Touchdown and related problems in electrostatic MEMS device equation


 Håvar Larssen
 13 dager siden
 Visninger:
Transkript
1 Nonlinear differ. equ. appl. 15 (28), /8/ , DOI 1.17/s c 28 Birkhäuser Verlag Basel/Switzerland Touchdown and related problems in electrostatic MEMS device equation Nikos I. Kavallaris, Tosiya Miyasita and Takashi Suzuki Abstract. We study the electrostatic MEMSdevice equation, u t u = λ x β, with Dirichlet boundary condition. First, we describe the touchdown of nonstationary solution in accordance with the total set of stationary (1 u) p solutions. Then, we classify radially symmetric stationary solutions and their radial Morse indices. Finally, we show the MorseSmale property for radially symmetric nonstationary solutions. Mathematics Subject Classification (2). Primary 35K55, 35J6; Secondary 74H35, 74G55, 74K15. Keywords. Electrostatic MEMS, touchdownquenching, Morse idex. 1. Introduction The purpose of the present paper is to study the globalintime behaviour of the solution to the parabolic problem u t u = λf(x), u<1 in (,T) (1 u) p u = on (,T) u t= = u (x) in, (1.1) where λ> is a constant, p>1, f(x), f(x) is a continuous function, R n is a bounded domain with smooth boundary, and u = u (x) [, 1) is a continuous function. This equation models the dynamic deflection of an elastic mebrane inside a microelectro mechanical system (MEMS). This kind of systems combine electronics with various types of microsize mechanical devices and could be found in accelerometers for airbag deployment in automobiles, in ink jet printer heads, in optical switches, in chemical sensors and so on, for more details see [21] and the references therein. On leave to Department of Statistics and Actuarial Science, University of the Aegean, current
2 364 N. I. Kavallaris, T. Miyasita and T. Suzuki NoDEA The structure of the set of stationary solutions has been studied for p =2 ([8, 11]). We obtain, similarly, an upper bound of λ for the existence of the solution denoted by λ> for general p>1. Thus, if <λ<λ and λ>λ, then C λ and C λ =, respectively, where C λ = { u C 2 () C () u solves (1.1) }. Also, if C λ there is a unique minimal element, denoted by u λ. The globalintime behaviour of the nonstationary solution has been studied also for p = 2 ([9]). More precisely, for λ<λ and λ>λ, there is a solution to (1.1) converging to u λ uniformly and any solution must touchdown in finite time, i.e., T<+, lim t T u(,t) =1, (1.2) respectively. Here and henceforth, T> denotes the existence time of the solution. We can refine these results on the globalintime behaviour of the solution even for general p>1. Theorem 1.1. We have the following. 1. If λ>λ, then (1.2) occurs. 2. If there is u C λ \{u λ } and u u, u u or C λ = {u λ } and u u λ, then T =+ and u(,t) u λ uniformly as t If there is u C λ \{u λ }, u u,u u, then (1.2) occurs. Next, we take the case of =B {x R n x < 1}, f(x) = x β, β, (1.3) and prove existence of nonminimal radially symmetric stationary solutions. To state the results, we set Cr λ = { u C 2 () C () u = u( x ) solves (1.1) } C r = {λ} Cr λ λ> and β = p{2(p 1) (p +1)n} +(n 2)(p +1) p(p +1) 2p p = (n2 8n +4) 8 n 1, (n 2)(n 1)
3 Vol. 15 (28) Touchdown and related problems in MEMS 365 where β and p are the maximum solutions of the equations f(p, n, β) =and f(p, n, ) = respectively, for f(p, n, β) =(p +1) 2 n 2 4(p +1)(βp +3p +1)n 4 { β 2 p +2pβ (1 p) ( 5p 2 +2p +1 )}. Note that β, p are defined for n 7 and n =7, 8, 9 respectively. Theorem 1.2. If n 2, then C r is homeomorphic to R and has end points (, ) and (λ, 1 x β+2 ), where In the cases of λ = (β +2){β +2+(n 2)(p +1)} (p +1) 2. n [2, 6], p > 1, β >, n [7, 9], p > p, β >, or p (1, p), β > β, n 1, p > 1, β > β, whenever β>, C r bends infinitely many times with respect to λ around λ, while at most a finite number of bendings occur to C r when n [7, 9], n 1, p (1, p], β (, β], p > 1, β (, β]. The case p = 2 has also been studied on C = λ> {λ} Cλ. First, the above profile of C r is observed numerically [8]. Next, some estimates on λ are given for nonradially symmetric case [8, 9]. Finally, existence of the nonminimal nonradially symmetric stationary solution is proven by the variational method [4]. Given u C λ, the linearized eigenvalue problem is defined as follows: φ pλf(x) φ = µφ in, φ = on. (1.4) (1 u) Then, the number of its negative eigenvalues, denoted by i = i(λ, u), is called Morse index. In case (1.3) the number of negative eigenvalues corresponding to radially symmetric eigenfunctions, denoted by i R = i R (λ, u), is called radial Morse index. Theorem 1.3. Under the assumptions of Theorem 1.2, i R = i R (λ, u) increases one by one at each bending point. A general theorem guarantees i(u, λ) =i R (u, λ) for u = λf(r, u) in B, u = on B,
4 366 N. I. Kavallaris, T. Miyasita and T. Suzuki NoDEA if f r, where r = x, see [16]. This is not the case of (1.3) with β >. In fact, general theorems of radial symmetry of the solution to the stationary problem are not valid for β> ([1, 2]). There may be nonradial bifurcation and i(λ, u) =i R (λ, u) would not be always valid. This paper is composed of five sections. Theorems 1.1, 1.2, and 1.3 are proven in sections 2, 3, and 4, respectively. In the final section, 5, we confirm the role of radial Morse indices in the formation of connecting orbits. 2. Proof of Theorem 1.1 The original proof for p = 2 to the first case of this theorem is quite technical [9], it provides with some bounds of the blowup time though. Here we give a simpler proof by applying Kaplan s method adopted by [2]. The proof of the second case is obtained by using classical techniques like maximumprinciple or dynamicalsystems aguments, [7, 9], and we omit it. The final case is obtained by the method of [15]. Before proceeding to these cases, we recall that Kaplan s method guarantees touchdown for any λ> if the initial value is close to 1 and f>on (see also [3]). In fact, let µ 1 > be the principal eigenvalue of, and take the solution to ϕ 1 = µ 1 ϕ 1, ϕ 1 > in, ϕ 1 = on, normalized by ϕ 1 = 1. Then, using m = inf f > and A = uϕ 1dx, we obtain via Jensen s inequality da dt µ 1 + λm (1 u) p ϕ 1 dx µ 1 A + λm(1 A) p. Let H(s) = µ 1 s + λm(1 s) p,s (, 1). Then there exists a (, 1) such that H(s) > in(a, 1). Hence if we choose u close enough to 1 so that A() >a, > for all t when u is global. But, then we have then da dt t = t dσ A(t) A() ds H(s) 1 A() ds H(s) < + which is a contradiction. Kaplan s method could be also used to prove touchdown for big enough λ, see [11]. Now, we show the first case of Theorem 1.1, i.e., λ>λimplies (1.2). It suffices to assume u = for this purpose. Then, it holds that v t v = pλf(x) v in (,T) (1 u) v = on (,T) v t= = λf(x) in,
