The occurrence of a finite time singularity is shown for a free boundary problem modeling microelectromechanical systems (MEMS) when the applied voltage exceeds some value. The model involves a singular nonlocal reaction term and a nonlinear curvature term accounting for large deformations

Brubaker

Christoph Walker

Cimatti

Escher

Esposito

Flores

Joachim Escher

Laurençot

Pelesko

Philippe Laurençot

arXiv

Escher, Joachim

Laurencot, Philippe

Walker, Christoph

HAL-INSA Toulouse

Finite time singularity in a free boundary problem modeling MEMS

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Finite time singularity in a free boundary problemmodeling MEMSJoachim Escher, Philippe Laurencot, Christoph WalkerTo cite this version:Joachim Escher, Philippe Laurencot, Christoph Walker. Finite time singularity in a freeboundary problem modeling MEMS. Comptes rendus de l’Acade´mie des sciences. Se´rieI, Mathe´matique, Elsevier, 2013, 351, pp.807-812. <10.1016/j.crma.2013.10.004>. <hal-00828702>HAL Id: hal-00828702https://hal.archives-ouvertes.fr/hal-00828702Submitted on 31 May 2013HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.FINITE TIME SINGULARITY IN A FREE BOUNDARY PROBLEMMODELING MEMSJOACHIM ESCHER, PHILIPPE LAURENC¸OT, AND CHRISTOPH WALKERAbstract. The occurrence of a finite time singularity is shown for a free boundary problem mod-eling microelectromechanical systems (MEMS) when the applied voltage exceeds some value. Themodel involves a singular nonlocal reaction term and a nonlinear curvature term accounting for largedeformations.1. IntroductionAn idealized electostatically actuated microelectromechanical system (MEMS) consists of a fixedhorizontal ground plate held at zero potential above which an elastic membrane held at potential Vis suspended. A Coulomb force is generated by the potential difference across the device and resultsin a deformation of the membrane, thereby converting electrostatic energy into mechanical energy,see [1, 4, 7] for a more detailed account and further references. After a suitable scaling and assuminghomogeneity in transversal horizontal direction, the ground plate is assumed to be located at z = −1and the membrane displacement u = u(t, x) ∈ (−1,∞) with t > 0 and x ∈ I := (−1, 1) evolvesaccording to∂tu− ∂x(∂xu√1 + ε2(∂xu)2)= −λ(ε2 |∂xψ(t, x, u(t, x))|2 + |∂zψ(t, x, u(t, x))|2), (1)for t > 0 and x ∈ I with boundary conditionsu(t,±1) = 0 , t > 0 , (2)and initial conditionu(0, x) = u0(x) , x ∈ I . (3)The electrostatic potential ψ = ψ(t, x, z) satisfies a rescaled Laplace equation in the regionΩ(u(t)) := {(x, z) ∈ I × (−1,∞) : −1 < z < u(t, x)}between the plate and the membrane which readsε2 ∂2xψ + ∂2zψ = 0 , (x, z) ∈ Ω(u(t)) , t > 0 , (4)ψ(t, x, z) =1 + z1 + u(t, x), (x, z) ∈ ∂Ω(u(t)) , t > 0 , (5)Date: May 31, 2013.Partially supported by the Deutscher Akademischer Austausch Dienst (DAAD).12 JOACHIM ESCHER, PHILIPPE LAURENC¸OT, AND CHRISTOPH WALKERwhere ε > 0 denotes the aspect ratio of the device and λ > 0 is proportional to the square of theapplied voltage. The dynamics of (u, ψ) is thus given by the coupling of a quasilinear parabolicequation for u and an elliptic equation in a moving domain for ψ, the latter being only well-definedas long as the membrane does not touch down on the ground plate, that is, u does not reach thevalue −1. To