dynamics of a free boundary problem with curvature modeling electrostatic mems the dynamics of a free boundary problem for electrostatically actuated microelectromechanical systems mems is investigated the model couples the electric potential to the deformation of the membrane the deflection of the membrane being caused by application of a voltage difference across the device more precisely the electrostatic potential is a harmonic function in the angular domain that is partly bounded by the deformable membrane the gradient trace of the electric potential on this free boundary part acts as a source term in the governing equation for the membrane deformation the main feature of the model considered herein is that unlike most previous research the small deformation assumption is dropped and curvature for the deformation of the membrane is taken into account what leads to a quasilinear parabolic equation the free boundary problem is shown to be well posed locally in time for arbitrary voltage values and globally in time for small voltages values furthermore existence of asymptotically stable steady state configurations is proved in case of small voltage values as well as non existence of steady states if the applied voltage difference is large finally convergence of solutions of the free boundary problem to the solutions of the well established small aspect ratio model is shown

on the partial differential equations of electrostatic mems devices iii refined touchdown behavior abstract this paper is a continuation of n ghoussoub y guo on the partial differential equations of electrostatic mems devices stationary case siam j math anal 38 2007 1423 1449 and n ghoussoub y guo on the partial differential equations of electrostatic mems devices ii dynamic case nodea nonlinear differential equations appl 2008 in press where we analyzed nonlinear parabolic problem u t u f x 1 u 2 on a bounded domain of r n with dirichlet boundary conditions this equation models a simple electrostatic micro electromechanical system mems device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at 1 here u is modeled to describe dynamic deflection of the elastic membrane when a voltage represented here by is applied the membrane deflects towards the ground plate and a snap through touchdown must occur when it exceeds a certain critical value pull in voltage creating a so called pull in instability which greatly affects the design of many devices in an effort to achieve better mems design the material properties of the membrane can be technologically fabricated with a spatially varying dielectric permittivity profile f x in this work some a priori estimates of touchdown behavior are established based on which the refined touchdown profiles are obtained by adapting self similar method and center manifold analysis applying various analytical and numerical techniques some properties of touchdown set such as compactness location and shape are also discussed for different classes of varying permittivity profiles

criteria for copositive matrices abstract some finite criteria for copositive copositive plus and strictly copositive matrices are proposed and compared with existing determinantal tests the basic mathematical tool is principal pivoting

steady state responses of poroelastic half space soil medium to a moving rectangular load the dynamic steady state responses of a poroelastic half space soil medium subjected to a moving rectangular load are investigated analytically numerically the full dynamic poroelastic theory of biot is employed under the assumption of an incompressible solid grain and neglecting the apparent mass density using the fourier transform the governing equations of motion are then reduced to a system of four coupled ordinary differential equations which are solved semi analytically soil vertical displacements accelerations and pore water pressures induced by moving load are calculated computed result shows that load velocity and intrinsic permeability of the soil medium shows an apparent effect on its dynamic responses and pore water pressures

on the critical dimension of a fourth order elliptic problem with negative exponent we study the regularity of the extremal solution of the semilinear biharmonic equation 2u u 1 u 2 on a ball b rn under navier boundary conditions u u 0 on b where 0 is a parameter while 0 0 are fixed constants it is known that there exists such that for there is no solution while for 0 and it is singular supbu 1 for n 9 0 and 0 with small our proof for the singularity of extremal solutions in dimensions n 9 is based on certain improved hardy rellich inequalities

asymptotic first eigenvalue estimates for the biharmonic operator on a rectangle abstract we find an asymptotic expression for the first eigenvalue of the biharmonic operator on a long thin rectangle this is done by finding lower and upper bounds which become increasingly accurate with increasing length the lower bound is found by algebraic manipulation of the operator and the upper bound is found by minimising the quadratic form for the operator over a test space consisting of separable functions these bounds can be used to show that the negative part of the groundstate is small