5 Vol. 15 (28) Touchdown and related problems in MEMS 367 for v = u t and hence u t. In particular, lim t T u(,t) = 1 follows from T<+, because C λ =. We assume T =+ and multiply the first equation of (1.1) by ϕ 1 : t+1 t Then, it holds that λ t+1 t u t ϕ 1 dxds = = t+1 f(x)ϕ 1 (1 u) p dxds t ( u)ϕ 1 dxds + λ u(,t+1)ϕ 1 dx t+1 t u(,t)ϕ 1 dx + µ 1 u(,t+1) + µ 1 u(,t) 1+µ 1 by u(x, t) < 1in (, + ), and therefore, f(x)ϕ 1 λ (1 u(x, t)) p dx 1+µ 1 by u t. This implies sup t f(x)ϕ 1 (1 u) p dxds. t+1 f(x)ϕ 1 (1 u(,t)) p dx 1+µ 1 < +. λ From the monotone convergence theorem, and u(,t) 1 = u(x, t) dx < +, t uϕ 1 dxds it follows that u(,t) has a limit w in L 1 () as t, while K(x, t) = f(x) (1 w) p t t t f(x) (1 u) p converges to K(x) = in L 1 (,δ(x)dx)ast +, where δ(x) = dist(x, ). Testing (1.1) by ϕ C 2 () with ϕ =, we obtain t+1 t+1 t+1 uϕdx f(x)ϕ = u ϕdxds + λ (1 u) p dxds. Letting t we derive w( ϕ)dx = λf(x)ϕ dx. (2.1) (1 w) p
6 368 N. I. Kavallaris, T. Miyasita and T. Suzuki NoDEA This means that w is a weak solution to w = λf(x) (1 w) p in, w = on, (2.2) and we obtain a contradiction by the following lemma, see also [2, 9]. Lemma 2.1. Problem (2.2) has no weak solutions for λ>λ. Proof. We define g(w) = 1 w (1 w) p, h(w) = ds g(s) = g ε (w) =(1 ε)g(w), h ε (w) = w w (1 s) p ds ds g ε (s) = 1 1 ε h(w) and put Φ ε (w) = h 1 ε (h(w)), where ε >. The following properties of this mapping w [, 1) Φ ε (w) are obtained by g > : Φ ε ()=, Φ ε (w) <w, Φ ε(w) =(1 ε)g(φ ε (w))g(w) 1 >, Φ ε (w) =(1 ε)g(φ ε (w))((1 ε)g (Φ ε (w)) g (w))g(w) 2 < (1 ε)g(φ ε (w))((1 ε)g (w) g (w))g(w) 2. Given a weak solution w = w(x) [, 1] to (2.2) satisfying f(x) (1 w) p L 1 (,δ(x)dx), we define w ε = Φ ε (w). This means h(w ε ) = (1 ε)h(w) and there is δ> such that w ε 1 δ. It also holds by the concavity of Φ ɛ that in weak sense, i.e., w ε Φ ε(w) λf(x) (1 w) p in, w ε = on w ε Jdx λ Φ f(x) ε(w) (1 w) p Jdx f(x)j = λ(1 ε) (1 w ε ) p dx for any J C 2 () satisfying J and J =. Thus, w ε is a weak supersolution to (2.2) and therefore, the iteration sequence {v k } k= defined by v k+1 = λ(1 ε)f(x) (1 v k ) p in, v k+1 = on (2.3)
7 Vol. 15 (28) Touchdown and related problems in MEMS 369 with v = w ε is monotone decreasing. It also satisfies v k and converges uniformly to a solution to (2.2) for λ(1 ε) by Dini s theorem. This implies C λ(1 ε), a contradiction by the definition of λ since ε> is arbitrary. The third case of Theorem 1.1 is a consequence of the following theorem. Theorem 2.2. If there is a pair of super and subsolutions to (2.2), v, v C 2 () C () satisfying v v < 1 and v v, and if the initial value satisfies v u < 1 and v u in (1.1), then it holds that T < + and lim t T u(,t) =1. Proof. Since v is a supersolution, the iteration sequence {v k } k= defined by (2.3) with v = v(x) [, 1) is monotone decreasing. It satisfies v k, and converges uniformly to a solution to (2.2) by Dini s theorem. Thus, we can assume that v is a classical solution to (2.2). Let u 1 and u 2 be the local in time solutions to (1.1) with u = v and u = v, respectively. Then, by the strong maximum principle and the Hopf s lemma we have u(,t ) u 1 (,t ) u 2 (,t ) for <t 1, i.e., u(,t ) >u 1 (,t ) >u 2 (,t ) in, u ν (,t ) < u 1 ν (,t ) < u 2 ν (,t ) on, where ν denotes the outer unit normal vector. Since v is a subsolution of (2.2) we obtain, by the maximum principle, that u 1 is increasing in time, i.e., u 1t >, and hence u 1 (,t ) is a strict subsolution of problem (2.2). On the other hand, we obtain u 2 (,t)=v because v is supposed to be a stationary solution. Therefore, we may assume that v and v are a strict subsolution and a solution of (2.2), respectively, and that u,v, v are C 2 functions on satisfying u v v. In this case, we can take a constant θ>1 such that u θv +(1 θ)v >v, and therefore, we can assume u = θv +(1 θ)v by the comparison theorem. This u = u (x) [, 1) is a strict subsolution because g(u) =(1 u) p is convex. In fact, u = θ v (1 θ) v < λθf(x) λ(1 θ)f(x) + (1 v) p (1 v) p = λf(x)[θg(v)+(1 θ)g(v)] <λf(x)g(u ). Then, u(x, t) is strict increasing in t for each x and so we have a measurable function v(x) such that lim u(x, t) =v(x) (v(x), 1] for x (2.4) t +
8 37 N. I. Kavallaris, T. Miyasita and T. Suzuki NoDEA under the assumption T =+. Then, similarly to the first case, it holds that f(x)ϕ 1 (1 v) p dx 1+µ 1 < +. λ Using Duhamel s principle, we can write the solution of problem (1.1) in the form u(x, t) = t U(x, y; t)u (y)dy + ds λf(y) U(x, y; s) (1 u(y, t s)) p dy, where U(x, y; t) > is the fundamental solution of the heat operator. The second term of the righthand side is written as λf(y) U(x, y; s) (1 u(y, t s)) p χ [,t](s)dyds, where the monotone convergence theorem is applicable by (2.4). Since lim U(x, y; t)u (y)dy = t + it holds that v(x) = ds λf(y) U(x, y; s) (1 v(y)) p dy for each x. Here, using Fubini s theorem, we obtain f(y) v(x) =λ G(x, y) dy (2.5) (1 v(y)) p by U(x, y; t)dt = G(x, y), where G(x, y) denotes the Green s function of. This implies that v = v(x) (v(x), 1] is a weak solution to (2.2). We have v u u(,t ) v and therefore, there is β (, 1) such that u βv +(1 β)v z.
9 Vol. 15 (28) Touchdown and related problems in MEMS 371 Here, v is a classical solution to (2.2), while v = v(x) [v(x), 1] is a weak solution f(x) to (2.2) satisfying (1 v) L 1 (,δ(x)dx). Then, we can define the minimal p solution, denoted by ũ to (1.1) with the initial value z by the iteration scheme ũ k+1 (x, t) = ũ (x, t) = t U(x, y; t)z(y)dy + U(x, y; t)z(y)dy. U(x, y; s)λf(y) (1 ũ k (y, t s)) p dyds In fact, this {ũ k } k= is monotone increasing in k and satisfies ũ k (x, t) z(x) because g(u) =(1 u) p is convex, and then ũ(x, t) is defined by If {u k } k= is defined by lim k ũk(x, t) =ũ(x, t). u k+1 (x, t) = u (x, t) = t U(x, y; t)u (y)dy + U(x, y; t)u (y)dy, U(x, y; s)λf(y) (1 u k (y, t s)) p dyds then the monotonicity of g(s) implies that u k ũ k for each k and hence lim k u k (x, t) = u(x, t). This implies v βv +(1 β)v in by letting k and then t +. Then, it follows that v v, a contradiction to (2.4). In the case that problem (2.2) has only a weak solution w, i.e. w = 1 and w satisfies (2.2) in the sence of (2.1), we can prove that an infinitetime touchdown occurs for the solution of problem (1.1). More precisely we have the following. Theorem 2.3. If proplem (2.2) has only a weak solution w, then for u (x) < w (x) there exists a unique gloabal solution u of (1.1) such that u (x, t) w (x) in for all <t<, while u (x, t) converges pointwise to w as t. In particular, lim t u (,t) =1. The proof of Theorem 2.3 follows the same steps with the proof of Theorem 2.5, for p = 2, in [9] and uses some arguments introduced in [2].