guarantee optimal operating conditions of the device, this touchdown phenomenon hasto be controlled and its occurrence is obviously related to the value of λ.The main difficulty to be overcome in the analysis of (1)-(5) is the nonlocal and nonlinear implicitdependence on u of the right-hand side of (1) which is also singular if u approaches −1. Except forthe singularity, these features disappear when setting ε = 0 in (1)-(5), a commonly made assumptionwhich reduces (1)-(5) to a singular semilinear reaction-diffusion equation. This so-called small aspectratio model has received considerable attention in recent years, see [4, 7] and the references therein.In this simplified situation, it has been established that touchdown does not take place if λ is belowa certain threshold value λ∗ > 0, but occurs if λ exceeds this value [4, 5, 6].We have recently investigated the well-posedness of (1)-(5) and established the following result [3].Theorem 1 (Local Well-Posedness). Let q ∈ (2,∞), ε > 0, λ > 0, and consider an initial valueu0 ∈ W 2q (I) such that u0(±1) = 0 and 0 ≥ u0(x) > −1 for x ∈ I . (6)Then there is a unique maximal solution (u, ψ) to (1)-(5) on the maximal interval of existence [0, T εm)in the sense thatu ∈ C1([0, T εm), Lq(I))∩ C([0, T εm),W2q (I))satisfies (1)-(3) together with0 ≥ u(t, x) > −1 , (t, x) ∈ [0, T εm)× I , (7)and ψ(t) ∈ W 22(Ω(u(t)))solves (4)-(5) for each t ∈ [0, T εm).We have also shown in [3] that, if λ and u0 are sufficiently small, the solution (u, ψ) to (1)-(5)exists for all times (i.e. T εm =∞) and touchdown does not take place, not even in infinite time.Theorem 2 (Global Existence). Let q ∈ (2,∞), ε > 0, and consider an initial value u0 sat-isfying (6). Given κ ∈ (0, 1), there are λ∗(κ) > 0 and r(κ) > 0 such that, if λ ∈ (0, λ∗(κ)) and‖u0‖W 2q(I) ≤ r(κ), the maximal solution (u, ψ) to (1)-(5) exists for all times and u(t, x) ≥ −1+ κ for(t, x) ∈ [0,∞)× I.On the other hand, we have been able to prove that no stationary solution to (1)-(5) existsprovided λ is sufficiently large. However,whether or not T εm is finite in this case has been left as anopen question. The purpose of this note is to show that – as expected on physical grounds – T εm isindeed finite for λ sufficiently large.Theorem 3 (Finite time singularity). Let q ∈ (2,∞), ε > 0, and consider an initial value u0satisfying (6). If λ > 1/ε and (u, ψ) denotes the maximal solution to (1)-(5) defined on [0, T εm), thenT εm <∞.FINITE TIME SINGULARITY IN A FREE BOUNDARY PROBLEM MODELING MEMS 3The criterion λ > 1/ε is likely to be far from optimal. As we shall see below, improving it wouldrequire to have a better control on ∂xu(±1). The proof of Theorem 3 relies on the derivation of achain of estimates which allow us to obtain a lower bound on the L1-norm of the right-hand sideof (1) depending only on u. The lower bound thus obtained is in fact the mean value of a convexfunction of u, and we may then end the proof with the help of Jensen’s inequality, an argumentwhich has already been used for the small aspect ratio model, see [5, 6].We shall point out that, in contrast to the small aspect ratio model, the finiteness of T εm does notguarantee that the touchdown phenomenon really takes place as t → T εm. Indeed, according to [3,Theorem 1.1 (ii)], the finiteness of T εm implies that min[−1,1] u(t) −→ −1 or ‖u(t)‖W 2q (I) −→ ∞ ast → T εm. While the former corresponds to the touchdown behaviour, the latter is more likely to beinterpreted as the membrane being no longer the graph of a function at time T εm.2. Proof of Theorem 3Let q ∈ (2,∞), ε > 0, λ > 0 and consider an initial value u0 satisfying (6). We denote themaximal solution to (1)-(5) defined on [0, T εm) by (u, ψ). Differentiating the boundary conditions (5),we readily obtain∂xψ(t, x,−1) = ∂xψ(t, x, u(t, x)) + ∂xu(t, x) ∂zψ(t, x, u(t, x)) = 0 , (t, x) ∈ (0, Tεm)× I , (8)and∂zψ(t,±1, z) = 1 , (t, z) ∈ (0, Tεm)× (−1, 0) . (9)Additional information on the boundary behaviour of ψ is provided by the next lemma.Lemma 4. For t ∈ (0, T εm),1 + z ≤ ψ(t, x, z) ≤ 1 , (x, z) ∈ Ω(u(t)) , (10)±∂xψ(t,±1, z) ≤ 0 , z ∈ (−1, 0) . (11)Proof. Fix t ∈ (0, T εm). The upper bound in (10) readily follows from the maximum principle. Next,the function σ, defined by σ(x, z) = 1 + z, obviously satisfies ε2∂2xσ + ∂2zσ = 0 in Ω(u(t)) as well asσ(±1, z) = 1 + z = ψ(t,±1, z) , z ∈ (−1, 0) ,σ(x,−1) = 0 = ψ(t, x,−1) , x ∈ (−1, 1) .Owing to the non-positivity (7) of u(t), it also satisfiesσ(x, u(t, x)) = 1 + u(t, x) ≤ 1 = ψ(t, x, u(t, x)) , x ∈ (−1, 1) ,and we infer from the comparison principle that ψ(t, x, z) ≥ σ(x, z) for (x, z) ∈ Ω(u(t)). It thenfollows from (10) that ψ(t, x, z) ≥ 1 + z = ψ(t,±1, z) for (x, z) ∈ Ω(u(t)) which readily implies (11).To simplify notations, we setγm(t, x) := ∂zψ(t, x, u(t, x)) , γg(t, x) := ∂zψ(t, x,−1) , (t, x) ∈ (0, Tεm)× (−1, 1) , (12)4 JOACHIM ESCHER, PHILIPPE LAURENC¸OT, AND CHRISTOPH WALKERand first derive an upper bound of the L1-norm of the right-hand side of (1), observing that, due to(8), it also reads−λε2 |∂xψ(t, x, u(t, x))|2 + |∂zψ(t, x, u(t, x))|2 = −λ(1 + ε2(∂xu(t, x))2)γm(t, x)2 .Lemma 5. For t ∈ (0, T εm),∫ 1−1(1 + ε2(∂xu(t, x))2)γm(t, x)2 dx ≥ 2∫ 1−1(1 + ε2(∂xu(t, x))2)γm(t, x) dx− 2 . (13)Proof. Fix t ∈ (0, T εm). We multiply (4) by ∂zψ(t) − 1 and integrate over Ω(u(t)). Using (8), (9),and Green’s formula we obtain0 =− ε2∫Ω(u)∂x∂zψ ∂xψ d(x, z) + ε2∫ 1−1(∂xu)2 γm (γm − 1) dx−12∫ 1−1(γ2g − 2γg)dx+12∫ 1−1(γ2m − 2γm)dx .Since ∫Ω(u)∂x∂zψ ∂xψ d(x, z) =12∫ 1−1(∂xu)2 γ2m dxby (8) and since γ2g − 2γg ≥ −1, we end up with (13). We again use (4) to obtain a lower bound for the boundary integral of the right-hand side of (13)which depends on the Dirichlet energy of ψ.Lemma 6. For t ∈ (0, T εm),∫ 1−1(1 + ε2(∂xu(t, x))2)γm(t, x) dx ≥∫Ω(u(t))(ε2|∂xψ(t, x, z)|2 + |∂zψ(t, x, z)|2)d(x, z) . (14)Proof. Fix t ∈ (0, T εm). We multiply (4) by ψ(t) and integrate over Ω(u(t)). Using (5), (8), andGreen’s formula we obtain0 =−∫Ω(u(t))(ε2|∂xψ(t, x, z)|2 + |∂zψ(t, x, z)|2)d(x, z) + ε2∫ 0−1(1 + z) ∂xψ(t, 1, z) dz− ε2∫ 0−1(1 + z) ∂xψ(t,−1, z) dz + ε2∫ 1−1(∂xu(t, x))2 γm(t, x) dx+∫ 1−1γm(t, x) dx .Owing to (11), the second and third terms of the right-hand side of the above equality are non-positive, whence (14). We finally argue as in [2, Lemma 9] to establish a connection between the Dirichlet energy of ψand u.FINITE TIME SINGULARITY IN A FREE BOUNDARY PROBLEM MODELING MEMS 5Lemma 7. For t ∈ (0, T εm),∫Ω(u(t))(ε2|∂xψ(t, x, z)|2 + |∂zψ(t, x, z)|2)d(x, z) ≥∫ 1−1dx1 + u(t, x). (15)Proof. Let t ∈ (0, T εm) and x ∈ (−1, 1). We deduce from (5) and the Cauchy-Schwarz inequalitythat11 + u(t, x)=(ψ(t, x, u(t, x))− ψ(t, x,−1))21 + u(t, x)=11 + u(t, x)(∫ u(t,x)−1∂zψ(t, x, z) dz)2≤∫ u(t,x)−1(∂zψ(t, x, z))2 dz . (16)Integrating the above inequality with respect to x ∈ (−1, 1) readily gives (15). Remark 8. Observe that (16) provides a quantitative estimate on the singularity of ∂zψ generated byu when touchdown occurs.Combining the three lemmas above with Jensen’s inequality give the following estimate.Proposition 9. For t ∈ (0, T εm),∫ 1−1(1 + ε2(∂xu(t, x))2)γm(t, x)2 dx ≥ 4ϕ(12∫ 1−1u(t, x) dx)− 2 , (17)where ϕ(r) := 1/(1 + r), r ∈ (−1,∞).Proof. Fix t ∈ (0, T εm). We infer from Lemma 5, Lemma 6, and Lemma 7 that∫ 1−1(1 + ε2(∂xu(t, x))2)γm(t, x)2 dx ≥ 2∫ 1−1ϕ(u(t, x)) dx− 2 .To complete the proof, we argue as in [5, 6] and use the convexity of ϕ and Jensen’s inequality toobtain (17). Proof of Theorem 3. IntroducingE(t) := −12∫ 1−1u(t, x) dx , t ∈ [0, T εm) ,the bounds (7) ensure that0 ≤ E(t) < 1 , t ∈ [0, T εm) . (18)6 JOACHIM ESCHER, PHILIPPE LAURENC¸OT, AND CHRISTOPH WALKERIt follows from (1), (8), and Proposition 9 thatdEdt(t) =−12 ∂xu(t, x)√1 + ε2 (∂xu(t, x))2x=1x=−1+λ2∫ 1−1(1 + ε2(∂xu(t, x))2)γm(t, x)2 dx≥ Fλ(E) := 2λϕ(−E)− λ−1ε. (19)If λ > 1/ε, we note that Fλ(0) > 0 and thus Fλ(r) ≥ Fλ(0) > 0 for r ∈ [0, 1) due to the monotonicityof Fλ. Since E(0) ≥ 0 by (18), it follows from (19) and the properties of Fλ that t 7→ E(t) isincreasing on [0, T εm). Consequently,dEdt(t) ≥ Fλ(E(0)) ≥ Fλ(0) , t ∈ [0, Tεm) .Integrating the previous inequality with respect to time and using (18), we end up with the inequality1 ≥ E(0) + Fλ(0)Tεm which provides the claimed finiteness of Tεm. AcknowledgementsThis research was done while Ph.L. was enjoying the kind hospitality of the Institut fu¨r AngewandteMathematik of the Leibniz Universita¨t Hannover.References[1] N.D. Brubaker and J.A. Pelesko. Non-linear effects on canonical MEMS models. European J. Appl. Math 22(2011), 455–470.[2] J. Escher, Ph. Laurenc¸ot, and Ch. Walker. A parabolic free boundary problem modeling electrostatic MEMS.Arch. Rational Mech. Anal. (to appear).[3] J. Escher, Ph. Laurenc¸ot, and Ch. Walker. Dynamics of a free boundary problem with curvature modelingelectrostatic MEMS. arXiv:1302.6026 (submitted).[4] P. Esposito, N. Ghoussoub, and Y. Guo.Mathematical Analysis of Partial Differential Equations Modeling Electro-static MEMS, volume 20 of Courant Lecture Notes in Mathematics. Courant Institute of Mathematical Sciences,New York, 2010.[5] G. Flores, G. Mercado, J. A. Pelesko, and N. Smyth. Analysis of the dynamics and touchdown in a model ofelectrostatic MEMS. SIAM J. Appl. Math. 67 (2007), 434–446.[6] Y. Guo, Z. Pan, and M.J. Ward. Touchdown and pull-in voltage behavior of a MEMS device with varying dielectricproperties. SIAM J. Appl. Math. 66 (2005), 309–338.[7] J.A. Pelesko and D.H. Bernstein. Modeling MEMS and NEMS. Chapman & Hall/CRC, Boca Raton, FL, 2003.FINITE TIME SINGULARITY IN A FREE BOUNDARY PROBLEM MODELING MEMS 7Leibniz Universita¨t Hannover, Institut fu¨r Angewandte Mathematik, Welfengarten 1, D–30167Hannover, GermanyE-mail address : escher@ifam.uni-hannover.deInstitut de Mathe´matiques de Toulouse, CNRS UMR 5219, Universite´ de Toulouse, F–31062Toulouse Cedex 9, FranceE-mail address : laurenco@math.univ-toulouse.frLeibniz Universita¨t Hannover, Institut fu¨r Angewandte Mathematik, Welfengarten 1, D–30167Hannover, GermanyE-mail address : walker@ifam.uni-hannover.de

