improved hardy and rellich inequalities on riemannian manifolds in this paper we establish improved hardy and rellich type inequalities on a riemannian manifold m furthermore we also obtain sharp constants for improved hardy and rellich type inequalities on the hyperbolic space n

viscous cahn hilliard equation ii analysis the viscous cahn hilliard equation may be viewed as a singular limit of the phase field equations for phase transitions it contains both the allen cahn and cahn hilliard models of phase separation as particular cases by specific choices of parameters it may be formulated as a one parameter say homotopy connecting the cahn hilliard 0 and allen cahn 1 models the limit 0 is singular in the sense that the smoothing property of the analytic semigroup changes from being of the type associated with second order operators to the type associated with fourth order operators the properties of the gradient dynamical system generated by the viscous cahn hilliard equation are studied as varies in 0 1 continuity of the phase portraits near equilibria is established independently of 0 1 and using this a piecewise uniform in time perturbation result is proved for trajectories finally the continuity of the attractor is established and in one dimension the existence and continuity of inertial manifolds shown and the flow on the attractor detailed

best constants for higher order rellich inequalities in l p ω we obtain a series improvement to higher order lp rellich inequalities on a riemannian manifold m the improvement is shown to be sharp as each new term of the series is added

level set approach to mean curvature flow in arbitrary codimension we develop a level set theory for the mean curvature evolution of surfaces with arbitrary co dimension thus generalizing the previous work 8 15 on hypersurfaces the main idea is to surround the evolving surface of co dimension k in r by a family of hypersurfaces the level sets of a function evolving with normal velocity equal to the sum of the d k smallest principal curvatures the existence and the uniqueness of a weak level set solution is easily established using mainly the results of 8 and the theory of viscosity solutions for second order nonlinear parabolic equations the level set solutions coincide with the classical solutions whenever the latter exist the proof of this connection uses a careful analysis of the squared distance from the surfaces it is also shown that varifold solutions constructed by brakke 7 are included in the level set solutions the idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of de giorgi an introduction to the theory of barriers and his connection to the level set solutions is also provided

on some higher order hardy rellich type inequalities with boundary terms abstract we determine boundary remainder terms for some higher order hardy rellich inequalities involving the polyharmonic operator m the results are proved by studying suitable auxiliary boundary eigenvalue problems the optimal constants found may not be the classical hardy rellich ones

on an application of tikhonov s fixed point theorem to a nonlocal cahn hilliard type system modeling phase separation abstract this paper investigates a nonlocal version of a model for phase separation on an atomic lattice that was introduced by p podio guidugli 2006 36 the model consists of an initial boundary value problem for a nonlinearly coupled system of two partial differential equations governing the evolution of an order parameter and the chemical potential singular contributions to the local free energy in the form of logarithmic or double obstacle potentials are admitted in contrast to the local model which was studied by p podio guidugli and the present authors in a series of recent publications in the nonlocal case the equation governing the evolution of the order parameter contains in place of the laplacian a nonlocal expression that originates from nonlocal contributions to the free energy and accounts for possible long range interactions between the atoms it is shown that just as in the local case the model equations are well posed where the technique of proving existence is entirely different it is based on an application of tikhonov s fixed point theorem in a rather unusual separable and reflexive banach space

grusin operators riesz transforms and nilpotent lie groups we establish that the riesz transforms of all orders corresponding to the grusin operator h n nabla x 2 x 2n nabla y 2 and the first order operators nabla x x nu nabla y where x in mathbf r n y in mathbf r m n in mathbf n and nu in 1 ldots n n are bounded on l p mathbf r n m for all p in langle 1 infty rangle and are also weak type 1 1 moreover the transforms of order less than or equal to n 1 corresponding to h n and the operators nabla x x n nabla y are bounded on l p mathbf r n m for all p in langle 1 infty rangle but if n is odd all transforms of order n 2 are bounded if and only if p in langle 1 n rangle the proofs are based on the observation that the nabla x x nu nabla y generate a finite dimensional nilpotent lie algebra the corresponding connected simply connected nilpotent lie group is isometrically represented on the spaces l p mathbf r n m and h n is the corresponding sublaplacian