10 372 N. I. Kavallaris, T. Miyasita and T. Suzuki NoDEA An natural question arises, from the above theorem, if the rate of convergence of u towards the singular steady state w could be determined. Since this is a very interesting question and the answer requires a delicate analysis it will be addressed in a forthcoming paper. The problem of the characterization of the set of the touchdown points, for the case p =2, has been first tackled in [9], while some more refined results were given in [12]. In particular, in [12], it is proven, for problem v t v = λf(x) v 2, v<1 in (,T) v =1 on (,T) v t= = 1 in, (2.6) that the finitetime touchdown set Q := {x :v(x, T )=,T = T max < } A := {x :f(x) } under the hypothesis that Q is a compact set, which is the case when is convex and f satisfies some extra condition. Under the same hypothesis some upper and lower estimates of the touchdown profile are given as well as some refined touchdown profiles are provided, in the radial symmetric case, coming out by classical arguments like similarityvariables and centralmanifolds techniques. In the case that is not convex the question of determining the touchdown rate is still open, and probably some scaling arguments, which have been intoduced for general domains, could work in this case. Now, for the radial symmetric case, it is proven in [9], by using ideas from [5], that the origin r = is the only touchdown point provided that f (r). Most of the above results, existing in [9] for p = 2, could be easily extended to the case p>1. 3. Proof of Theorem 1.2 We study the structure of the set of radially symmetric stationary solutions in the case of (1.3). This problem is reduced to (r n 1 u r ) r + λrn+β 1 (1 u) p =, u<1, for r (, 1) u(1)=, u r () = (3.1) where u = u(r) for r = x. We apply the phaseplane analysis [13, 17, 18, 19]. First, putting r = y 2 we obtain for a = 2n+β 2 β+2. (y a u y ) y + u(1)=, 4λ y a (β +2) 2 (1 u) p =, u<1 for y (, 1) β+2, lim y β β+2 uy = (3.2) y
11 Vol. 15 (28) Touchdown and related problems in MEMS 373 We note that every positive solution of problem (3.2) can be obtained as a solution of the following initial value problem with certain positive constant A : (y a u y ) y + u() = A (, 1), 4λ y a (β +2) 2 (1 u) p =, u<1, for y (, 1) lim y β β+2 uy =. (3.3) y Putting {β +2+(n 2) (p +1)} (β +2)(1 A) k = λ (p +1) 2, we apply the Emden transformation u(y) =1 (1 A) e 2 t w(t), y = ke t. Then, (3.3) is reduced to 2(np + n 2p +2β +2) 4(np + n 2p + β) ẅ + ẇ + (w (β +2)(p +1) (β +2)(p +1) 2 1w ) p = <w< A e t for t R lim e 2 t w =1, lim e 2 t ẇ = 2 t t p +1. (3.4) The profile of this orbit is studied in detail in [13]. First, we show the following lemma. Lemma 3.1. Problem (3.4) admits a unique solution for t 1. Proof. Through the transformation, v(z) =e 2 t w(t), z = e t, (3.4) is reduced to (z b 4(np + n 2p + β) v z ) z (p +1) 2 z b v p = (β +2) <v< 1 1 A, z >, v()=1, lim zv z =, z where b = pβ+β+2np+2n 2p 2 ()(β+2) and then we obtain the integral equation 4(np + n 2p + β) v(z) =1+ (p +1) 2 (β +2) z σ b dσ σ ζ b v(ζ) p dζ. (3.5)
12 374 N. I. Kavallaris, T. Miyasita and T. Suzuki NoDEA Now, we consider the Banach space X = {v = v(z) v C([,ε]) and v()=1}, equipped with the supremum norm, where ε> is a constant to be determined later. Let Ψ(v) X be the righthand side of (3.5) defined for v X. For, given { v i Y v X 1 v 1 }, (i =1, 2), 1 A we have Ψ(v 1 ) Ψ(v 2 ) 4(np + n 2p + β) (p +1) 2 (β +2) z z σ b dσ σ ζ b v 1 (ζ) p v 2 (ζ) p dζ 4(np + n 2p + β) (p +1) 2 σ b dσ (β +2) σ 1 ζ b p θv 1 (ζ)+(1 θ)v 2 (ζ) p 1 dθ v 1(ζ) v 2 (ζ) v 1 (ζ) p v 2 (ζ) p dζ 4p (np + n 2p + β) (p +1) 2 (β +2)(1 A) p 1 z σ b dσ σ ζ b v 1 v 2 dζ Therefore 4p (np + n 2p + β) ε (p +1) 2 (β +2)(1 A) p 1 σdσ v 1 v 2 2p (np + n 2p + β) (p +1) 2 (β +2)(1 A) p 1 ε2 v 1 v 2. Ψ(v 1 ) Ψ(v 2 ) Similarly, we obtain 4(np + n 2p + β) Ψ(v) 1+ (p +1) 2 (β +2) 2p (np + n 2p + β) (p +1) 2 (β +2)(1 A) p 1 ε2 v 1 v 2. z 2(np + n 2p + β) 1+ (p +1) 2 ε 2, (β +2) σ b dσ σ ζ b v(ζ) p dζ
13 Vol. 15 (28) Touchdown and related problems in MEMS 375 and hence 2(np + n 2p + β) Ψ(v) 1+ (p +1) 2 ε 2. (β +2) For ε min (p +1) 2 (β +2)(1 A) p 1, 4p (np + n 2p + β) it holds that Ψ(v i ) Y for i =1, 2 and Ψ(v 1 ) Ψ(v 2 ) 1 2 v 1 v 2. (p +1) 2 (β +2)A 2(np + n 2p + β)(1 A) Hence Ψ has a unique fixed point v X by the contraction mapping principle, and the proof is complete. Now we write (3.4) as an autonomous system ( ) ( d w ẇ = dt ẇ 2(np+n 2p+2β+2) ()(β+2) ẇ 4(np+n 2p+β) () 2 (β+2) ( w 1 w p ) Note that (w,ẇ) =(1, ) is a singularfixed point of this system. The linearized equation around this fixed point is given by ( ) ( ) ( ) d Ẇ 1 W =, dt W Ẇ 4(np+n 2p+β) ()(β+2) 2(np+n 2p+2β+2) ()(β+2) ). and the eigenvalues µ ± of the coefficient matrix ( 1 4(np+n 2p+β) ()(β+2) 2(np+n 2p+2β+2) ()(β+2) ), are given as µ ± = (np + n 2p +2β +2)± f(p, n, β) (p +1)(β +2) where f(p, n, β) =(p +1) 2 n 2 4(p + 1)(βp +3p +1)n 4{β 2 p +2pβ(1 p) (5p 2 +2p +1)}. Since n 2, we obtain np + n 2p +2β +2 >. Thus, (w, ẇ) =(1, ) is an attractor.
14 376 N. I. Kavallaris, T. Miyasita and T. Suzuki NoDEA The curve ẇ = ϕ(w) defined for ϕ(w) = a (w 1w ) p a = (p + 1)(β +2) 2(np + n 2p + β) np + n 2p +2β +2 (p +1) 2 > (β +2) passes through (1, ) and the righthalf plane {(w, ẇ) w>} of the phase space is separated into four regions: I = {(w, ẇ) w>, ẇ>, ẇ>ϕ(w)}, II = {(w, ẇ) w>, ẇ<, ẇ>ϕ(w)}, III = {(w, ẇ) w>, ẇ<, ẇ<ϕ(w)}, IV = {(w, ẇ) w>, ẇ>, ẇ<ϕ(w)}. According to these cases, the directions of the orbits O are as follows: (w, ẇ) O I dw dt >, dẇ dt <, (w, ẇ) O II dw dt <, dẇ dt <, (w, ẇ) O III dw dt <, dẇ dt >, (w, ẇ) O IV dw dt >, dẇ dt >. If (w, ẇ) Oapproaches w =, then dẇ dt +, and therefore it enters region IV. Thus, O in the righthalf plane does not touch w = in finite time. If (w, ẇ) O I with w 1, then it must eventually come into region II because dw dẇ dt = ẇ> and dt = γ( ẇ + ϕ(w)) γ( ẇ aw) <, where γ =2(np + n 2p +2β +2)/(p + 1)(β +2). From those considerations, we see that the orbit O is compact in the righthalf space, and eventually approaches the singular point (w, ẇ) =(1, ) as t +. The following elementary lemma controls the (local) behaviour of the orbit O around the singularfixed point (w,ẇ) =(1, ). Lemma 3.2. We have the following. 1. If n [2, 6], then f(p, n, β) < for any p>1 and β>. 2. If n [7, 9], f(p, n, β) for p (1, p] and β (, β], and f(p, n, β) < otherwise, i.e. for p>p and β>or for p (1, p) and β>β, whenever β>. 3. If n 1, f(p, n, β) for p>1 and β (, β], and f(p, n, β) < otherwise.
15 Vol. 15 (28) Touchdown and related problems in MEMS 377 Proof. We have ( f(p, n, β) = 4p β ) 2 2(p 1) (p +1)n +(p +1) 3 (n 2) 2. 2 Since 2(p 1) (p +1)n = (p +1)(n 2) 4 <, f(p, n, β) achives the maximum at β = for fixed p>1 and n 2 with the value f(p, n, ) = (p +1) 2 n 2 4(p +1)(3p +1)n +4 ( 5p 2 +2p +1 ) =(n 2) (n 1) p 2 +2 ( n 2 8n +4 ) p +(n 2) 2. By a simple computation, we have f(p, n, ) < for n =2,, 6 and p>1, and f(p, n, ) for p (1, p] and n =7, 8, 9 f(p, n, ) < for p>p and n =7, 8, 9. Then, the first and second cases of the lemma are obtained by the definion of β. Since f(1,n,)=4 ( n 2 8n +8 ), and f(p, n, ) is increasing with respect to p>for n 1, we obtain f(p, n, β) for p 1, n 1, and β (, β], while f(p, n, β) < for n 1, p > 1 and β>β. This completes the proof of the lemma. Lemma 3.2 guarantees that the singular point (1, ) is a spiral attractor if and only if n [2, 6], p > 1, β>, n [7, 9], p > p, β >, or p (1, p), β>β n 1, p > 1, β>β, while in the secondary case, i.e. when n [7, 9], p (1, p], β (, β], n 1, p > 1, β (, β] the orbit O is absorbed into (1, ) exponentially.
16 378 N. I. Kavallaris, T. Miyasita and T. Suzuki NoDEA Through the above transformation, the boundary condition u(1) = in (3.1) corresponds to w(τ) = { τ = 1 2 } 1 [β +2+(n 2) (p + 1)] (β +2) λ (p +1) 2 log {β +2+(n 2) (p +1)} (1 A) λ(p +1) 2 = log 1 k. In other words, for any τ R, (λ τ,u τ ) defined by β+2 w(τ + u τ (r) =1 2 log r) w(τ) r β+2 {β +2+(n 2) (p +1)} (β +2) λ τ = (p +1) 2 w(τ) 1 A τ =1 e 2 τ w(τ) (3.6) satisfies (3.1), and conversely, every solution of (3.1) is written in the form of (3.6). Hence C r is homeomorphic to O. Since the singular point (w, ẇ) =(1, ) corresponds to (λ, u) =(λ, 1 x β+2 ), we obtain the conclusion of Theorem Proof of Theorem 1.3 Theorem 1.3 is proven by the method of [19]. Each (λ, u) C r corresponds to (w(τ), ẇ(τ)) Oand therefore, it is parametrized by τ R: (λ, u) =(λ(τ),u(τ)). We denote by µ l τ,l=1, 2,...,the l th eigenvalue of the linearized problem (1.4) corresponding to a radially symmetric eigenfuction, i.e. a function satisfying φ rr n 1 pλr β φ r r φ = µφ ( <r<1) (1 u) φ r () = φ(1)=. (4.1) Each µ l τ is simple. If (λ(τ),u(τ)) is on the turning point of C r, then, by the implicit function theorem, there is l 1 such that µ l τ =. If µ l τ = holds for some l 1 with (λ(τ),u(τ)) C r not on the turning point, then, by the bifurcation theory on the critical point of odd multiplicity, [23, 24], it is actually the bifurcation point of C r. But, this is impossible by Theorem 1.2, and therefore, (λ(τ),u(τ)) is on the turning point of C r if and only if (4.1) has the eigevalue. Denoting the turning points of C r by T k =(λ(τ k ),u(τ k )) (τ 1 <τ 2 < ), we only have to show µ l(k) τ=τ k < for k 1 and l = l(k) such that µ l(k) τ k =, to prove Theorem 1.3.
