diffusion versus absorption in semilinear elliptic equations we study the limit behaviour of a sequence of singular solutions of a nonlinear elliptic equation with a strongly degenerate absorption term at the boundary of the domain we give sharp conditions on the rate of degeneracy in terms of the distance to the boundary in order the pointwise singularity not to propagate along the boundary

reduction of courant algebroids and generalized complex structures we present a theory of reduction for courant algebroids as well as dirac structures generalized complex and generalized kahler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction the enhanced symmetry group of a courant algebroid leads us to define extended actions and a generalized notion of moment map key examples of generalized kahler reduced spaces include new explicit bi hermitian metrics on cp2

generalized complex geometry generalized complex geometry encompasses complex and symplectic ge ometry as its extremal special cases we explore the basic properties of this geometry including its enhanced symmetry group elliptic deforma tion theory relation to poisson geometry and local structure theory we also dene and study generalized complex branes which interpolate be tween at bundles on lagrangian submanifolds and holomorphic bundles on complex submanifolds

stabilization of the variable coefficient structural acoustic model with curved middle surface and delay effects in the structural component abstract this paper is concerned with the uniform energy decay rates of a structural acoustic model which describes the interactions between the acoustic medium and the elastic structural in an acoustic chamber where one wall is flexible and curved the coupled system considered consists of a variable coefficient wave equation and a variable coefficient plate equation defined on the riemannian manifold with the coupling on the interface between the acoustic medium and the flexible wall both the components of the dynamics are subject to nonlinear boundary damping furthermore a nonlinear delay term acting in the boundary feedbacks of the structure component is considered the riemannian geometry method is applied to derive the energy estimates of the nonlinear coupled system with variable coefficients the uniform energy decay rates of the nonlinear variable coefficient structural acoustic system with delay effects on the riemannian manifold are quantified by a solution to a constructed nonlinear ode

existence and energy decay rates of solutions to the variable coefficient euler bernoulli plate with a delay in localized nonlinear internal feedback abstract this paper is concerned with the existence and energy decay estimates of solutions to the variable coefficient euler bernoulli plate with a delay in localized nonlinear internal feedback the existence of solutions to the nonlinear system is obtained by using galerkin method combined with some energy estimates furthermore applying riemannian geometry method we establish uniform decay rates of the plate equation with variable coefficients and a nonlinear delay term

on the observability inequalities for exact controllability of wave equations with variable coefficients this paper deals with boundary exact controllability for the dynamics governed by the wave equation with nonconstant coefficients in the principal part subject to dirichlet or neumann boundary controls the observability inequalities are established by the riemannian geometry method under some geometric condition for the dirichlet problem and for the neumann problem respectively next a number of nontrivial examples are presented to verify the observability inequality in particular a counterexample is given without boundary exact controllability where the control is exerted on the whole boundary

stabilization of the transmission wave plate equation with variable coefficients abstract we consider the stabilization of the transmission wave plate equation with variable coefficients and a general transmission boundary by virtue of the riemannian geometric approach we obtain some stability results for nonlinear feedbacks acting on the regions under suitable geometric conditions

stabilization of a transmission wave plate equation abstract we consider a stabilization problem for a model arising in the control of noise coupling the damped wave equation with a damped kirchhoff plate equation we prove an exponential stability result under some geometric condition our method is based on an identity with multipliers that allows to show an appropriate energy estimate

nonlinear boundary stabilization for a transmission problem in elasticity in this article we study the uniform stability for the solutions of a transmission problem in non homogeneous anisotropic elasticity we consider a boundary dissipation given for a frictional damping and explicit rates of decay are obtained

inverse observability estimates for second order hyperbolic equations with variable coefficients abstract we consider a general second order hyperbolic equation defined on an open bounded domain r n with variable coefficients in both the elliptic principal part and in the first order terms as well at first no boundary conditions b c are imposed our main result theorem 3 5 is a reconstruction or inverse estimate for solutions w under checkable conditions on the coefficients of the principal part the h1 l2 energy at time t t or at time t 0 is dominated by the l2 norms of the boundary traces w a and wt modulo an interior lower order term once homogeneous b c are imposed our results yield under a uniqueness theorem needed to absorb the lower order term continuous observability estimates for both the dirichlet and neumann case with an explicit sharp observability time hence by duality exact controllability results moreover no artificial geometrical conditions are imposed on the controlled part of the boundary in the neumann case in contrast with existing literature the first step of our method employs a riemann geometry approach to reduce the original variable coefficient principal part problem in r n to a problem on an appropriate riemann manifold determined by the coefficients of the principal part where the principal part is the laplacian in our second step we employ explicit carleman estimates at the differential level to take care of the variable first order energy level terms in our third step we employ micro local analysis yielding a sharp trace estimate to remove artificial geometrical conditions on the controlled part of the boundary in the neumann case

stabilization of euler bernoulli plate equation with variable coefficients by nonlinear boundary feedback abstract the aim of this paper is to investigate the uniform stabilization of euler bernoulli plate equation with variable coefficients in the principle part subject to nonlinear boundary feedback laws the exponential or rational energy decay rate is obtained by the multiplier method and the riemannian geometry method

general stability of a viscoelastic variable coefficients plate equation with time varying delay in localized nonlinear internal feedback abstract this paper is concerned with the decay properties of a memory type euler bernoulli plate equation with variable coefficients and time varying delay in localized nonlinear internal feedback by using piecewise multiplier method and riemannian geometric analysis we establish the explicit and general decay rates of system whose nonlinear internal feedbacks act on a suitable subregion

