the fokker planck equation with absorbing boundary conditions we study the initial boundary value problem for the fokker planck equation in an interval with absorbing boundary conditions we develop a theory of well posedness of classical solutions for the problem we also prove that the resulting solutions decay exponentially for long times to prove these results we obtain several crucial estimates which include hypoellipticity away from the singular set for the fokker planck equation with absorbing boundary conditions as well as the holder continuity of the solutions up to the singular set

spectral convergence and nonlinear dynamics of reaction diffusion equations under perturbations of the domain abstract in this paper we obtain the continuity of attractors for semilinear parabolic problems with neumann boundary conditions relatively to perturbations of the domain we show that if the perturbations on the domain are such that the convergence of eigenvalues and eigenfunctions of the neumann laplacian is granted then we obtain the upper semicontinuity of the attractors if moreover every equilibrium of the unperturbed problem is hyperbolic we also obtain the continuity of attractors we also give necessary and sufficient conditions for the spectral convergence of neumann problems under perturbations of the domain

on the sharp effect of attaching a thin handle on the spectral rate of convergence abstract consider two domains connected by a thin tube it can be shown that the resolvent of the dirichlet laplacian is continuous with respect to the channel section parameter this in particular implies the continuity of isolated simple eigenvalues and the corresponding eigenfunctions with respect to domain perturbation under an explicit nondegeneracy condition we improve this information providing a sharp control of the rate of convergence of the eigenvalues and eigenfunctions in the perturbed domain to the relative eigenvalue and eigenfunction in the limit domain as an application we prove that again under an explicit nondegeneracy condition the case of resonant domains features polynomial splitting of the two eigenvalues and a clear bifurcation of eigenfunctions

mean exit times for particles driven by weakly colored noise consider a particle whose velocity is an ornstein uhlenbeck process with correlation time varepsilon 2 frac dx dt frac sigma z varepsilon qquad frac dz dt frac z varepsilon 2 frac sqrt 2 xi t varepsilon where xi t is gaussian white noise this represents brownian motion with memory i e a non markovian brownian motion singular perturbation techniques reduce the mean exit time problem to a boundary layer problem which turns out to be the half range expansion problem for the fokker planck operator this is solved via complex variable theory it is found that the spatial density has a milne extrapolation length varepsilon sigma zeta tfrac 1 2 varepsilon sigma 1 46035 cdots where zeta is the riemann zeta function it is also found that the average time at which x t first escapes an interval a 0 b ven that it starts at x 0 0 and z 0 z 0 is e t text ex frac 1 2 sigma 2 left a varepsilo

natural frequencies of orthotropic rectangular plates with stepped thickness abstract it is shown how one can derive an approximate formula for estimating a natural frequency of an orthotropic rectangular plate with stepped thickness by using several natural frequencies of the corresponding isotropic plate reduced from the orthotropic one to justify the method an orthotropic two part rectangular plate with simply supported sides is discussed and an approximate formula is proposed for estimating the fundamental natural frequency

isotropic hypoellipticity and trend to equilibrium for the fokker planck equation with a high degree potential we consider the fokker planck equation with a confining or anti confining potential which behaves at infinity like a possibly high degree homogeneous function hypoellipticity techniques provide the well posedness of the weak cauchy problem in both cases as well as instantaneous smoothing and exponential trend to equilibrium lower and upper bounds for the rate of convergence to equilibrium are obtained in terms of the lowest positive eigenvalue of the corresponding witten laplacian with detailed applications

characterization of the shape stability for nonlinear elliptic problems abstract we characterize all geometric perturbations of an open set for which the solution of a nonlinear elliptic pde of p laplacian type with dirichlet boundary condition is stable in the l norm the necessary and sufficient conditions are jointly expressed by a geometric property associated to the p convergence if the dimension n of the space satisfies n 1 p n and if the number of the connected components of the complements of the moving domains are uniformly bounded a simple characterization of the uniform convergence can be derived in a purely geometric frame in terms of the hausdorff complementary convergence several examples are presented

free vibration of stiffened rectangular plates using green s functions and integral equations a new method for the free vibration analysis of stiffened rectangular plates based on the use of green s functions and the solution of a system of fredholm integral equations of the second kind is demonstrated the lateral forces and twisting moments of constraint between the plate and beam stiffeners are accounted for for plates with simply supported edges perpendicular to the stiffeners the integral equations are solved exactly to yield the characteristic equations for the natural frequencies the characteristic equations are then solved to yield the exact natural frequencies within the context of the mathematical model chosen the pertinent parameters of the problem all appear explicitly in the characteristic equation which then has implications for its use as a mathematical design tool the exact natural modes can be calculated and the orthogonality relation for the natural modes is given which implies that the forced response of the stiffened plate including the effects of damping can be determined by modal analysis as an example a table and figures show the natural frequencies for a stiffened simply supported rectangular plate with a single stiffener