17 Vol. 15 (28) Touchdown and related problems in MEMS 379 Using (3.6), we have λ(τ) = {β +2+(n 2) (p +1)} (β +2) p +1 ẇ(τ) w(τ) p+2 {β +2+(n 2) (p +1)} (β +2) ẅ(τ)w(τ) (p +2)ẇ(τ) λ(τ) 2 = p +1 w(τ) p+3, (4.2) where here and above = d dτ, stands for the differentiation with respect to τ. On the other hand, from (1.1) it holds that Due to (3.6) we have u rr + n 1 u r + λr β pλr β r (1 u) p + u =, ( <r<1) (1 u) u r () = u(1)=. (4.3) β log r) ẇ(τ + β+2 2 log r)w(τ) u(r, τ )=ẇ(τ)w(τ w(τ) 2 r β+2 and since O does not contain a segment we derive that u(r, τ ). Moreover, since λ(τ k ) = in (4.3), we can take φ(,τ k )= u(,τ k ). Thus, the perturbation theory, see [14], guarantees the existence of φ = φ(,τ) and µ = µ(τ) smooth in τ for τ τ k 1 satisfying (4.1), φ(,τ k )= u(,τ k ), and µ(τ k )=µ l(k) τ k µ l(k) τ k is simple. Differentiating (4.3) and (4.1) with respect to τ, we have = since and r ü + λr β p (1 u) p + λr β (1 u) u + p λr β p(p +1)λrβ u + (1 u) (1 u) p+2 u2 pλr β + ü = in, ü = on, (4.4) (1 u) p r φ λr β p(p +1)λrβ φ (1 u) (1 u) p+2 uφ pλr β φ (1 u) = µφ + µ φ in, φ = on, (4.5) where r = 2 / r 2 + n 1 r / r. Multiplying (4.4) and (4.5) by u, integrating over, and subtracting each other, we obtain r λ(τ β k ) (1 u k ) p φ(τ k )dx = µ(τ k ) φ(τ k ) 2 dx in the bending point τ = τ k, where u k = u(,τ k ).
18 38 N. I. Kavallaris, T. Miyasita and T. Suzuki NoDEA Let I k = The following lemma implies µ(τ k ) φ(τ k ) 2 dx = λ(τ k )I k r β (1 u k ) p φ(τ k )dx. = {β +2+(n 2) (p +1)} (β +2)2 ω n ẅ(τ k ) 2 2(p +1) 2 λ k w(τ k ) p+3 by (4.2), where ω n denotes the area of the unit sphere in R n, and hence µ(τ k ) <. Lemma 4.1. It holds that I k = (β +2)ω n 2(p +1)λ k ẅ(τ k ) w(τ k ), where λ k = λ(τ k ). Proof. We have r β λ (p 1) (1 u) p udx + λ r β d p 1 dx = λr β (1 u) dτ (1 u) = d ( r u)(u 1)dx = r udx + d u r udx dτ dτ = λ r udx d dτ λ r β udx pλ (1 u) p r β (1 u) p udx = r β u udx (1 u) and therefore, (p 1) λ k I k = r u k dx λ k I k pλ k r udx λ p 1 dx r β (1 u) p udx r β (1 u k ) u k u k dx, by putting τ = τ k, where u k = u(,τ k ). This means r β pλ k I k = r u k dx pλ k (1 u k ) u k u k dx. (4.6)
19 Vol. 15 (28) Touchdown and related problems in MEMS 381 while implies Since u k = u(r, τ k ), we have r u k dx = ω n ( u k ) r (1), β+2 w(τ + u(r, τ )=1 2 log r) w(τ) r β+2 u r (r, τ )= β +2 w(τ)ẇ(τ + β+2 2 log r) ẇ(τ)w(τ + β+2 2 log r) p +1 w(τ) 2 r β+2 1 β +2 w(τ)ẅ(τ + β+2 2 log r) ẇ(τ)ẇ(τ + β+2 2 log r) 2 w(τ) 2 r β+2 1. Here, we have ẇ(τ k ) = by (4.2) and λ(τ k ) = and hence r u k dx = (β +2)ω n ẅ(τ k ) 2 w(τ k ). (4.7) and hence Multiplying (4.1) by u and integrating over, we obtain pλr β u r φdx φudx = µ φudx (1 u) Putting τ = τ k,wehave pλr β φudx = µ φudx r uφdx (1 u) r β = µ φudx + λ (1 u) p φdx. pλ k r β (1 u k ) u ku k dx = λ k I k. (4.8) Hence substituting (4.7) and (4.8) into (4.6), we finally derive I k = (β +2)ω n ẅ(τ k ) 2(p +1)λ k w(τ k ) and the proof is complete.
20 382 N. I. Kavallaris, T. Miyasita and T. Suzuki NoDEA 5. Connecting orbits Problem (1.1) generates a local semiflow in X = W 1,q () for q>n, and each ϕ C λ is regarded as an equilibrium. It is said to be hyperbolic if (1.4) does not admit µ = as an eigenvalue. Given ϕ, ψ C λ, we define the stable manifold W s (ϕ) ofϕ and the unstable manifold W u (ψ) ofψ by W s (φ) ={ξ X there exists an orbit u :[, + ) X to (1.1) such that u() = ξ and u(t) φ in X as t + }, W u (ψ) ={ξ X there exists an orbit u :(, ] X to (1.1) such that u() = ξ and u(t) ψ in X as t }. If there is a solution u = u(,t) to (1.1) for t R satisfying lim t u(t) =ψ and lim t + u(t) =ϕ in X, we call {u(,t) t R} a connecting orbit between ϕ and ψ. In the case of (1.3), we can define radially symmetric versions of the above notions. Here, we just note that ϕ Cr λ is called radially hyperbolic if (4.1) does not admit µ = as an eigenvalue, and that radial stable and unstable manifolds of ϕ Cr λ are denoted by Wr s (ϕ) and Wr u (ψ), respectively. Then, we obtain the MorseSmale property of radially symmetric orbits [1, 6, 22]. Theorem 5.1. Let ϕ, ψ C λ r be radially hyperbolic and assume the existence of a radial connecting orbit {u(,t) t R} between ϕ and ψ. Then W s r (ϕ) and W u r (ψ) intersect transversally in X r = {v X v = v( x )}, and T u W s r (ϕ)+t u W u r (ψ) =X r for u = u(, ), where T u W denotes the tangential space of W at u. It also holds that i R (λ, ϕ) <i R (λ, ψ). To confirm this, we denote by µ k the kth eigenvalue of the linearized problem (4.1) around ψ C λ r. For given <k<i R (λ, ψ) and µ k <µ<min{,µ k+1 },we consider a finer invariant manifold than the unstable manifold as Wr,k(ψ) u ={u Wr u (ψ) lim t e µt u(,t) ψ Xr =, where u(,t)(t ) is a solution to (1.1) satisfying u(, ) = u }. Theorem 5.1 is obtained by the discrete Lyapunov function. To state this, for given Ψ C[, 1], we denote by l(ψ) the supremum of k such that there is r 1 <r 2 < <r k 1 satisfying Ψ(r j )Ψ(r j+1 ) < for j =1, 2,,k. Then, Lemma 2.9 of [6] is stated as follows. Lemma 5.2. Given radially hyperbolic ϕ, ψ C λ r, we obtain the following.
