2y(x) −[β∫Ω|∇y(x)|2 dx γ]∆y(x)+λ f(x)[1− y(x)]σ{1 + χ∫Ω1[1− y(x)]σ−1}dx;(5)X = R+ × R+ × R+ × R+×{f ∈ L∞(Ω) : |x : f(x) > 0| 6= 0}×{λ : 0 ≤ λ < λ∗ < +∞};B(y) = ∆2y(x);Y = Yd ={u ∈ H4(Ω) : u = ∆u − d ∂u∂ν= 0,a.e. in ∂Ω, d ≥ 0, 0 < u < 1,∫Ω1[1− u(x)]8σ dx < M1,∫Ω|∆u(x)|2dx < M2};Z = L2(Ω).(6)We consider (x0,y0), belonging to X × Y , which enjoysF (x0,y0) = 0, where x0 = (1, 0, 0, 0, f0, λ) e y0 =y0(x0) = u0(x) is the solution ofProceedings of the World Congress on Engineering 2013 Vol II, WCE 2013, July 3 - 5, 2013, London, U.K.ISBN: 978-988-19252-8-2 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)WCE 2013∆2 u0(x) =λ f0(x)[1− u0(x)]20 < u0(x) < 1, in Ωu0(x) = ∆u0 − d ∂u0(x)∂ν= 0, on ∂Ω(7)as provided by Theorem 3. Next we sketch how assumptionsof Theorem 2 turn out to be verified.Assumption (i):F (x0,y0) = ∆2 u0(x) − λ f0(x)[1− u0(x)]2 = 0follows directly from the existence results for problem (Sλ)subject to Steklov boundary conditions as proved in [2].Condition (ii) is verified since the dependence of Fthrough parameters is continuous.The assumptions (iv) e (v) follow by the properties of theoperator ∆2.To verify the (iii) we show that there exists k1 ∈ (0, 1)such that for every (α, β, γ, χ, f, λ) ∈ X one has∫Ω|α∆2y1(x) − α∆2y2(x)− [F (α, β, γ, χ, f, λ, y1) − F (α, β, γ, χ, f, λ, y2)] |2 dx≤ k1∫Ω|α∆2y1(x) − α∆2y2(x)|2 dx (8)and in turn we have∫Ω|[G(β, γ, y1(x)) + H(λ, χ, f(x), y1(x))]− [G(β, γ, y2(x)) + H(λ, χ, f(x), y2(x))]|2 dx≤ 2∫Ω|G(β, γ, y1(x)) − G(β, γ, y2(x))|2 dx+2∫Ω|H(λ, χ, f(x), y1(x))]−H(λ, χ, f(x), y2(x))|2 dx≤ k1∫Ω|α∆2y1(x) − ∆2y2(x)|2 dx (9)where we have setG(β, γ, u(x)) =[β∫Ω|∇u(x)|2 dx + γ]∆u(x) (10)andH(λ, χ, f(x), u(x))=λ f(x)[1− u(x)]2[1 + χ∫Ω1[1− u(x, t)]2] dxObserve that we obtain the existence result globally in thepositive parameters α, β, γ, χ as well as for any σ ≥ 2,f ∈ L∞, λ ∈ (0, λ∗) as consequence of the last claim inTheorem 2.III. NONLOCAL TIME DEPENDENT PROBLEMSRecently in [7] the authors obtain existence, uniquenessand regularity results for a model which takes into accountthe dynamic of the problem as follows:α∆2 u(x, t) + c u′(x, t) + p u′′(x, t)=[β∫Ω |∇u(x, t)|2 dx + γ]∆u(x, t)+λ(t)f(x)[1− u(x, t)]σ[1 + χ∫Ω1[1− u(x, t)]σ−1 dx] ,0 ≤ u(x, t) < 1, in Ω× [0, T ]u(x, 0) = u0, in Ωu′(x, 0) = 0, in Ωu(x, t) = ∆u(x, t) − d ∂u(x, t)∂ν= 0, on ∂Ω× [0, T ](11)Theorem 4: Let Ω ⊂ RN , 1 ≤ N ≤ 3, be a boundeddomain with sufficiently small diameter, σ ≥ 2, non negativeconstants β, γ, χ and 0 ≤ d < d0, where d0 is the firstboundary eigenvalue of the biharmonic operator subjectto Steklov boundary conditions. Let also p, c be boundedfunctions and λ ∈ C1((0, T );L2(Ω)) such that ‖λ‖∞ < λ∗,u0 ∈ H2 ∩ H10 (Ω) (satisfying suitable compatibility con-ditions) and u1 ∈ L2(Ω). Then, problem (11) possesses aunique solution u ∈ C0([0, T ];H2(Ω)) ∩C1([0, T ];L2(Ω)).The same conclusion holds if d =∞ and Ω is a ball.Theorem 5: Letu ∈ C0([0, T ];H20 (Ω)) ∩ C1([0, T ];L2(Ω))be the solution to problem (11) given by Theorem 4. Assumeu0, u1 ∈ H2 ∩H10 (Ω) and c ∈W 1,∞((0, T );L2(Ω)). Then,the solution enjoys the following regularity:u ∈ C0([0, T ];H4(Ω)) ∩ C1([0, T ];H2 ∩H10 (Ω))∩ C2([0, T ];L2(Ω))Theorems (4) and (5) follows by means of a non straight-forward extension of the technique used in the stationarycase to the dynamic setting and again this approach enablesus to prove existence and uniqueness of the solution locallyin time but globally in the physical parameters involved inthe problem.Moreover, differently from the stationary case, here theproblem of regularity is somehow delicate as the equationmanifests itself through an hyperbolic nature. Then we areconcerned with proving regularity of solutions by adaptingand further developing abstract results of [1] and [13]. Inthis respect, it is worth to mention that standard interpolationtheory does not suite optimal regularity results even with theaid of higher order (operator) perturbations in a penalizedframework.Finally, let us mention that the inverse identificationproblem of identifying the pull-in voltage λ(t) in (11),under a suitable (accessible and measurable) supplementaryinformation on the solution, has been studied in [5, Cassani-Kaltenbacher-Lorenzi ’09].Proceedings of the World Congress on Engineering 2013 Vol II, WCE 2013, July 3 - 5, 2013, London, U.K.ISBN: 978-988-19252-8-2 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)WCE 2013REFERENCES[1] C. Baiocchi, Soluzioni ordinarie e generalizzate del problema diCauchy per equazioni differenzilai astratte del secondo ordine in spazidi Hilbert, Ricerche Mat. 16 (1967), 27–95.[2] E. Berchio, D. Cassani and F. 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Methods Nonlinear Anal. 11(1998), 115–133.Proceedings of the World Congress on Engineering 2013 Vol II, WCE 2013, July 3 - 5, 2013, London, U.K.ISBN: 978-988-19252-8-2 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)WCE 2013

Nonlocal Singular Problems and Applications to MEMS

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