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

a procedure for the positive definiteness of forms of even order a positivity method for forms of even order is reported it is shown that testing for positivity of such forms can be reduced to a test for positive definiteness of parametric matrices special emphasis is placed on quartic and sixth order forms examples are given to illustrate the proposed criterion

the sub harmonic bifurcation of stokes waves steady periodic water waves on the free surface of an infinitely deep irrotational flow under gravity without surface tension stokes waves can be described in terms of solutions of a quasi linear equation which involves the hilbert transform and which is the euler lagrange equation of a simple functional the unknowns are a 2 periodic function w which gives the wave profile and the froude number a dimensionless parameter reflecting the wavelength when the wave speed is fixed and vice versa

variational approach to mems model with fringing field abstract we study semilinear elliptic equations which are called mems model with fringing field using the variational method and dual approach we show the multiple existence of positive solutions

global bifurcation diagrams and exact multiplicity of positive solutions for a one dimensional prescribed mean curvature problem arising in mems abstract we study global bifurcation diagrams and exact multiplicity of positive solutions for the one dimensional prescribed mean curvature problem arising in mems u x 1 u x 2 1 u p u 1 l x l u l u l 0 where 0 is a bifurcation parameter and p l 0 are two evolution parameters we determine the exact number of positive solutions by the values of p l and moreover for p 1 the bifurcation diagram undergoes fold and splitting bifurcations while for 0 p 1 the bifurcation diagram undergoes fold splitting and segment shrinking bifurcations our results extend and improve those of brubaker and pelesko n d brubaker j a pelesko analysis of a one dimensional prescribed mean curvature equation with singular nonlinearity nonlinear anal 75 2012 5086 5102 and pan and xing h pan r xing exact multiplicity results for a one dimensional prescribed mean curvature problem related to a mems model nonlinear anal rwa 13 2012 2432 2445 by generalizing the nonlinearity 1 u 2 to 1 u p with general p 1 we also answer an open question raised by brubaker and pelesko on the extension of global bifurcation diagram results to general p 0 concerning this open question we find and prove that global bifurcation diagrams for 0 p 1 are different to and more complicated than those for p 1

a comparison of the sherali adams lov 341sz schrijver and lasserre relaxations for 0 1 programming sherali and adams sa90 lov asz and schrijver ls91 and recently lasserre las01b have proposed lift and project methods for constructing hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p subseteq or n converging to p in n steps lasserre s approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments we present the three methods in a common elementary framework and show that the lasserre construction provides the tightest relaxations of p as an application this gives a direct simple proof for the convergence of the lasserre s hierarchy we describe applications to the stable set polytope and to the cut polytope

estimates for the quenching time of a mems equation with fringing field abstract in this paper we consider a zero initial boundary value parabolic problem in mems with fringing field u t u g u 1 u 2 in a bounded domain of r n here 0 is a parameter related to the applied voltage 0 g is a positive nondecreasing convex function diverging as u 1 firstly we show that the solvability of the stationary problem w g w 1 w 2 with dirichlet boundary condition is characterized by a parameter meanwhile it is shown that for any solution to the parabolic equation will quench u 1 at a finite time secondly we focus on estimating the quenching time t in terms of i e the quenching time t o 1 2 as

recent improvements in the complexity of the effective nullstellensatz we bring up to date the estimates on the complexity of the effective nullstellensatz and the membership problem

transient responses of porous media under moving surface impulses three dimensional transient responses of porous media under moving surface impulses of finite frequency components are theoretically studied we discuss three free surface stiffness conditions such as fully permeable open pore fully impermeable closed pore and partially permeable boundaries that are not explicitly discussed before the transient responses of the solid vertical displacement and the pore fluid pressure triggered by the moving impulses on the surface are particularly investigated in different typical surface stiffness moving impulse velocities material permeabilities and impulse peak frequencies it is concluded that the r1 surface wave carries the strongest energy as that for stationary source configurations moreover it is more sensitive to surface stiffness condition than body waves represented in the responses of the corresponding wave forms of obvious different amplitudes and arrival time furthermore the apparent velocity of the moving impulse pointing toward the fixed receiver may cause blue shift in frequency the higher velocity triggers more obvious frequency shift for the moving impulse of low peak frequency this shift becomes much serious the lateral velocity of the moving impulse to the receiver may also twist the received wave forms especially for the impulse of low peak frequency