21 Vol. 15 (28) Touchdown and related problems in MEMS If u,u 1 Wr,k u (ψ), w T u Wr,k u (ψ), u u 1, and w, then it holds that l(u u 1 ) k and l(w) k. 2. If u,u 1 Wr s (ϕ), w T u Wr s (ϕ), u u 1, and w, then it holds that l(u u 1 ) i R (λ, ϕ)+1 and l(w) i R (λ, ϕ)+1. Proof of Theoerem 5.1. The proof is similar to that of Lemma 3.1 in [6]. First, we note u W s r (ϕ) W u r (ψ) and hence i R (λ, ϕ) <i R (λ, ψ) by Lemma 5.2. This implies W u r, i R (λ,ϕ) (ψ) W u r, i R (λ,ψ) (ψ), where the dimension of T u W u r (ϕ) and the codimension of T u W s r (ϕ) are equal to i R (λ, ϕ). Therefore it is sufficient to show T u W u r, i R (λ, ϕ) (ψ) T u W s r (ϕ) ={}, but this is an immediate consequence of Lemma 5.2, hence T u W s r (ϕ)+t u W u r (ψ) =X r. Acknowledgement N.I. Kavallaris was supported by the 21st Century Center of Excellence Program Towards a New Basic Science : Depth and Synthesis. He would like also to thank the Division of Mathematical Science of Graduate School of Engineers Science of Osaka University of its hospitality during the preparation of this manuscript. References [1] S. B. Angenent, The MorseSmale property for a semilinear parabolic equation. J. Differential Equations. 62 (1986), [2] H. Brezis, T. Cazenave, Y. Martel, and A. Ramiandrisoa, Blow up for u t u = g(u) revisited. Adv. Differential Equations 1 (1996), [3] KS Chou and GF Zhenh, Some dichotomy results for the quenching problem, preprint. [4] P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the first branch of unstable solutions for an elliptic problem with a singular nonlinearity. Comm. Pure Appl. Math. 6 (27), [5] A. Friedman and B. McLeod, Blowup of positive solutions of semilinear heat equations. Indiana Uni. Math. J. 34 (1985),
22 384 N. I. Kavallaris, T. Miyasita and T. Suzuki NoDEA [6] M. Fila, H. Matano and P. Poláčik, Existence of L 1 connection between equilibria of a semilinear parabolic equation. J. Dynamics and Differential Equations. 14 (22), [7] H. Fujita, On the nonlinear equations u + e u =and v/ t = v + e v. Bull. Amer. Math. Soc. 75 (1969), [8] N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: stationary case. SIAM J. Math. Anal. 38 (27), [9] N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices II: dynamic case. Nonlinear Diff. Eqns. Appl. in press. [1] B. Gidas, W.M. Ni and L. Niereberg, Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), [11] Y. Guo, Z. Pan and M. J. Ward, Touchdown and pullin voltage behavior of a MEMS device with varying dielectric properties, SIAM J. Appl. Math. 166 (26), [12] Y. Guo, On the partial differential equations of electrostatic MEMS devices III:Refined Touchdown Behaviour, submitted in J. Differential Equations. [13] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources. Arch. Rational Mech. Anal. 49 (1973), [14] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, [15] A. Kohda and T. Suzuki, Blowup criteria for semilinear parabolic equations. J. Math. Anal. Appl. 243 (2), [16] C.S. Lin and W.M. Ni, A counterexample to the nodal domain conjecture and a related semilinear equation. Proc. Am. Math. Soc. 12 (1988), [17] T. Miyasita and T. Suzuki, Nonlocal Gel fand problem in higher dimension, Nonlocal Elliptic and Parabolic Problems, Banach Center Publ. 66 (24), [18] K. Nagasaki and T. Suzuki, Radial solutions for u + λe u =on annuli in higher dimensions. J. Differential Equations. 1 (1992), [19] K. Nagasaki and T. Suzuki, Spectral and related properties about the Emden Fowler equation u = λe u on circular domains. Math. Ann. 299 (1994), [2] Y. Naito and T. Suzuki, Radial symmetry of positive solutions for semilinear elliptic equations on the unit ball in R n. Funkcial Ekvac. 41 (1998), [21] J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS, Chapman Hall and CRC Press, 22.
23 Vol. 15 (28) Touchdown and related problems in MEMS 385 [22] P. Poláčik, Transversal and nontransversal intersections of stable and unstable manifolds in reaction diffusion equations on symmetric domains. Differential and Integral Equations. 7 (1994), [23] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems. J. Func. Anal. 7 (1971), [24] P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), Nikos I. Kavallaris Division of Mathematical Science Graduate School of Engineering Science Osaka University Toyonaka Japan Tosiya Miyasita Department of Mathematics Graduate School of Science Kyoto University Kyoto Japan Takashi Suzuki Division of Mathematical Science Graduate School of Engineering Science Osaka University Toyonaka Japan Received: 22 May 27 Accepted: 7 January 28 Published Online First 3 October 28
24 .
UNIVERSITETET I OSLO
UNIVERSITETET I OSLO Det matematisknaturvitenskapelige fakultet Eksamen i MAT2400 Analyse 1. Eksamensdag: Onsdag 15. juni 2011. Tid for eksamen: 09.00 13.00 Oppgavesettet er på 6 sider. Vedlegg: Tillatte
DetaljerPhysical origin of the Gouy phase shift by Simin Feng, Herbert G. Winful Opt. Lett. 26, (2001)
by Simin Feng, Herbert G. Winful Opt. Lett. 26, 485487 (2001) http://smos.sogang.ac.r April 18, 2014 Introduction What is the Gouy phase shift? For Gaussian beam or TEM 00 mode, ( w 0 r 2 E(r, z) = E
DetaljerSlopeIntercept Formula
LESSON 7 Slope Intercept Formula LESSON 7 SlopeIntercept Formula Here are two new words that describe lines slope and intercept. The slope is given by m (a mountain has slope and starts with m), and intercept
DetaljerUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postponed exam: ECON420 Mathematics 2: Calculus and linear algebra Date of exam: Tuesday, June 8, 203 Time for exam: 09:00 a.m. 2:00 noon The problem set covers
DetaljerGraphs similar to strongly regular graphs
Joint work with Martin Ma aj 5th June 2014 Degree/diameter problem Denition The degree/diameter problem is the problem of nding the largest possible graph with given diameter d and given maximum degree
DetaljerUniversitetet i Bergen Det matematisknaturvitenskapelige fakultet Eksamen i emnet Mat131  Differensiallikningar I Onsdag 25. mai 2016, kl.
1 MAT131 Bokmål Universitetet i Bergen Det matematisknaturvitenskapelige fakultet Eksamen i emnet Mat131  Differensiallikningar I Onsdag 25. mai 2016, kl. 0914 Oppgavesettet er 4 oppgaver fordelt på
DetaljerUNIVERSITETET I OSLO ØKONOMISK INSTITUTT
UNIVERSITETET I OSLO ØKONOMISK INSTITUTT Eksamen i: ECON30/40 Matematikk : Matematisk analyse og lineær algebra Exam: ECON30/40 Mathematics : Calculus and Linear Algebra Eksamensdag: Tirsdag 0. desember
DetaljerUNIVERSITETET I OSLO ØKONOMISK INSTITUTT
UNIVERSITETET I OSLO ØKONOMISK INSTITUTT Eksamen i: ECON320/420 Matematikk 2: Matematisk analyse og lineær algebra Exam: ECON320/420 Mathematics 2: Calculus and Linear Algebra Eksamensdag: Tirsdag 7. juni
DetaljerAn oscillatory bifurcation from infinity, and
Nonlinear differ. equ. appl. 15 (28), 335 345 121 9722/8/3335 11, DOI 1.17/s387241 c 28 Birkhäuser Verlag Basel/Switzerland An oscillatory bifurcation from infinity, and from zero Philip Korman Abstract.
DetaljerTrigonometric Substitution
Trigonometric Substitution Alvin Lin Calculus II: August 06  December 06 Trigonometric Substitution sin 4 (x) cos (x) dx When you have a product of sin and cos of different powers, you have three different
DetaljerSolutions #12 ( M. y 3 + cos(x) ) dx + ( sin(y) + z 2) dy + xdz = 3π 4. The surface M is parametrized by σ : [0, 1] [0, 2π] R 3 with.
Solutions #1 1. a Show that the path γ : [, π] R 3 defined by γt : cost ı sint j sint k lies on the surface z xy. b valuate y 3 cosx dx siny z dy xdz where is the closed curve parametrized by γ. Solution.
DetaljerNeural Network. Sensors Sorter
CSC 302 1.5 Neural Networks Simple Neural Nets for Pattern Recognition 1 AppleBanana Sorter Neural Network Sensors Sorter Apples Bananas 2 Prototype Vectors Measurement vector p = [shape, texture, weight]
DetaljerMathematics 114Q Integration Practice Problems SOLUTIONS. = 1 8 (x2 +5x) 8 + C. [u = x 2 +5x] = 1 11 (3 x)11 + C. [u =3 x] = 2 (7x + 9)3/2
Mathematics 4Q Name: SOLUTIONS. (x + 5)(x +5x) 7 8 (x +5x) 8 + C [u x +5x]. (3 x) (3 x) + C [u 3 x] 3. 7x +9 (7x + 9)3/ [u 7x + 9] 4. x 3 ( + x 4 ) /3 3 8 ( + x4 ) /3 + C [u + x 4 ] 5. e 5x+ 5 e5x+ + C
DetaljerSplitting the differential Riccati equation
Splitting the differential Riccati equation Tony Stillfjord Numerical Analysis, Lund University Joint work with Eskil Hansen Innsbruck Okt 15, 2014 Outline Splitting methods for evolution equations The
DetaljerOppgave 1. ( xφ) φ x t, hvis t er substituerbar for x i φ.