critical behavior for the polyharmonic operator with hardy potential abstract let us consider the dirichlet problem l u m u u x 2 m u 2 1 u u 0 in b d u b 0 for m 1 where b is the unit ball in r n n 2 m 2 2 n n 2 m we find that whatever n may be this problem is critical in the sense of pucci serrin and grunau depending on the value of 0 being the best constant in rellich inequality the present work extends to the perturbed operator m x 2 m i a well known result by grunau regarding the polyharmonic operator see grunau 1996

sharp poincare hardy and poincare rellich inequalities on the hyperbolic space abstract we study hardy type inequalities associated to the quadratic form of the shifted laplacian h n n 1 2 4 on the hyperbolic space h n n 1 2 4 being as it is well known the bottom of the l 2 spectrum of h n we find the optimal constant in a resulting poincare hardy inequality which includes a further remainder term which makes it sharp also locally the resulting operator is in fact critical in the sense of 17 a related improved hardy inequality on more general manifolds under suitable curvature assumption and allowing for the curvature to be possibly unbounded below is also shown it involves an explicit curvature dependent and typically unbounded potential and is again optimal in a suitable sense furthermore with a different approach we prove rellich type inequalities associated with the shifted laplacian which are again sharp in suitable senses

optimal boundary control of a nonstandard viscous cahn hilliard system with dynamic boundary condition abstract in this paper we study an optimal boundary control problem for a model for phase separation that was introduced by podio guidugli 2006 the model consists of a strongly coupled system of nonlinear parabolic differential equations in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically in contrast to the existing control literature about this pde system we consider here a dynamic boundary condition involving the laplace beltrami operator for the order parameter of the system which models an additional nonconserving phase transition occurring on the surface of the domain we show the frechet differentiability of the associated control to state operator in appropriate banach spaces and derive results on the existence of optimal controls and on first order necessary optimality conditions in terms of a variational inequality and the adjoint state system

phase segregation dynamics in particle systems with long range interactions ii interface motion we study properties of the solutions of a family of second order integrodifferential equations which describe the large scale dynamics of a class of microscopic phase segregation models with particle conserving dynamics we first establish existence and uniqueness as well as some properties of the instantonic solutions then we concentrate on formal asymptotic sharp interface limits we argue that the obtained interface evolution laws a stefan like problem and the mullins sekerka solidification model coincide with the ones which can be obtained in the analogous limits from the cahn hilliard equation the fourth order pde which is the standard macroscopic model for phase segregation with one conservation law

on a nonlocal phase separation model abstract an alternative to the cahn hilliard model of phase separation for two phase systems in a simplified isothermal case is given the model is derived from a free energy with a nonlocal interacting term and allows reasonable bounds for the concentrations using the free energy as lyapunov functional the asymptotic state of the system is investigated and characterized by a variational principle

on birman s sequence of hardy rellich type inequalities abstract in 1961 birman proved a sequence of inequalities i n for n n valid for functions in c 0 n 0 l 2 0 in particular i 1 is the classical integral hardy inequality and i 2 is the well known rellich inequality in this paper we give a proof of this sequence of inequalities valid on a certain hilbert space h n 0 of functions defined on 0 moreover f h n 0 implies f h n 1 0 as a consequence of this inclusion we see that the classical hardy inequality implies each of the inequalities in birman s sequence we also show that for any finite b 0 these inequalities hold on the standard sobolev space h 0 n 0 b furthermore in all cases the birman constants 2 n 1 2 2 2 n in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in l 2 0 resp l 2 0 b we also show that these birman constants are related to the norm of a generalized continuous cesaro averaging operator whose spectral properties we determine in detail

global existence and uniqueness for a singular degenerate cahn hilliard system with viscosity abstract existence and uniqueness are investigated for a nonlinear diffusion problem of phase field type consisting of a parabolic system of two partial differential equations complemented by neumann homogeneous boundary conditions and initial conditions this system aims to model two species phase segregation on an atomic lattice podio guidugli 2006 19 in the balance equations of microforces and microenergy the two unknowns are the order parameter and the chemical potential a simpler version of the same system has recently been discussed in colli et al 2011 8 in this paper a fairly more general phase field equation for is coupled with a genuinely nonlinear diffusion equation for the existence of a global in time solution is proved with the help of suitable a priori estimates in the case of a constant atom mobility a new and rather unusual uniqueness proof is given based on a suitable combination of variables