Laurençot, Philippe

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 351 (2013) 807–812Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comPartial differential equationsFinite time singularity in a free boundary problem modelingMEMSSingularité en temps fini pour un modèle de microsystèmeélectromécanique à frontière libreJoachim Escher a, Philippe Laurençot b,1, Christoph Walker aa Leibniz Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1, 30167 Hannover, Germanyb Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université de Toulouse, 31062 Toulouse cedex 9, Francea r t i c l e i n f o a b s t r a c tArticle history:Received 30 May 2013Accepted after revision 7 October 2013Available online 22 October 2013Presented by the Editorial BoardThe occurrence of a finite time singularity is shown for a free boundary problem modelingmicroelectromechanical systems (MEMS) when the applied voltage exceeds some value.The model involves a singular nonlocal reaction term and a nonlinear curvature term takinginto account large deformations.© 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éL’apparition d’une singularité en temps fini est établie pour un problème à frontière libredécrivant l’évolution spatio-temporelle d’un microsystème électromécanique lorsque latension appliquée est suffisamment élevée. Le modèle inclut un terme de réaction singulieret un terme non linéaire de courbure, prenant en compte les grandes déformations.© 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeUn transducteur électrostatique ou capacitif est un microsystème électromécanique constitué de deux électrodes, l’uneétant rigide et fixe et l’autre étant assimilée à une membrane élastique et fixée à ses deux extrémités. L’application d’unetension électrique entre les électrodes engendre une force électrostatique qui induit une déformation de la membraneélastique et convertit ainsi l’énergie électrique en énergie mécanique. La dynamique du dispositif est alors décrite par ladéformation verticale u de la membrane élastique et le potentiel électrostatique entre les deux électrodes ψ . Si on supposequ’il n’y a pas de variation dans la direction horizontale transverse, un changement d’échelle permet de supposer quel’électrode fixe est située en z = −1 et que la déformation de la membrane u = u(t, x) ∈ (−1,∞) à l’instant t > 0 et à laposition x ∈ I := (−1,1) vérifie l’équation (1) avec les conditions limites (2) et initiale (3). Le potentiel électrostatique estsolution de l’équation de Laplace renormalisée (4) dans la région située entre les deux électrodes :Ω(u(t)) := {(x, z) ∈ I × (−1,∞): −1 < z < u(t, x)}E-mail addresses: escher@ifam.uni-hannover.de (J. Escher), laurenco@math.univ-toulouse.fr (P. Laurençot), walker@ifam.uni-hannover.de (C. Walker).1 Partially supported by the Deutscher Akademischer Austausch Dienst (DAAD).1631-073X/$ – see front matter © 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.http://dx.doi.org/10.1016/j.crma.2013.10.004808 J. Escher et al. / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 807–812avec les conditions limites (5), le paramètre ε > 0 désignant le rapport entre les dimensions verticale et horizontales dudispositif et le paramètre λ > 0 étant proportionnel au carré de la tension électrique appliquée. L’évolution spatio-temporellede (u,ψ) est donc régie par le couplage d’une équation parabolique quasilinéaire pour la déformation u et d’une équationelliptique pour le potentiel électrostatique ψ . Cette dernière est posée dans le domaine dépendant du temps Ω(u(t)) etn’est bien définie que tant que la membrane élastique ne touche pas l’électrode rigide, c’est-à-dire tant que u > −1. Cettesituation (collage) peut effectivement se produire et dépend fortement de la valeur de λ. Ce phénomène doit être contrôléafin de garantir un fonctionnement optimal du dispositif.La relation entre le terme source dans (1) et la déformation u est clairement implicite, non locale et non linéaire, et c’estl’une des difficultés rencontrées dans l’analyse mathématique du système (1)–(5), l’autre étant le comportement singulierdu terme source dans (1) lorsque u → −1.Nous avons récemment montré dans [5] le caractère localement bien posé de (1)–(5) pour des données initiales suf-fisamment régulières. Plus précisément, étant donnés q > 2 et une condition initiale