a numerical model for the isolation of moving load induced vibrations by pile rows embedded in layered porous media abstract a numerical model is developed to analyse the isolation of moving load induced vibrations using pile rows embedded in a layered poroelastic half space based on biot s theory and the transmission and reflection matrices trm method the free wave field solution for a moving load applied on the surface of a layered poroelastic half space and the fundamental solution for an harmonic circular patch load are determined using muki and sternberg s method the second kind of frequency domain fredholm integral equations for the dynamic interaction between pile rows and the layered poroelastic half space are derived the time domain solution is recovered via inverse fourier transform in order to obtain the amplitude reduction ratio and thus assess the vibration isolation efficiency of pile rows a special case of the present model shows good agreement with an existing solution numerical results of this study show that the speed of moving loads has an important influence on the isolation of vibrations by pile rows the same pile rows can achieve better isolation efficiencies for higher speed loads than for lower speed loads pile rows embedded in a two layered poroelastic half space with a softer overlying layer usually generate better vibration isolation effects than those with a stiffer overlying layer finally better isolation vibration may be realized by increasing the pile length and decreasing the net spacing between neighboring piles in a pile row

the existence and dynamic properties of a parabolic nonlocal mems equation abstract let r n be a c 2 bounded domain and 0 be a constant we will prove the existence of constants n n 1 d x 1 w 2 for the nonlocal mems equation v 1 v 2 1 1 1 v d x 2 in v 0 on such that a solution exists for any 0 n and no solution exists for any n where is the pull in voltage and w is the limit of the minimal solution of v 1 v 2 in with v 0 on as moreover n if is a strictly convex smooth bounded domain we will prove the local existence and uniqueness of the solution of the parabolic nonlocal mems equation u t u 1 u 2 1 1 1 u d x 2 in 0 u 0 on 0 u x 0 u 0 in we prove the existence of a unique global solution and the asymptotic behaviour of the global solution of the parabolic nonlocal mems equation under various boundedness conditions on we also obtain the quenching behaviour of the solution of the parabolic nonlocal mems equation when is large

further study of a fourth order elliptic equation with negative exponent we continue to study the nonlinear fourth order problem t u d 2 u l u 2 l u u 0 u 0 on where n is a bounded smooth domain and 0 is a parameter when n 2 and is a convex domain we know that there is c 0 such that for 0 c the problem possesses at least two regular solutions we will see that the convexity assumption on can be removed i e the main results are still true for a general bounded smooth domain the main technique in the proofs of this paper is the blow up argument and the main difficulty is the analysis of touch down behaviour

multiplicity of solutions for a fourth order equation with power type nonlinearity let b be the unit ball in mathbb r n n 3 and n be the exterior unit normal vector on the boundary we consider radial solutions to delta 2 u lambda 1 rm sign p u p quad rm in b quad u 0 quad frac partial u partial n 0 quad rm on partial b where 0 for positive p we assume 5 n 12 and p frac n 4 n 4 or n 13 and frac n 4 n 4 0 such that if s there exists a radial weakly singular solution for s there exist infinitely many regular radial solutions and the number of radial regular solutions goes to infinity as s

decay estimates for some semilinear damped hyperbolic problems let be a bounded open domain in rn g r r a non decreasing continuous function such that g 0 0 and h e lloc1 r l2 under suitable assumptions on g and h the rate of decay of the difference of two solutions is studied for some abstract evolution equations of the general form u lu g u h t x as t the results obtained by use of differential inequalities can be applied to the case of the semilinear wave equation u u delta u g text u t text h text in r times omega text u text 0 on r times partial omega in r u 0 onr for instance if g s c left s right p 1 s d left s right q 1 s with c d 0 and 1 p q n 2 q n 2 then if h in l infty r l 2 omega all solutions are bounded in the energy space for t 0 and if u v are two such solutions the energy norm of u t v t decays like t 1 p 1 as t

the regularity and local bifurcation of steady periodic water waves steady periodic water waves on infinite depth satisfying exactly the kinematic and dynamic boundary conditions on the free surface with or without surface tension are given by solutions of a rather tidy nonlinear pseudo differential operator equation for a 2 periodic function of a real variable being an euler lagrange equation this formulation has the advantage of gradient structure but is complicated by the fact that it involves a non local operator namely the hilbert transform and is quasi linear