singular elliptic operators of second order with purely discrete spectra the friedrichs extension of a second order singular elliptic operator is considered on a weighted p2 2 space the region q is not necessarily bounded necessary conditions and sufficient conditions on the coefficients that will insure a discrete spectrum are given with a certain degree of sharpness achieved the boundary conditions include the dirichlet neumann and mixed dirichlet neumann boundary value problems

on the sobolev and hardy constants for the fractional navier laplacian abstract we prove the coincidence of the sobolev and hardy constants relative to the dirichlet and navier fractional laplacians of any real order m 0 n 2 over bounded domains in r n

global existence of weak solutions to a nonlocal cahn hilliard navier stokes system abstract a well known diffuse interface model consists of the navier stokes equations nonlinearly coupled with a convective cahn hilliard type equation this system describes the evolution of an incompressible isothermal mixture of binary fluids and it has been investigated by many authors here we consider a variant of this model where the standard cahn hilliard equation is replaced by its nonlocal version more precisely the gradient term in the free energy functional is replaced by a spatial convolution operator acting on the order parameter while the potential f may have any polynomial growth therefore the coupling with the navier stokes equations is difficult to handle even in two spatial dimensions because of the lack of regularity of we establish the global existence of a weak solution in the two dimensional case we also prove that such a solution satisfies the energy identity and a dissipative estimate provided that f fulfills a suitable coercivity condition

bessel pairs and optimal hardy and hardy rellich inequalities we give necessary and sufficient conditions on a pair of positive radial functions v and w on a ball b of radius r in rn n 1 so that the following inequalities hold for all u in c 0 infty b label one int limits b v x nabla u 2 dx geq int limits b w x u 2dx label two int limits b v x delta u 2 dx geq int limits b w x nabla u 2 dx n 1 int limits b left frac v x x 2 frac v r x x right nabla u 2dx this characterization makes a very useful connection between hardy type inequalities and the oscillatory behaviour of certain ordinary differential equations and helps in the identification of a large number of such couples v w that we call bessel pairs as well as the best constants in the corresponding inequalities this allows us to improve extend and unify many results old and new about hardy and hardy rellich type inequalities such as those obtained by caffarelli et al compos math 53 259 275 1984 brezis and vazquez revista mat univ complutense madrid 10 443 469 1997 wang and willem j funct anal 203 550 568 2003 adimurthi et al proc am math soc 130 489 505 2002 and many others

phase segregation dynamics in particle systems with long range interactions i macroscopic limits we present and discuss the derivation of a nonlinear nonlocal integrodifferential equation for the macroscopic time evolution of the conserved order parameter r t of a binary alloy undergoing phase segregation our model is ad dimensional lattice gas evolving via kawasaki exchange with respect to the gibbs measure for a hamiltonian which includes both short range local and long range nonlocal interactions the nonlocal part is given by a pair potential dj x y 0 x and y in d in the limit 0 the macroscopic evolution is observed on the spatial scale 1 and time scale 2 i e the density r t is the empirical average of the occupation numbers over a small macroscopic volume element centered atr x a rigorous derivation is presented in the case in which there is no local interaction in a subsequent paper part ii we discuss the phase segregation phenomena in the model in particular we argue that the phase boundary evolutions arising as sharp interface limits of the family of equations derived in this paper are the same as the ones obtained from the corresponding limits for the cahn hilliard equation

optimal distributed control of a nonlocal convective cahn hilliard equation by the velocity in three dimensions in this paper we study a distributed optimal control problem for a nonlocal convective cahn hilliard equation with degenerate mobility and singular potential in three dimensions of space while the cost functional is of standard tracking type the control problem under investigation cannot easily be treated via standard techniques for two reasons the state system is a highly nonlinear system of pdes containing singular and degenerating terms and the control variable which is given by the velocity of the motion occurring in the convective term is nonlinearly coupled to the state variable the latter fact makes it necessary to state rather special regularity assumptions for the admissible controls which while looking a bit nonstandard are however quite natural in the corresponding analytical framework in fact they are indispensable prerequisites to guarantee the well posedness of the associated state system in this paper we employ recently proved existence uniqueness and regularity results f