Oppgave 1 Beviskalklen i læreboka inneholder sluttningsregelen QR: {ψ φ}, ψ ( xφ). En betingelse for å anvende regelen er at det ikke finnes frie forekomste av x i ψ. Videre så inneholder beviskalklen
DetaljerSVM and Complementary Slackness
SVM and Complementary Slackness David Rosenberg New York University February 21, 2017 David Rosenberg (New York University) DSGA 1003 February 21, 2017 1 / 20 SVM Review: Primal and Dual Formulations
DetaljerUnit Relational Algebra 1 1. Relational Algebra 1. Unit 3.3
Relational Algebra 1 Unit 3.3 Unit 3.3  Relational Algebra 1 1 Relational Algebra Relational Algebra is : the formal description of how a relational database operates the mathematics which underpin SQL
DetaljerUNIVERSITETET I OSLO ØKONOMISK INSTITUTT
UNIVERSITETET I OSLO ØKONOMISK INSTITUTT Exam: ECON320/420 Mathematics 2: Calculus and Linear Algebra Eksamen i: ECON320/420 Matematikk 2: Matematisk analyse og lineær algebra Date of exam: Friday, May
DetaljerMoving Objects. We need to move our objects in 3D space.
Transformations Moving Objects We need to move our objects in 3D space. Moving Objects We need to move our objects in 3D space. An object/model (box, car, building, character,... ) is defined in one position
DetaljerUNIVERSITETET I OSLO ØKONOMISK INSTITUTT
UNIVERSITETET I OSLO ØKONOMISK INSTITUTT Utsatt ksamen i: ECON3120/4120 Matematikk 2: Matematisk analyse og lineær algebra Postponed exam: ECON3120/4120 Mathematics 2: Calculus and linear algebra Eksamensdag:
DetaljerUNIVERSITETET I OSLO ØKONOMISK INSTITUTT
UNIVERSITETET I OSLO ØKONOMISK INSTITUTT Eksamen i: ECON20/420 Matematikk 2: Matematisk analyse og lineær algebra Exam: ECON20/420 Mathematics 2: Calculus and Linear Algebra Eksamensdag: Fredag 2. mai
DetaljerTrust region methods: global/local convergence, approximate January methods 24, / 15
Trust region methods: global/local convergence, approximate methods January 24, 2014 Trust region methods: global/local convergence, approximate January methods 24, 2014 1 / 15 Trustregion idea Model
DetaljerExistence of resistance forms in some (non selfsimilar) fractal spaces
Existence of resistance forms in some (non selfsimilar) fractal spaces Patricia Alonso Ruiz D. Kelleher, A. Teplyaev University of Ulm Cornell, 12 June 2014 Motivation X Fractal Motivation X Fractal Laplacian
DetaljerStationary Phase Monte Carlo Methods
Stationary Phase Monte Carlo Methods Daniel Doro Ferrante G. S. Guralnik, J. D. Doll and D. Sabo HET Physics Dept, Brown University, USA. danieldf@het.brown.edu www.het.brown.edu Introduction: Motivations
DetaljerUNIVERSITETET I OSLO ØKONOMISK INSTITUTT
UNIVERSITETET I OSLO ØKONOMISK INSTITUTT Eksamen i: ECON3120/4120 Mathematics 2: Calculus an linear algebra Exam: ECON3120/4120 Mathematics 2: Calculus an linear algebra Eksamensag: Tirsag 3. juni 2008
DetaljerUNIVERSITETET I OSLO ØKONOMISK INSTITUTT
UNIVERSITETET I OSLO ØKONOMISK INSTITUTT Eksamen i: ECON320/420 Matematikk 2: Matematisk analyse og lineær algebra Exam: ECON320/420 Mathematics 2: Calculus and Linear Algebra Eksamensdag: Mandag 8. desember
DetaljerKneser hypergraphs. May 21th, CERMICS, Optimisation et Systèmes
Kneser hypergraphs Frédéric Meunier May 21th, 2015 CERMICS, Optimisation et Systèmes Kneser hypergraphs m, l, r three integers s.t. m rl. Kneser hypergraph KG r (m, l): V (KG r (m, l)) = ( [m]) l { E(KG
DetaljerUNIVERSITETET I OSLO ØKONOMISK INSTITUTT
UNIVERSITETET I OSLO ØKONOMISK INSTITUTT Utsatt eksamen i: ECON420 Matematikk 2: Matematisk analyse og lineær algebra Postponed exam: ECON420 Mathematics 2: Calculus and Linear Algebra Eksamensdag: Mandag
DetaljerDynamic Programming Longest Common Subsequence. Class 27
Dynamic Programming Longest Common Subsequence Class 27 Protein a protein is a complex molecule composed of long singlestrand chains of amino acid molecules there are 20 amino acids that make up proteins
DetaljerUNIVERSITETET I OSLO ØKONOMISK INSTITUTT
UNIVERSITETET I OSLO ØKONOMISK INSTITUTT Eksamen i: ECON320/420 Matematikk 2: Matematisk analyse og lineær algebra Exam: ECON320/420 Mathematics 2: Calculus and Linear Algebra Eksamensdag: Onsdag 6. desember
DetaljerCall function of two parameters
Call function of two parameters APPLYUSER USER x fµ 1 x 2 eµ x 1 x 2 distinct e 1 0 0 v 1 1 1 e 2 1 1 v 2 2 2 2 e x 1 v 1 x 2 v 2 v APPLY f e 1 e 2 0 v 2 0 µ Evaluating function application The math demands
DetaljerGradient. Masahiro Yamamoto. last update on February 29, 2012 (1) (2) (3) (4) (5)
Gradient Masahiro Yamamoto last update on February 9, 0 definition of grad The gradient of the scalar function φr) is defined by gradφ = φr) = i φ x + j φ y + k φ ) φ= φ=0 ) ) 3) 4) 5) uphill contour downhill
Detaljer5 E Lesson: Solving Monohybrid Punnett Squares with Coding
5 E Lesson: Solving Monohybrid Punnett Squares with Coding Genetics Fill in the Brown colour Blank Options Hair texture A field of biology that studies heredity, or the passing of traits from parents to
DetaljerOn the Existence of Strong Solutions to a Fluid Structure Interaction Problem with Navier Boundary Conditions
J. Math. Fluid Mech. 2019 21:36 c 2019 Springer Nature Switzerland AG https://doi.org/10.1007/s0002101904407 Journal of Mathematical Fluid Mechanics On the Existence of Strong Solutions to a Fluid Structure
DetaljerRingvorlesung Biophysik 2016
Ringvorlesung Biophysik 2016 BornOppenheimer Approximation & Beyond Irene Burghardt (burghardt@chemie.unifrankfurt.de) http://www.theochem.unifrankfurt.de/teaching/ 1 Starting point: the molecular Hamiltonian
DetaljerUNIVERSITETET I OSLO
UNIVERSITETET I OSLO Det matematisknaturvitenskapelige fakultet Eksamen i INF 3230 Formell modellering og analyse av kommuniserende systemer Eksamensdag: 4. april 2008 Tid for eksamen: 9.00 12.00 Oppgavesettet
DetaljerUNIVERSITETET I OSLO ØKONOMISK INSTITUTT
UNIVERSITETET I OSLO ØKONOMISK INSTITUTT Eksamen i: ECON320/420 Mathematics 2: Calculus and linear algebra Exam: ECON320/420 Mathematics 2: Calculus and linear algebra Eksamensdag: Tirsdag 30. mai 207
DetaljerUNIVERSITETET I OSLO ØKONOMISK INSTITUTT
UNIVERSITETET I OSLO ØKONOMISK INSTITUTT Eksamen i: ECON3120/4120 Matematikk 2: Matematisk analyse og lineær algebra Exam: ECON3120/4120 Mathematics 2: Calculus and Linear Algebra Eksamensdag: Tirsdag
DetaljerTMA4329 Intro til vitensk. beregn. V2017
Norges teknisk naturvitenskapelige universitet Institutt for Matematiske Fag TMA439 Intro til vitensk. beregn. V17 ving 4 [S]T. Sauer, Numerical Analysis, Second International Edition, Pearson, 14 Teorioppgaver
DetaljerFYSMEK1110 Eksamensverksted 23. Mai :1518:00 Oppgave 1 (maks. 45 minutt)
FYSMEK1110 Eksamensverksted 23. Mai 2018 14:1518:00 Oppgave 1 (maks. 45 minutt) Page 1 of 9 Svar, eksempler, diskusjon og gode råd fra studenter (30 min) Hva får dere poeng for? Gode råd fra forelesere
Detaljer0:7 0:2 0:1 0:3 0:5 0:2 0:1 0:4 0:5 P = 0:56 0:28 0:16 0:38 0:39 0:23
UTKAST ENGLISH VERSION EKSAMEN I: MOT100A STOKASTISKE PROSESSER VARIGHET: 4 TIMER DATO: 16. februar 2006 TILLATTE HJELPEMIDLER: Kalkulator; Tabeller og formler i statistikk (Tapir forlag): Rottman: Matematisk
DetaljerLevel Set methods. Sandra AllaartBruin. Level Set methods p.1/24
Level Set methods Sandra AllaartBruin sbruin@win.tue.nl Level Set methods p.1/24 Overview Introduction Level Set methods p.2/24 Overview Introduction Boundary Value Formulation Level Set methods p.2/24