exact controllability stabilization and perturbations for distributed systems exact controllability is studied for distributed systems of hyperbolic type or for petrowsky systems like plate equations the control is a boundary control or a local distributed control exact controllability consists in trying to drive the system to rest in a given finite time the solution of the problems depends on the function spaces where the initial data are taken and also depends on the function space where the control can be chosen a systematic method named hum for hilbert uniqueness method is introduced as the terminology indicates it is based on uniqueness results classical or new and on hilbert spaces constructed in infinitely many ways by using uniqueness a number of applications are indicated having a general method for exact controllability implies having a general method for stabilization this leads to new and complicated nonlinear riccati s type pdes to be compared with direct methods when available in the last part of the paper we consider how all this behaves fo

decay of solutions of wave equations in a bounded region with boundary dissipation abstract an energy decay rate is obtained for solutions of wave type equations in a bounded region in rn whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface

stabilization and controllability for the transmission wave equation we address the problem of control of the transmission wave equation in particular we consider the case where due to total internal reflection of waves at the interface the system may not be controlled from exterior boundaries we show that such a system can be controlled by introducing both boundary control along the exterior boundary and distributed control near the transmission boundary and give a physical explanation why the additional control near the transmission boundary might be needed for some domains

local classification of generalize complex structures we give a local classification of generalized complex structures about a point a generalized complex structure is equivalent to a product of a symplectic manifold with a holomorphic poisson manifold we use a nash moser type argument in the style of conn s linearization theorem

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

observability inequalities for shallow shells we consider some observability inequalities from boundary for a general shallow shell with a middle surface of any shape at first an estimate is established by the geometric multiplier method in the case that no boundary conditions are imposed under some checkable geometric conditions then our results yield continuous observability estimates for two kinds of boundary conditions which have a physical meaning with an explicit observability time and hence by duality exact controllability results

scalar curvature and conformal deformation of hyperbolic space abstract let h m gh m 3 be the m dimensional hyperbolic space with its riemannian metric gh of sectional curvature 1 and let k be a smooth function on h m in the first part of this article we establish sufficient conditions for k to be respectively not to be the scalar curvature of some complete metric u4 m 2 gh pointwise conformal to gh in the second part we prove results for the two dimensional case singularity and uniquens questions

uniform decay of the solution to a wave equation with memory conditions on the boundary abstract a wave equation with variable coefficients in principal part and memory conditions on the boundary is considered the riemannian geometry method is applied to prove the exponential decay of the energy provided the relaxation function also decays exponentially

uniform stability of transmission of wave plate equations with source on riemannian manifold abstract this paper is concerned with the semilinear transmission of wave plate system with source term on riemannian manifold we prove the existence of weak solutions by using faedo galerkin s method furthermore by introducing nonlinear boundary feedbacks acting only on plate we establish the explicit and general decay rates of the system our proofs are based on the geometric multiplier method and the riemannian geometry method

controllability for transmission wave plate equations on riemannian manifolds abstract in this paper we consider controllability for transmission system of coupling wave equations with euler bernoulli equations on riemannian manifolds we show that such a system is exactly controllable by boundary controls only along the exterior boundary which means there is no control on the interface of the transmission system our proofs rely on the hilbert uniqueness method hum and geometric multiplier method

regularity and asymptotic behavior for a damped plate membrane transmission problem abstract we consider a transmission problem where a structurally damped plate equation is coupled with a damped or undamped wave equation by transmission conditions we show that exponential stability holds in the damped damped situation and polynomial stability but no exponential stability holds in the damped undamped case additionally we show that the solutions first defined by the weak formulation in fact have higher sobolev space regularity

blow ups in generalized complex geometry generalized complex geometry is a theory that unifies complex geometry and symplectic geometry into one single framework it was introduced by hitchin and gualtieri around 2002 in this thesis we address the following question given a generalized complex manifold together with a submanifold does the blow up of that submanifold admit again a generalized complex structure we give the following answers to this question if the directions that are normal to the submanifold are purely complex then the blow up admits a generalized complex structure for which the blow down map is generalized holomorphic if and only if the induced lie algebra on the conormal bundle of the submanifold is degenerate this is a rather restrictive condition but it is always satisfied if the submanifold is of complex codimension two on the other extreme if all the normal directions are symplectic then we show that the blow up always admits a generalized complex structure provided the submanifold is compact we then also give an example of a submanifold that is not of the above two types and show that it can not be blown up finally we also address the analogous question in generalized kahler geometry given a generalized kahler manifold together with a submanifold we show that under certain geometric constraints on the submanifold the blow up is again generalized kahler

stabilization of transmission coupled wave and euler bernoulli equations on riemannian manifolds by nonlinear feedbacks abstract we consider the transmission system of coupling wave equations with euler bernoulli equations on riemannian manifolds by introducing nonlinear boundary feedback controls we establish the exponential and rational energy decay rate for the problem our proofs rely on the geometric multiplier method

generalized kahler geometry generalized kahler geometry is the natural analogue of kahler geometry in the context of generalized complex geometry just as we may require a complex structure to be compatible with a riemannian metric in a way which gives rise to a symplectic form we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure we prove that generalized kahler geometry is equivalent to the bi hermitian geometry on the target of a 2 dimensional sigma model with 2 2 supersymmetry we also prove the existence of natural holomorphic courant algebroids for each of the underlying complex structures and that these split into a sum of transverse holomorphic dirac structures finally we explore the analogy between pre quantum line bundles and gerbes in the context of generalized kahler geometry

boundary feedback stabilization of the transmission problem of naghdi s model we consider the stabilization of the transmission problem of naghdi s model by boundary feedbacks where the model has a middle surface of any shape the exponential decay rate for the problem is established under some checkable geometric conditions on the middle surface