a parabolic free boundary problem modeling electrostatic mems the evolution problem for a membrane based model of an electrostatically actuated microelectromechanical system is studied the model describes the dynamics of the membrane displacement and the electric potential the latter is a harmonic function in an angular domain the deformable membrane being a part of the boundary the former solves a heat equation with a right hand side that depends on the square of the trace of the gradient of the electric potential on the membrane the resulting free boundary problem is shown to be well posed locally in time furthermore solutions corresponding to small voltage values exist globally in time while global existence is shown not to hold for high voltage values it is also proven that for small voltage values there is an asymptotically stable steady state solution finally the small aspect ratio limit is rigorously justified

on the quenching of a nonlocal parabolic problem arising in electrostatic mems control abstract we consider a nonlocal parabolic model for a micro electro mechanical system specifically for a radially symmetric problem with monotonic initial data it is shown that the solution quenches so that touchdown occurs in the device in a situation where there is no steady state it is also shown that quenching occurs at a single point and a bound on the approach to touchdown is obtained numerical simulations illustrating the results are given

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

singular behavior of an electrostatic elastic membrane system with an external pressure abstract we analyze nonnegative solutions of the nonlinear elliptic problem u f x u 2 p where 0 and p 0 are constants on a bounded domain of r n n 1 with a dirichlet boundary condition this equation models an electrostatic elastic membrane system with an external pressure p 0 where 0 denotes the applied voltage first we completely address the existence and nonexistence of positive solutions the classification of all possible singularities at x 0 for nonnegative solutions u x satisfying u 0 0 is then analyzed for the special case where b 1 0 r 2 and f x x with 2 in particular we show that for some u x admits only the isotropic singularity at x 0 and otherwise u x may admit the anisotropic singularity at x 0 when u x admits the isotropic singularity at x 0 the refined singularity of u x at x 0 is further investigated depending on whether p 0 by applying fourier analysis

sharp effective nullstellensatz the usual proofs of this result however give no information about the g1 s for instance they give no bound on their degrees this question was first considered by g hermann h she used elimination theory to get a bound on the degree of the gi s which was doubly exponential in the number of variables her results were later improved in mw and in th all these produce bounds that are doubly exponential in the number of variables a major breakthrough was achieved by brownawell b1 who proved the following result

some singular equations modeling mems in the past fifteen years mathematical models for microelectromechanical systems mems have been the subject of several studies in particular due to the interesting qualitative properties they feature still most research is devoted to an illustrative but simplified model which is deduced from a more complex model when the aspect ratio of the device vanishes the so called vanishing or small aspect ratio model the analysis of the aforementioned complex model involving a moving boundary has started only recently and an outlook of the results obtained so far in this direction is provided in this survey

convergence of solutions of a nonlocal biharmonic mems equation with the fringing field abstract we study a nonlocal biharmonic mems equation with the fringing field it arises in the micro electro mechanical system mems devices first we establish the local solution and extend it globally in time next we discuss the dynamical properties of omega limit set especially we show that the omega limit set is included in the set of the stationary solution finally we consider the convergence rate to the stationary solution

lmi based estimation of scene points in vision systems with generalized cameras this paper considers the problem of estimating the position of a scene point from its image projections onto generalized cameras i e cameras that can be modeled by a spherical projection followed by a perspective one the sought estimate is defined through a geometric criterion specifically the minimization of the angles between the projections on the sphere of the available image points and the corresponding projections of the estimate a solution based on convex optimization with linear matrix inequalities lmis is proposed for addressing this problem which provides a candidate of the sought estimate moreover a condition is provided for establishing exactness of the found candidate i e establishing whether the found candidate is a minimizer of the considered geometric criterion

analysis of a one dimensional prescribed mean curvature equation with singular nonlinearity abstract in this paper the classical solution set u of the one dimensional prescribed mean curvature equation u 1 u 2 1 u 2 l x l u l u l 0 for 0 and l 0 is analyzed via a time map it is shown that the solution set depends on both parameters and l and undergoes two bifurcations the first is a standard saddle node bifurcation which happens for all l at l the second is a splitting bifurcation namely there exists a value l such that as l transitions from greater than or equal to l to less than l the upper branch of the bifurcation diagram of problem splits into two parts in contrast the solution set of the semilinear version of problem is independent of l and exhibits only a saddle node bifurcation therefore as this analysis suggests the splitting bifurcation is a byproduct of the mean curvature operator coupled with the singular forcing

global solution of the initial boundary value problem to a hyperbolic nonlocal mems equation we are concerned with the multi dimensional mems wave equation with nonlocal and singular nonlinearity the global solution of the initial boundary value problem is obtained in the case of the small initial data