u0 ∈ W 2q (I) vérifiant u0(±1) = 0et 0  u0(x) > −1, x ∈ I , il existe une unique solution maximale (u,ψ) de (1)–(5) définie pour t ∈ [0, T εm) et telle que0  u(t, x) > −1, (t, x) ∈ [0, T εm) × I (Théorème 1). Nous avons, de plus, établi que cette solution existe globalement pourvuque λ et ‖u0‖W 2q (I) soient suffisamment petits. Dans ce cas, il existe κ ∈ (0,1) tel que u(t, x) > −1+κ pour (t, x) ∈ [0,∞)× I(Théorème 2).Dans cette note, nous étudions le comportement des solutions de (1)–(5) lorsque λ est suffisamment grand et montronsla formation d’une singularité en temps fini (Théorème 3). Plus précisément, T εm < ∞ dès que λ > 1/ε. D’une part, cerésultat complète un résultat de non-existence de solutions stationnaires pour les grandes valeurs de λ obtenu dans notrearticle précédent [5]. D’autre part, une conséquence du résultat d’existence local [5, Theorem 1.1 (ii)] est que le caractèrefini de T εm peut correspondre à deux types de singularité lorsque t → T εm : le phenomène de collage de la membrane surl’électrode rigide, c’est-à-dire min[−1,1] u(t) −→ −1 lorsque t → T εm , ou l’explosion au temps T εm de la norme W 2q (I) de u.1. IntroductionAn idealized electrostatically actuated microelectromechanical system (MEMS) consists of a fixed horizontal ground plateheld at zero potential, above which an elastic membrane held at potential V is suspended. A Coulomb force is generatedby the potential difference across the device and results in a deformation of the membrane, thereby converting electrostaticenergy into mechanical energy, see [1,6,10] for a more detailed account and further references. After a suitable scaling andassuming homogeneity in transversal horizontal direction, the ground plate is assumed to be located at z = −1 and themembrane displacement u = u(t, x) ∈ (−1,∞) with t > 0 and x ∈ I := (−1,1) evolves according to:∂t u − ∂x(∂xu√1 + ε2(∂xu)2)= −λ(ε2∣∣∂xψ(t, x, u(t, x))∣∣2 + ∣∣∂zψ(t, x, u(t, x))∣∣2), (1)for t > 0 and x ∈ I with boundary conditions:u(t,±1) = 0, t > 0, (2)and the initial condition:u(0, x) = u0(x), x ∈ I. (3)The electrostatic potential ψ = ψ(t, x, z) satisfies a rescaled Laplace equation in the region:Ω(u(t)) := {(x, z) ∈ I × (−1,∞): −1 < z < u(t, x)}between the plate and the membrane, which reads:ε2∂2x ψ + ∂2z ψ = 0, (x, z) ∈ Ω(u(t)), t > 0, (4)ψ(t, x, z) = 1 + z1 + u(t, x) , (x, z) ∈ ∂Ω(u(t)), t > 0, (5)where ε > 0 denotes the aspect ratio of the device and λ > 0 is proportional to the square of the applied voltage. Thedynamics of (u,ψ) is thus given by the coupling of a quasilinear parabolic equation for u and an elliptic equation in amoving domain for ψ , the latter being only well defined as long as the membrane does not touch down on the groundplate, that is, u does not reach the value −1. To guarantee optimal operating conditions of the device, this touchdownphenomenon has to be controlled and its occurrence is obviously related to the value of λ.The main difficulty to be overcome in the analysis of (1)–(5) is the nonlocal and nonlinear implicit dependence on u ofthe right-hand side of (1), which is also singular if u approaches −1. Except for the singularity, these features disappearwhen setting ε = 0 in (1)–(5), a commonly made assumption that reduces (1)–(5) to a singular semilinear reaction–diffusionequation (for a quasilinear variant thereof, see [2]). This so-called small aspect ratio model has received considerable atten-tion in recent years, see [6,10] and the references therein. In this simplified situation, it has been established that touchdownJ. Escher et al. / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 807–812 809does not take place if λ is below a certain threshold value λ∗ > 0, but occurs if λ exceeds this value [6–8]. Another simpli-fication is the small deformation approximation, which reduces the second-order differential term on the right-hand side of(1) to ∂2x u and has been studied in [3,4,9].We have recently investigated the well-posedness of the evolution problem (1)–(5) and established the following re-sult [5].Theorem 1 (Local Well-Posedness). Let q ∈ (2,∞), ε > 0, λ > 0, and consider an initial value:u0 ∈ W 2q (I) such that u0(±1) = 0 and 0  u0(x) > −1 for x ∈ I. (6)Then there is a unique solution (u,ψ) to (1)–(5) on the maximal interval of existence [0, T εm) in the sense that:u ∈ C1([0, T εm), Lq(I)) ∩ C([0, T εm), W 2q (I))satisfies (1)–(3) together with:0  u(t, x) > −1, (t, x) ∈ [0, T εm) × I, (7)and ψ(t) ∈ W 22 (Ω(u(t))) solves (4)–(5) for each t ∈ [0, T εm).We have also shown in [5] that, if λ and u0 are sufficiently small, the solution (u,ψ) to (1)–(5) exists for all times (i.e.T εm = ∞) and touchdown does not take place, not even in infinite time.Theorem 2 (Global Existence). Let q ∈ (2,∞), ε > 0, and consider an initial value u0 satisfying (6). Given κ ∈ (0,1), there areλ∗(κ) > 0 and r(κ) > 0 such that, if λ ∈ (0, λ∗(κ)) and ‖u0‖W 2q (I)  r(κ), the solution (u,ψ) to (1)–(5) exists for all times andu(t, x) −1 + κ for (t, x) ∈ [0,∞) × I .On the other hand, we have been able to prove that no stationary solution to (1)–(5) exists provided that λ is sufficientlylarge. However, whether or not T εm is finite in this case has been left as an open question. The purpose of this note is toshow that – as expected on physical grounds – T εm is indeed finite for λ sufficiently large.Theorem 3 (Finite-time singularity). Let q ∈ (2,∞), ε > 0, and consider an initial value u0 satisfying (6). If λ > 1/ε and (u,ψ)denotes the maximal solution to (1)–(5) defined on [0, T εm), then T εm < ∞.The criterion λ > 1/ε is likely to be far from optimal. As we shall see below, improving it would require to have a bettercontrol on ∂xu(±1). The proof of Theorem 3 relies on the derivation of a chain of estimates that allows us to obtain a lowerbound on the L1-norm of the right-hand side of (1), depending only on u. The lower bound thus obtained is in fact themean value of a convex function of u, and we may then end the proof with the help of Jensen’s inequality, an argumentthat has already been used for the small aspect ratio model, see [7,8].We shall point out that, in contrast to the small aspect ratio model, the finiteness of T εm does not guarantee that thetouchdown phenomenon really takes place as t → T εm . Indeed, according to [5, Theorem 1.1 (ii)], the finiteness of T εm impliesthat min[−1,1] u(t) −→ −1 or ‖u(t)‖W 2q (I) −→ ∞ as t → T εm . While the former corresponds to the touchdown behaviour, thelatter is more likely to be interpreted as the membrane being no longer the graph of a function at time T εm .2. Proof of Theorem 3Let q ∈ (2,∞), ε > 0, λ > 0 and consider an initial value u0 satisfying (6). We denote the maximal solution to (1)–(5)defined on [0, T εm) by (u,ψ). Differentiating the boundary conditions (5), we readily obtain:∂xψ(t, x,−1) = ∂xψ(t, x, u(t, x)) + ∂xu(t, x)∂zψ(t, x, u(t, x)) = 0, (t, x) ∈ (0, T εm) × I, (8)and∂zψ(t,±1, z) = 1, (t, z) ∈(0, T εm) × (−1,0). (9)Additional information on the boundary behaviour of ψ is provided by the next lemma.Lemma 4. For t ∈ (0, T εm),1 + z ψ(t, x, z)  1, (x, z) ∈ Ω(u(t)), (10)±∂xψ(t,±1, z) 0, z ∈ (−1,0). (11)810 J. Escher et al. / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 807–812Proof. Fix t ∈ (0, T εm). The upper bound in (10) readily follows from the maximum principle. Next, the function σ , definedby σ(x, z) = 1 + z, obviously satisfies ε2∂2x σ + ∂2z σ = 0 in Ω(u(t)) as well as:σ(±1, z) = 1 + z = ψ(t,±1, z), z ∈ (−1,0),σ (x,−1) = 0 = ψ(t, x,−1), x ∈ (−1,1).Owing to the non-positivity (7) of u(t), it also satisfies:σ(x, u(t, x)) = 1 + u(t, x)  1 = ψ(t, x, u(t, x)), x ∈ (−1,1),and we infer from the comparison principle that ψ(t, x, z)  σ(x, z) for (x, z) ∈ Ω(u(t)). It then follows from (10) thatψ(t, x, z)  1 + z = ψ(t,±1, z) for (x, z) ∈ Ω(u(t)), which readily implies (11). To simplify notations, we set:γm(t, x) := ∂zψ(t, x, u(t, x)), γg(t, x) := ∂zψ(t, x,−1), (t, x) ∈(0, T εm) × (−1,1), (12)and first derive an upper bound of the L1-norm of the right-hand side of (1), observing that, due to (8), it also reads:−λ(ε2∣∣∂xψ(t, x, u(t, x))∣∣2 + ∣∣∂zψ(t, x, u(t, x))∣∣2) = −λ(1 + ε2(∂xu(t, x))2)γm(t, x)2.Lemma 5. For t ∈ (0, T εm),1∫−1(1 + ε2(∂xu(t, x))2)γm(t, x)2 dx  21∫−1(1 + ε2(∂xu(t, x))2)γm(t, x)dx − 2. (13)Proof. Fix t ∈ (0, T εm). We multiply (4) by ∂zψ(t) − 1 and integrate over Ω(u(t)). Using (8), (9), and Green’s formula weobtain:0 = −ε2∫Ω(u)∂x∂zψ ∂xψ d(x, z) + ε21∫−1(∂xu)2 γm(γm − 1)dx− 121∫−1(γ 2g − 2γg)dx + 121∫−1(γ 2m − 2γm)dx.Since∫Ω(u)∂x∂zψ ∂xψ d(x, z) = 121∫−1(∂xu)2γ 2m dxby (8) and since γ 2g − 2γg  −1, we end up with (13). We again use (4) to obtain a lower bound for the boundary integral of the right-hand side of (13), which depends onthe Dirichlet energy of ψ .Lemma 6. For t ∈ (0, T εm),1∫−1(1 + ε2(∂xu(t, x))2)γm(t, x)dx ∫Ω(u(t))(ε2∣∣∂xψ(t, x, z)∣∣2 + ∣∣∂zψ(t, x, z)∣∣2)d(x, z). (14)Proof. Fix t ∈ (0, T εm). We multiply (4) by ψ(t) and integrate over Ω(u(t)). Using (5), (8), and Green’s formula we obtain:0 = −∫Ω(u(t))(ε2∣∣∂xψ(t, x, z)∣∣2 + ∣∣∂zψ(t, x, z)∣∣2)d(x, z) + ε20∫−1(1 + z)∂xψ(t,1, z)dz− ε20∫(1 + z)∂xψ(t,−1, z) dz + ε21∫ (∂xu(t, x))2γm(t, x)dx +1∫γm(t, x)dx.