on a class of rellich inequalities we prove the rellich and the improved rellich inequalities that involve the distance function from a hypersurface of codimension k under a certain geometric assumption in case the distance is taken from the boundary that assumption is the convexity of the domain we also discuss the best constant of these inequalities

the dirichlet boundary problem for a nonlocal cahn hilliard equation abstract we study the existence uniqueness and continuous dependence on initial data of the solution for a nonlocal cahn hilliard equation with dirichlet boundary condition on a bounded domain under a nondegeneracy assumption the solutions are classical but when this is relaxed the equation is satisfied in a weak sense also we prove that there exists a global attractor in some metric space

the neumann boundary problem for a nonlocal cahn hilliard equation we study the existence uniqueness and continuous dependence on initial data of the solution to a nonlocal cahn hilliard equation on a bounded domain the equation generates a gradient flow for a free energy functional with nonlocal interaction also we apply a nonlinear poincare inequality to show the existence of an absorbing set in each constant mass affine space

well posedness and long time behavior for a nonstandard viscous cahn hilliard system we study a diffusion model of phase field type consisting of a system of two partial differential equations encoding the balances of microforces and microenergy the two unknowns are the order parameter and the chemical potential by a careful development of uniform estimates and the deduction of certain useful boundedness properties we prove existence and uniqueness of a global in time smooth solution to the associated initial boundary value problem moreover we give a description of the relative omega limit set

hardy and rellich type inequalities on complete manifolds abstract we prove some hardy and rellich type inequalities on complete noncompact riemannian manifolds supporting a weight function which is not very far from the distance function in the euclidean space

on the initialization problem in the design of structurally constrained regulators abstract parameter optimization methods have recently proved to be an efficient tool for the design of structurally constrained multivariate regulators such as decentralized and pid however most of the underlying numerical algorithms require preliminarily selection of a regulator ensuring closed loop asymptotic stability solving this problem is particularly intriguing whenever robust zero error regulation must be guaranteed since the regulator cannot in this case be asymptotically stable in this paper constructive sufficient conditions are given for the existence of structurally constrained high gain regulators ensuring both closed loop asymptotic stability and robust zero error asymptotic regulation in the face of polynomial exogenous signals

hardy and rellich inequalities for submanifolds in hadamard spaces abstract some of the most known integral inequalities are the sobolev hardy and rellich inequalities on regions in euclidean spaces in the context of submanifolds the sobolev inequality was proved by michael simon 13 and hoffman spruck 12 since then a sort of applications to the submanifold theory has been derived from those inequalities years later carron 6 obtained a hardy inequality for submanifolds in hadamard spaces in this paper we prove the general hardy and rellich inequalities for submanifolds in hadamard spaces some applications are given and we also analyse the equality cases

a necessary and sufficient condition for the stabilisation of a matrix and its principal submatrices abstract we present a necessary and sufficient condition for the existence of a real diagonal matrix such that its product with a given real square matrix is hurwitz together with all its principal submatrices we also give an explicit formula for a matrix which in particular supplies all the involved matrices with negative distinct eigenvalues

reliable regulation in decentralised control systems this article addresses the design of decentralised regulators which supply the control systems with signal tracking and disturbance rejection this property has to be attained to the maximum possible extent even when instrumentation faults occur thus causing the opening of some feedback loops the problem is tackled for lti asymptotically stable plants subject to perturbations under the assumption that the laplace transforms of the exogenous signals have multiple poles on the imaginary axis the proposed regulator is composed of an lti nominal controller supervised by a reconfiguration block once the actions of the reconfiguration block have been settled the synthesis of the nominal controller is reformulated as a suitable regulation problem a constructive sufficient condition for its solvability is established this condition turns out to be also necessary if the exogenous signals are polynomial in time