Detaljermelting ECMI Modelling week 2008 Modelling and simulation of ice/snow melting Sabrina Wandl  University of Linz Tuomo MäkiMarttunen  Tampere UT
and and ECMI week 2008 Outline and Problem Description find model for processes consideration of effects caused by presence of salt point and numerical solution and and heat equations liquid phase: T L
DetaljerSuperlinear Ambrosetti Prodi problem for the plaplacian operator
Nonlinear Differ. Equ. Appl. 17 (010), 337 353 c 010 Birkhäuser Verlag Basel/Switzerland 10197/10/03033717 published online January 8, 010 DOI 10.1007/s000300100057 Nonlinear Differential Equations
DetaljerNO X chemistry modeling for coal/biomass CFD
NO X chemistry modeling for coal/biomass CFD Jesper Møller Pedersen 1, Larry Baxter 2, Søren Knudsen Kær 3, Peter Glarborg 4, Søren Lovmand Hvid 1 1 DONG Energy, Denmark 2 BYU, USA 3 AAU, Denmark 4 DTU,
DetaljerPeriodic solutions for a class of nonautonomous differential delay equations
Nonlinear Differ. Equ. Appl. 16 (2009), 793 809 c 2009 Birkhäuser Verlag Basel/Switzerland 10219722/09/06079317 published online July 4, 2009 DOI 10.1007/s0003000900358 Nonlinear Differential Equations
DetaljerUNIVERSITETET I OSLO ØKONOMISK INSTITUTT
UNIVERSITETET I OSLO ØKONOMISK INSTITUTT Eksamen i: ECON360/460 Samfunnsøkonomisk lønnsomhet og økonomisk politikk Exam: ECON360/460  Resource allocation and economic policy Eksamensdag: Fredag 2. november
DetaljerSecond Order ODE's (2P) Young Won Lim 7/1/14
Second Order ODE's (2P) Copyright (c) 20112014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
DetaljerUNIVERSITETET I OSLO ØKONOMISK INSTITUTT
UNIVERSITETET I OSLO ØKONOMISK INSTITUTT Utsatt eksamen i: ECON320/420 Matematikk 2: Matematisk analyse og lineær algebra Postponed exam: ECON320/420 Mathematics 2: Calculus and Linear Algebra Eksamensdag:
DetaljerUNIVERSITETET I OSLO ØKONOMISK INSTITUTT
UNIVERSIEE I OSLO ØKONOMISK INSIU Eksamen i: ECON320/420 Mathematics 2: Calculus and Linear Algebra Exam: ECON320/420 Mathematics 2: Calculus and Linear Algebra Eksamensdag:. desember 207 Sensur kunngjøres:
DetaljerTFY4170 Fysikk 2 Justin Wells
TFY4170 Fysikk 2 Justin Wells Forelesning 5: Wave Physics Interference, Diffraction, Young s double slit, many slits. Mansfield & O Sullivan: 12.6, 12.7, 19.4,19.5 Waves! Wave phenomena! Wave equation
Detaljer32.2. Linear Multistep Methods. Introduction. Prerequisites. Learning Outcomes
Linear Multistep Methods 32.2 Introduction In the previous Section we saw two methods (Euler and trapezium) for approximating the solutions of certain initial value problems. In this Section we will see
DetaljerFlows and Critical Points
Nonlinear differ. equ. appl. 15 (2008), 495 509 c 2008 Birkhäuser Verlag Basel/Switzerland 10219722/04049515 published online 26 November 2008 DOI 10.1007/s0003000870312 Nonlinear Differential Equations
DetaljerUNIVERSITETET I OSLO ØKONOMISK INSTITUTT
UNIVERSITETET I OSLO ØKONOMISK INSTITUTT Utsatt eksamen i: ECON420 Matematikk 2: Matematisk analyse og lineær algebra Postponed exam: ECON420 Mathematics 2: Calculus and Linear Algebra Eksamensdag: Mandag
DetaljerEndelig ikkerøyker for Kvinner! (Norwegian Edition)
Endelig ikkerøyker for Kvinner! (Norwegian Edition) Allen Carr Click here if your download doesn"t start automatically Endelig ikkerøyker for Kvinner! (Norwegian Edition) Allen Carr Endelig ikkerøyker
DetaljerHvor mye praktisk kunnskap har du tilegnet deg på dette emnet? (1 = ingen, 5 = mye)
INF247 Er du? Er du?  Annet Ph.D. Student Hvor mye teoretisk kunnskap har du tilegnet deg på dette emnet? (1 = ingen, 5 = mye) Hvor mye praktisk kunnskap har du tilegnet deg på dette emnet? (1 = ingen,
DetaljerVerifiable SecretSharing Schemes
Aarhus University Verifiable SecretSharing Schemes Irene Giacomelli joint work with Ivan Damgård, Bernardo David and Jesper B. Nielsen Aalborg, 30th June 2014 Verifiable SecretSharing Schemes Aalborg,
DetaljerOn time splitting for NLS in the semiclassical regime
On time splitting for NLS in the semiclassical regime Rémi Carles CNRS & Univ. Montpellier Rémi Carles (Montpellier) Splitting for semiclassical NLS 1 / 21 Splitting for NLS i t u + 1 2 u = f ( u 2) u,
DetaljerGeneralization of agestructured models in theory and practice
Generalization of agestructured models in theory and practice Stein Ivar Steinshamn, stein.steinshamn@snf.no 25.10.11 www.snf.no Outline How agestructured models can be generalized. What this generalization
DetaljerUNIVERSITETET I OSLO ØKONOMISK INSTITUTT
1 UNIVERSITETET I OSLO ØKONOMISK INSTITUTT BOKMÅL Utsatt eksamen i: ECON2915 Vekst og næringsstruktur Eksamensdag: 07.12.2012 Tid for eksamen: kl. 09:0012:00 Oppgavesettet er på 5 sider Tillatte hjelpemidler:
DetaljerOppgave 1a Definer følgende begreper: Nøkkel, supernøkkel og funksjonell avhengighet.
TDT445 Øving 4 Oppgave a Definer følgende begreper: Nøkkel, supernøkkel og funksjonell avhengighet. Nøkkel: Supernøkkel: Funksjonell avhengighet: Data i en database som kan unikt identifisere (et sett
DetaljerUNIVERSITETET I OSLO
UNIVERSITETET I OSLO Det matematisknaturvitenskapelige fakultet Eksamen i INF 3230 Formell modellering og analyse av kommuniserende systemer Eksamensdag: 4. juni 2010 Tid for eksamen: 9.00 12.00 Oppgavesettet
DetaljerHØGSKOLEN I NARVIK  SIVILINGENIØRUTDANNINGEN
HØGSKOLEN I NARVIK  SIVILINGENIØRUTDANNINGEN EKSAMEN I FAGET STE 6243 MODERNE MATERIALER KLASSE: 5ID DATO: 7 Oktober 2005 TID: 900200, 3 timer ANTALL SIDER: 7 (inklusiv Appendix: tabell og formler) TILLATTE
DetaljerLipschitz Metrics for Nonsmooth evolutions
Lipschitz Metrics for Nonsmooth evolutions Alberto Bressan Department of Mathematics, Penn State University bressan@math.psu.edu Alberto Bressan (Penn State) Nonsmooth evolutions 1 / 36 Wellposedness
DetaljerDatabases 1. Extended Relational Algebra
Databases 1 Extended Relational Algebra Relational Algebra What is an Algebra? Mathematical system consisting of: Operands  variables or values from which new values can be constructed. Operators 
DetaljerExercise 1: Phase Splitter DC Operation
Exercise 1: DC Operation When you have completed this exercise, you will be able to measure dc operating voltages and currents by using a typical transistor phase splitter circuit. You will verify your
DetaljerECON3120/4120 Mathematics 2, spring 2004 Problem solutions for the seminar on 5 May Old exam problems
Department of Economics May 004 Arne Strøm ECON0/40 Mathematics, spring 004 Problem solutions for the seminar on 5 May 004 (For practical reasons (read laziness, most of the solutions this time are in
DetaljerEksamensoppgave i TMA4320 Introduksjon til vitenskapelige beregninger
Institutt for matematiske fag Eksamensoppgave i TMA432 Introduksjon til vitenskapelige beregninger Faglig kontakt under eksamen: Anton Evgrafov Tlf: 453 163 Eksamensdato: 8. august 217 Eksamenstid (fra
DetaljerNumerical Simulation of Shock Waves and Nonlinear PDE
Numerical Simulation of Shock Waves and Nonlinear PDE Kenneth H. Karlsen (CMA) Partial differential equations A partial differential equation (PDE for short) is an equation involving functions and their
DetaljerKROPPEN LEDER STRØM. Sett en finger på hvert av kontaktpunktene på modellen. Da får du et lydsignal.