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

a fourth order model for mems with clamped boundary conditions the dynamical and stationary behaviors of a fourth order equation in the unit ball with clamped boundary conditions and a singular reaction term are investigated the equation arises in the modeling of microelectromechanical systems mems and includes a positive voltage parameter lambda it is shown that there is a threshold value lambda 0 of the voltage parameter such that no radially symmetric stationary solution exists for lambda lambda while at least two such solutions exist for lambda in 0 lambda local and global well posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as the occurrence of finite time singularities when lambda lambda

ground vibration in the vicinity of a rectangular load on a half space abstract the transmission of vibrations in the near field of the surface of the ground due to an harmonic load acting on a rectangle is investigated theoretically the interior of the ground is modelled as an elastic half space the solution is obtained by using the double fourier transform with numerical integration of the inverse double fourier transform the attenuation of the amplitude is presented as a function of distance from the applied force for various frequencies other results show the direct receptance at the load for various masses placed there and also the effect on vibration of placing a mass between the load and point of measurement

a new model for electrostatic mems with two free boundaries abstract a moving boundary problem with two free boundaries modeling a two dimensional idealized mems device with pull in instability is discussed we use a fixed point argument to show that the model possesses stationary solutions for small source voltages we also give a rigorous evidence that solutions of the model converge towards solutions of the associated small aspect ratio equation in the vanishing aspect ratio limit

on positivity for the biharmonic operator under steklov boundary conditions the positivity preserving property for the inverse of the biharmonic operator under steklov boundary conditions is studied it is shown that this property is quite sensitive to the parameter involved in the boundary condition moreover positivity of the steklov boundary value problem is linked with positivity under boundary conditions of navier and dirichlet type

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

on a hyperbolic equation arising in electrostatic mems abstract we consider the initial boundary value problem of a damped wave equation with singular nonlinearity which describes an electrostatic micro electro mechanical system mems device the results of the pull in voltage being the critical threshold for global existence and quenching are obtained if the applied voltage then the equation admits a unique global small solution that exponentially converges to the minimal steady state while large solution may quench in finite time if then any solution quenches in finite time finally in the sense of the viscosity dominated limit the asymptotic relation of solutions between the hyperbolic equation and the parabolic one is investigated also the related error estimates in arbitrary order are derived

abstract wave equations with a singular nonlinear forcing term abstract let v h be real separable hilbert spaces the abstract wave equation u a t u g u where u t v a t maps v to its dual v and g is a nonlinear map from the ball s r0 u v u

camera displacement via constrained minimization of the algebraic error this paper proposes a new approach to estimate the camera displacement of stereo vision systems via minimization of the algebraic error over the essential matrices manifold the proposed approach is based on the use of homogeneous forms and linear matrix inequality lmi optimizations and has the advantages of not presenting local minima and not introducing approximations of nonlinear terms numerical investigations carried out with both synthetic and real data show that the proposed approach provides significantly better results than svd methods as well as minimizations of the algebraic error over the essential matrices manifold via both gradient descent and simplex search algorithms

lmi techniques for optimization over polynomials in control a survey numerous tasks in control systems involve optimization problems over polynomials and unfortunately these problems are in general nonconvex in order to cope with this difficulty linear matrix inequality lmi techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems this survey aims to provide the reader with a significant overview of the lmi techniques that are used in control systems for tackling optimization problems over polynomials describing approaches such as decomposition in sum of squares positivstellensatz theory of moments polya s theorem and matrix dilation moreover it aims to provide a collection of the essential problems in control systems where these lmi techniques are used such as stability and performance investigations in nonlinear systems uncertain systems time delay systems and genetic regulatory networks it is expected that this survey may be a concise useful reference for all readers