−1 −1 −1J. Escher et al. / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 807–812 811Owing to (11), the second and third terms of the right-hand side of the above equality are non-positive, whence (14). We finally argue as in [4, Lemma 9] to establish a connection between the Dirichlet energy of ψ and u.Lemma 7. For t ∈ (0, T εm),∫Ω(u(t))(ε2∣∣∂xψ(t, x, z)∣∣2 + ∣∣∂zψ(t, x, z)∣∣2)d(x, z)1∫−1dx1 + u(t, x) . (15)Proof. Let t ∈ (0, T εm) and x ∈ (−1,1). We deduce from (5) and the Cauchy–Schwarz inequality that:11 + u(t, x) =(ψ(t, x, u(t, x)) − ψ(t, x,−1))21 + u(t, x) =11 + u(t, x)( u(t,x)∫−1∂zψ(t, x, z)dz)2u(t,x)∫−1(∂zψ(t, x, z))2dz. (16)Integrating the above inequality with respect to x ∈ (−1,1) readily gives (15). Remark 8. Observe that (16) provides a quantitative estimate on the singularity of ∂zψ generated by u when touchdownoccurs.Combining the three lemmas above with Jensen’s inequality gives the following estimate.Proposition 9. For t ∈ (0, T εm),1∫−1(1 + ε2(∂xu(t, x))2)γm(t, x)2 dx  4ϕ(121∫−1u(t, x)dx)− 2, (17)where ϕ(r) := 1/(1 + r), r ∈ (−1,∞).Proof. Fix t ∈ (0, T εm). We infer from Lemma 5, Lemma 6, and Lemma 7 that:1∫−1(1 + ε2(∂xu(t, x))2)γm(t, x)2 dx  21∫−1ϕ(u(t, x))dx − 2.To complete the proof, we argue as in [7,8] and use the convexity of ϕ and Jensen’s inequality to obtain (17). Proof of Theorem 3. Introducing:E(t) := −121∫−1u(t, x)dx, t ∈ [0, T εm),the bounds (7) ensure that:0  E(t) < 1, t ∈ [0, T εm). (18)It follows from (1), (8), and Proposition 9 that:dEdt(t) = −12[∂xu(t, x)√1 + ε2(∂xu(t, x))2]x=1x=−1+ λ21∫−1(1 + ε2(∂xu(t, x))2)γm(t, x)2 dx Fλ(E) := 2λϕ(−E) − λ − 1 . (19)ε812 J. Escher et al. / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 807–812If λ > 1/ε, we note that Fλ(0) > 0 and thus Fλ(r)  Fλ(0) > 0 for r ∈ [0,1) due to the monotonicity of Fλ . Since E(0)  0by (18), it follows from (19) and the properties of Fλ that t → E(t) is increasing on [0, T εm). Consequently,dEdt(t)  Fλ(E(0)) Fλ(0), t ∈[0, T εm).Integrating the previous inequality with respect to time and using (18), we end up with the inequality 1  E(0) + Fλ(0)T εm ,which provides the claimed finiteness of T εm . AcknowledgementsThis research was done while P.L. was enjoying the kind hospitality of the Institut für Angewandte Mathematik of theLeibniz Universität Hannover.References[1] N.D. Brubaker, J.A. Pelesko, Non-linear effects on canonical MEMS models, Eur. J. Appl. Math. 22 (2011) 455–470.[2] N.D. Brubaker, J.A. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity, Nonlinear Anal. 75 (2012)5086–5112.[3] G. Cimatti, A free boundary problem in the theory of electrically actuated microdevices, Appl. Math. Lett. 20 (2007) 1232–1236.[4] J. Escher, Ph. Laurençot, C. Walker, A parabolic free boundary problem modeling electrostatic MEMS, Arch. Ration. Mech. Anal. (2013) (in press).[5] J. Escher, Ph. Laurençot, C. Walker, Dynamics of a free boundary problem with curvature modeling electrostatic MEMS, arXiv:1302.6026 (submitted forpublication).[6] P. Esposito, N. Ghoussoub, Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, Courant Lect. Notes Math.,vol. 20, Courant Institute of Mathematical Sciences, New York, 2010.[7] G. Flores, G. Mercado, J.A. Pelesko, N. Smyth, Analysis of the dynamics and touchdown in a model of electrostatic MEMS, SIAM J. Appl. Math. 67 (2007)434–446.[8] Y. Guo, Z. Pan, M.J. Ward, Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties, SIAM J. Appl. Math. 66 (2005)309–338.[9] P. Laurençot, C. Walker, A stationary free boundary problem modeling electrostatic MEMS, Arch. Ration. Mech. Anal. 207 (2013) 139–158.[10] J.A. Pelesko, D.H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003.

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Finite time singularity in a free boundary problem modeling MEMS

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