KROPPEN LEDER STRØM Sett en finger på hvert av kontaktpunktene på modellen. Da får du et lydsignal. Hva forteller dette signalet? Gå flere sammen. Ta hverandre i hendene, og la de to ytterste personene
DetaljerSmart HighSide Power Switch BTS730
PGDSO20 RoHS compliant (green product) AEC qualified 1 Ω Ω µ Data Sheet 1 V1.0, 20071217 Data Sheet 2 V1.0, 20071217 Ω µ µ Data Sheet 3 V1.0, 20071217 µ µ Data Sheet 4 V1.0, 20071217 Data Sheet
DetaljerOppgave 1. Norges teknisknaturvitenskapelige universitet NTNU Institutt for fysikk EKSAMEN I: MNFFY 245 INNFØRING I KVANTEMEKANIKK
Norges teknisknaturvitenskapelige universitet NTNU Institutt for fysikk EKSAMEN I: MNFFY 45 INNFØRING I KVANTEMEKANIKK DATO: Fredag 4 desember TID: 9 5 Antall vekttall: 4 Antall sider: 5 Tillatte hjelpemidler:
DetaljerContinuity. Subtopics
0 Cotiuity Chapter 0: Cotiuity Subtopics.0 Itroductio (Revisio). Cotiuity of a Fuctio at a Poit. Discotiuity of a Fuctio. Types of Discotiuity.4 Algebra of Cotiuous Fuctios.5 Cotiuity i a Iterval.6 Cotiuity
DetaljerRadially symmetric growth of nonnecrotic tumors
Nonlinear Differ. Equ. Appl. 17 (1), 1 c 9 Birkhäuser Verlag Basel/Switzerland 1197/1/11 published online October, 9 DOI 1.17/s39376 Nonlinear Differential Equations and Applications NoDEA Radially
DetaljerMaple Basics. K. Cooper
Basics K. Cooper 2012 History History 1982 Macsyma/MIT 1988 Mathematica/Wolfram 1988 /Waterloo Others later History Why? Prevent silly mistakes Time Complexity Plots Generate LATEX This is the 21st century;
DetaljerGround states for fractional Kirchhoff equations with critical nonlinearity in low dimension
onlinear Differ. Equ. Appl. 07 4:50 c 07 Springer International Publishing AG DOI 0.007/s000300704737 onlinear Differential Equations and Applications odea Ground states for fractional Kirchhoff equations
DetaljerExam in Quantum Mechanics (phys201), 2010, Allowed: Calculator, standard formula book and up to 5 pages of own handwritten notes.
Exam in Quantum Mechanics (phys01), 010, There are 3 problems, 1 3. Each problem has several sub problems. The number of points for each subproblem is marked. Allowed: Calculator, standard formula book
DetaljerDu må håndtere disse hendelsene ved å implementere funksjonene init(), changeh(), changev() og escape(), som beskrevet nedenfor.
613 July 2013 Brisbane, Australia Norwegian 1.0 Brisbane har blitt tatt over av store, muterte wombater, og du må lede folket i sikkerhet. Veiene i Brisbane danner et stort rutenett. Det finnes R horisontale
DetaljerHan Ola of Han Per: A NorwegianAmerican Comic Strip/En Norskamerikansk tegneserie (Skrifter. Serie B, LXIX)
Han Ola of Han Per: A NorwegianAmerican Comic Strip/En Norskamerikansk tegneserie (Skrifter. Serie B, LXIX) Peter J. Rosendahl Click here if your download doesn"t start automatically Han Ola of Han Per:
DetaljerPerpetuum (im)mobile
Perpetuum (im)mobile Sett hjulet i bevegelse og se hva som skjer! Hva tror du er hensikten med armene som slår ut når hjulet snurrer mot høyre? Hva tror du ordet Perpetuum mobile betyr? Modell 170, Rev.
DetaljerEksamen i TMA4190 Mangfoldigheter Onsdag 4 juni, Tid :
Norges teknisknaturvitenskapelige universitet Institutt for matematiske fag SOLUTIONS Eksamen i TMA4190 Mangfoldigheter Onsdag 4 juni, 2013. Tid : 09.00 13.00 Oppgave 1 a) La U R n være enhetsdisken x
DetaljerMA2501 Numerical methods
MA250 Numerical methods Solutions to problem set Problem a) The function f (x) = x 3 3x + satisfies the following relations f (0) = > 0, f () = < 0 and there must consequently be at least one zero for
DetaljerSpeed Racer Theme. Theme Music: Cartoon: Charles Schultz / Jef Mallett Peanuts / Frazz. September 9, 2011 Physics 131 Prof. E. F.
September 9, 2011 Physics 131 Prof. E. F. Redish Theme Music: Speed Racer Theme Cartoon: Charles Schultz / Jef Mallett Peanuts / Frazz 1 Reading questions Are the lines on the spatial graphs representing
DetaljerSTILLAS  STANDARD FORSLAG FRA SEF TIL NY STILLAS  STANDARD
FORSLAG FRA SEF TIL NY STILLAS  STANDARD 1 Bakgrunnen for dette initiativet fra SEF, er ønsket om å gjøre arbeid i høyden tryggere / sikrere. Både for stillasmontører og brukere av stillaser. 2 Reviderte
DetaljerSelfimproving property of nonlinear higher order parabolic systems near the boundary
Nonlinear Differ Equ Appl 7 200), 2 54 c 2009 Birkhäuser Verlag Basel/Switzerland 029722/0/000234 published online October 2, 2009 DOI 0007/s0003000900385 Nonlinear Differential Equations and Applications
DetaljerTDT4117 Information Retrieval  Autumn 2014
TDT4117 Information Retrieval  Autumn 2014 Assignment 1 Task 1 : Basic Definitions Explain the main differences between: Information Retrieval vs Data Retrieval En samling av data er en godt strukturert
DetaljerOle Isak Eira Masters student Arctic agriculture and environmental management. University of Tromsø Sami University College
The behavior of the reindeer herd  the role of the males Ole Isak Eira Masters student Arctic agriculture and environmental management University of Tromsø Sami University College Masters student at Department
DetaljerEksamen i FY3452 GRAVITASJON OG KOSMOLOGI Lørdag 19. mai :00 13:00
NTNU Side 1 av 2 Institutt for fysikk Faglig kontakt under eksamen: Professor Kåre Olaussen Telefon: 45 43 71 70 Eksamen i FY3452 GRAVITASJON OG KOSMOLOGI Lørdag 19. mai 2012 09:00 13:00 Tillatte hjelpemidler:
DetaljerUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS English Postponed exam: ECON2915 Economic growth Date of exam: 11.12.2014 Time for exam: 09:00 a.m. 12:00 noon The problem set covers 4 pages Resources allowed:
DetaljerDen som gjør godt, er av Gud (Multilingual Edition)
Den som gjør godt, er av Gud (Multilingual Edition) Arne Jordly Click here if your download doesn"t start automatically Den som gjør godt, er av Gud (Multilingual Edition) Arne Jordly Den som gjør godt,
Detaljer1 Aksiomatisk definisjon av vanlige tallsystemer
Notat XX for MAT1140 1 Aksiomatisk definisjon av vanlige tallsystemer 1.1 Aksiomer Vi betrakter en mengde R, utstyrt med to avbild Algebraiske aksiomer. ninger: addisjon { R R R, (x, y) x + y. { R R R,
DetaljerUNIVERSITETET I OSLO
UNIVERSITETET I OSLO Det matematisknaturvitenskapelige fakultet Eksamen i INF 3230/4230 Formell modellering og analyse av kommuniserende systemer Eksamensdag: 24. mars 2006 Tid for eksamen: 13.30 16.30
DetaljerPSi Apollo. Technical Presentation
PSi Apollo Spreader Control & Mapping System Technical Presentation Part 1 System Architecture PSi Apollo System Architecture PSi Customer label On/Off switch Integral SD card reader/writer MENU key Typical
DetaljerEksamen i FY3466 KVANTEFELTTEORI II Tirsdag 20. mai :00 13:00
NTNU Side 1 av 3 Institutt for fysikk Faglig kontakt under eksamen: Professor Kåre Olaussen Telefon: 9 36 52 eller 45 43 71 70 Eksamen i FY3466 KVANTEFELTTEORI II Tirsdag 20. mai 2008 09:00 13:00 Tillatte
DetaljerUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS English Exam: ECON2915 Economic Growth Date of exam: 25.11.2014 Grades will be given: 16.12.2014 Time for exam: 09.00 12.00 The problem set covers 3 pages Resources
DetaljerSolvability and Regularity for an Elliptic System Prescribing the Curl, Divergence, and Partial Trace of a Vector Field on SobolevClass Domains
J. Math. Fluid Mech. 19 (2017), 375 422 c 2016 Springer International Publishing 14226928/17/03037548 DOI 10.1007/s000210160289y Journal of Mathematical Fluid Mechanics Solvability and Regularity
DetaljerSCE1106 Control Theory
Master study Systems and Control Engineering Department of Technology Telemark University College DDiR, October 26, 2006 SCE1106 Control Theory Exercise 6 Task 1 a) The poles of the open loop system is
DetaljerMethod validation for NO (10 ppm to 1000 ppm) in nitrogen using the Fischer Rosemount chemiluminescence analyser NOMPUMELELO LESHABANE
Method validation for NO (10 ppm to 1000 ppm) in nitrogen using the Fischer Rosemount chemiluminescence analyser NOMPUMELELO LESHABANE Presentation Outline Introduction Principle of operation Precision
DetaljerEmneevaluering GEOV272 V17
Emneevaluering GEOV272 V17 Studentenes evaluering av kurset Svarprosent: 36 % (5 av 14 studenter) Hvilket semester er du på? Hva er ditt kjønn? Er du...? Er du...?  Annet PhD Candidate Samsvaret mellom
DetaljerMotzkin monoids. Micky East. York Semigroup University of York, 5 Aug, 2016
Micky East York Semigroup University of York, 5 Aug, 206 Joint work with Igor Dolinka and Bob Gray 2 Joint work with Igor Dolinka and Bob Gray 3 Joint work with Igor Dolinka and Bob Gray 4 Any questions?
Detaljer