optimal control of a singular pde modeling transient mems with control or state constraints abstract a particular feature of certain microelectromechanical systems mems is the appearance of a so called pull in instability corresponding to a singularity in the underlying pde model we here consider a transient mems model and its optimal control via the dielectric properties of the membrane and or the applied voltage in contrast to the static case the control problem suffers from low dimensionality of the control compared to the state and hence requires different techniques for establishing first order optimality conditions for this purpose we here use a relaxation approach combined with a localization technique

multiple quenching solutions of a fourth order parabolic pde with a singular nonlinearity modeling a mems capacitor finite time singularity formation in a fourth order nonlinear parabolic partial differential equation pde is analyzed the pde is a variant of a ubiquitous model found in the field of microelectromechanical systems mems and is studied on a one dimensional 1d strip and the unit disc the solution itself remains continuous at the point of singularity while its higher derivatives diverge a phenomenon known as quenching for certain parameter regimes it is shown numerically that the singularity will form at multiple isolated points in the 1d strip case and along a ring of points in the radially symmetric two dimensional case the location of these touchdown points is accurately predicted by means of asymptotic expansions the solution itself is shown to converge to a stable self similar profile at the singularity point analytical calculations are verified by use of adaptive numerical methods which take advantage of symmetries exhibited by the underlying pde to accurately resolve solutions very close t

a half space saturated poro elastic medium subjected to a moving point load in this paper an analytical solution for the dynamic response of a half space porous medium subjected to a moving point load is derived in the model the displacements of the solid skeleton and the pore pressure are expressed in terms of two scalar potentials and one vectorial potential based on biot s theory the frequency domain holmholtz equations for the potentials are derived through the fourier transformation with respect to time the general solutions for the potentials are derived through the fourier transformation with respect to the horizontal coordinates numerical results suggest that moving loads have very complicated effects on the dynamic response of the porous medium generally speaking a moving load with a high speed will generate a larger response in the porous medium than a static or a lower speed load

ground vibration in the vicinity of a strip load a two dimensional half space model abstract the transmission of vibrations over a distance of 75 m of the surface of the ground is investigated theoretically the mathematical model is two dimensional and the surface is assumed to be subjected to a harmonic load acting on a strip of finite width the interior of the ground is modelled as a half space the solution is obtained by using a dynamic stiffness matrix analysis the attenuation of the amplitude is presented as a function of distance from the applied load for various frequencies

exact stiffness method for quasi statics of a multi layered poroelastic medium abstract a method is presented to study the three dimensional quasi static response of a multi layered poroelastic half space with compressible constituents the system under consideration consists of n layers of different thickness and material properties overlying a homogeneous half space fourier expansion laplace transforms and hankel transforms with respect to the circumferential time and radial coordinates respectively are used in the formulation laplace hankel transforms of displacements and pore pressure at layer interfaces are considered as the basic unknowns exact stiffness matrices describing the relationship between generalized displacement and force vectors of a finite layer and a half space are derived explicitly in the transform space the global stiffness matrix of a layered system is assembled by considering the continuity of tractions and fluid flow at layer interfaces the time histories of displacements stresses and pore pressure are obtained by solving the stiffness equation system for discrete values of laplace and hankel transform parameters and using numerical quadrature schemes for laplace and hankel transform inversions selected numerical results for different layered systems are presented to portray the influence of layering and poroelastic material properties the advantage of the present method is that for an n layered system it yields a numerically stable symmetric stiffness matrix of order 4 n 4 n when compared to the unsymmetric and numerically unstable coefficient matrix of order 8 n 8 n associated with the conventional method based on the determination of layer arbitrary coefficients

global solutions of singular parabolic equations arising from electrostatic mems abstract we study dynamic solutions of the singular parabolic problem p u t u x 1 u 2 x t b 0 u x 0 u 0 x 0 x b u x t 0 x b where 0 and 0 are two parameters and b is the unit ball x r n x 1 with n 2 our interest is focussed on p with 2 3 n 4 9 for which p admits a singular stationary solution in the form s x 1 x 2 3 this equation models dynamic deflection of a simple electrostatic micro electro mechanical system mems device under the assumption u 0 s x we address the existence uniqueness regularity stability or instability and asymptotic behavior of weak solutions for p given 4 6 n 3 6 n 2 4 in particular we show that if either n 8 and or 2 n 7 then the minimal compact stationary solution u of p is stable and while s x is unstable however for n 8 and 0 p has no compact minimal solution and s x is an attractor from below not from above furthermore the refined asymptotic behavior of global solutions for p is also discussed for the case where n 8 and 0 which is given by a certain matching of different asymptotic developments in the large outer region closer to the boundary and the thin inner region near the singularity

the critical dimension for a fourth order elliptic problem with singular nonlinearity we study the regularity of the extremal solution of the semilinear biharmonic equation delta 2 u frac lambda 1 u 2 which models a simple micro electromechanical system mems device on a ball b subset mathbb r n under dirichlet boundary conditions u partial nu u 0 on partial b we complete here the results of lin and yang 14 regarding the identification of a pull in voltage 0 such that a stable classical solution u with 0 our main result asserts that the extremal solution u lambda is regular rm sup b u lambda 1 provided n leqq 8 while u lambda is singular rm sup b u lambda 1 for n geqq 9 in which case 1 c 0 x 4 3 leqq u lambda x leqq 1 x 4 3 on the unit ball where c 0 left frac lambda overline lambda right frac 1 3 and bar lambda frac 8 9 left n frac 2 3 right left n frac 8 3 right

ground vibration due to a high speed moving harmonic rectangular load on a poroviscoelastic half space abstract the transmission of vibrations in the ground due to a high speed moving vertical harmonic rectangular load is investigated theoretically the problem is three dimensional and the interior of the ground is modelled as a totally or partially saturated porous viscoelastic half space using the complete biot theory the solutions in the transformed domain are obtained using a double fourier transform on the surface spatial variables a modified hysteretic damping model defined in the wavenumber domain is used first presented by lefeuve mesgouez et al lefeuve mesgouez g le houedec d peplow a t 2000 vibration in the vicinity of a high speed moving harmonic strip load journal of sound and vibration 231 5 1289 1309 numerical results for the displacements of the solid and fluid phases over the surface of the ground and in depth are presented for loads moving with speeds up to and beyond the rayleigh wave speed of the medium

a free boundary problem modeling electrostatic mems i linear bending effects the dynamical and stationary behaviors of a fourth order evolution equation with clamped boundary conditions and a singular nonlocal reaction term which is coupled to an elliptic free boundary problem in a non smooth domain are investigated the equation arises in the modeling of microelectromechanical systems and includes two positive parameters lambda and varepsilon related to the applied voltage and the aspect ratio of the device respectively local and global well posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as a criterion for global existence excluding the occurrence of finite time singularities which are not physically relevant existence of a stable steady state is shown for sufficiently small lambda non existence of steady states is also established when varepsilon is small enough and lambda is large enough depending on varepsilon

global existence for nonlocal mems we prove the existence of solutions for a nonlocal equation arising from the mathematical modeling of microelectromechanicalsystems mems the existence result obtained within a suitable implicit function theorem framework is established under rather general boundary conditions and for bounded domains whose diameter is fairly small

positivity and almost positivity of biharmonic green s functions under dirichlet boundary conditions in general for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or equivalently a positivity preserving property is available the problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being reasonably written as a system of second order boundary value problems it is shown that on the other hand for bounded smooth domains omega subset mathbb r n the negative part of the corresponding green s function is small when compared with its singular positive part provided n geqq 3 moreover the biharmonic green s function in balls b subset mathbb r n under dirichlet that is clamped boundary conditions is known explicitly and is positive it has been known for some time that positivity is preserved under small regular perturbations of the domain if n 2 in the present paper such a stability result is proved for n geqq 3

compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity we study the branch of semistable and unstable solutions i e those whose morse index is at most 1 of the dirichlet boundary value problem u f x 1 u 2 on a bounded domain n which models among other things a simple electrostatic microelectromechanical system mems device we extend the results of 11 relating to the minimal branch by obtaining compactness along unstable branches for 1 n 7 on any domain and for a large class of permittivity profiles f we also show the remarkable fact that powerlike profiles f x x can push back the critical dimension n 7 of this problem by establishing compactness for the semistable branch on the unit ball also for n 8 and as long as as a byproduct we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for in a natural range in all these results the conditions on the space dimension and on the power of the profile are essentially sharp 2007 wiley periodicals inc