quantum graphs ii some spectral properties of quantum and combinatorial graphs the paper deals with some spectral properties of mostly infinite quantum and combinatorial graphs quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics nanotechnology optics and other areas a schnol type theorem is proven that allows one to detect that a point belongs to the spectrum when a generalized eigenfunction with an sub exponential growth integral estimate is available a theorem on spectral gap opening for decorated quantum graphs is established its analogue is known for the combinatorial case it is also shown that if a periodic combinatorial or quantum graph has a point spectrum it is generated by compactly supported eigenfunctions scars

smoothing estimates for boltzmann equation with full range interactions spatially homogeneous case in this work we are concerned with the regularities of the solutions to the boltzmann equation with physical collision kernels for the full range of intermolecular repulsive potentials r p 1 with p 2 we give new and constructive upper and lower bounds for the collision operator in terms of standard weighted fractional sobolev norms as an application we get the global entropy dissipation estimate which is a little stronger than that described by alexandre et al arch rational mech anal 152 4 327 355 2000 as another application we prove the smoothing effects for the strong solutions constructed by desvillettes and mouhot arch rational mech anal 193 2 227 253 2009 of the spatially homogeneous boltzmann equation with true hard potential and true moderately soft potential

the radiative transfer equation in the forward peaked regime in this work we study the radiative transfer equation in the forward peaked regime in free space specifically it is shown that the equation is well posed by proving instantaneous regularization of weak solutions for arbitrary initial datum in l1 classical techniques for hypo elliptic operators such as averaging lemma are used in the argument among the interesting aspects of the proof are the use of the stereographic projection and the presentation of a rigorous expression for the scattering operator given in terms of a fractional laplace beltrami operator on the sphere or equivalently a weighted fractional laplacian analog in the projected plane such representations may be used for accurate numerical simulations of the model as a bonus given by the methodology we show convergence of henyey greenstein scattering models and vanishing of the solution at time algebraic rate due to scattering diffusion

on the trend to global equilibrium in spatially inhomogeneous entropy dissipating systems the linear fokker planck equation we study the long time behavior of kinetic equations in which transport and spatial confinement in an exterior potential or in a box are associated with a degenerate collision operator acting only in the velocity variable we expose a general method based on logarithmic sobolev inequalities and the entropy to overcome the well known problem due to the degeneracy in the position variable of the existence of infinitely many local equilibria this method requires that the solution be somewhat smooth in this paper we apply it to the linear fokker planck equation and prove decay to equilibrium faster than o t 1 for all 0 2001 john wiley sons inc

asymptotic behaviour and self similarity for the three dimensional vlasov poisson fokker planck system abstract the aim of this work is to study the asymptotic behaviour of global in time solutions of the vlasov poisson fokker planck system in three dimensions we consider both cases with gravitational and electrostatic interaction but disregard friction it is proved that the distribution of particles tends for large time to the fundamental solution of the linear operator in l 1 norm which means that the effect of the interaction potential vanishes comparatively at t in quantitative terms the result assures that the total nonlinear interaction force decays for large time with a decay rate of order t 3 and the potential energy behaves like o t 3 2 the asymptotic result is independent of the repulsive or attractive character of the interaction field the main idea is to use the self similarity of the fundamental solution of the linear part of the equation and the regularity of the fokker planck operator in order to study the large time distribution of particles

generic global classical solutions of the vlasov fokker planck poisson system in three dimensions abstract it is shown that global classical solutions of the vlasov fokker planck poisson system for which the field decays in an appropriate way for large times are generic in the following sense sufficiently small perturbations of the initial data yield another global solution with the same type of asymptotic behaviour as a corollay one obtains global existence for small and for nearly neutral initial data on the way to the global perturbation result sufficient conditions for global existence and a local perturbation result are established

eigenvalues and eigenfunctions of the laplacian via inverse iteration with shift in this paper we present an iterative method inspired by the inverse iteration with shift technique of finite linear algebra designed to find the eigenvalues and eigenfunctions of the laplacian with homogeneous dirichlet boundary condition for arbitrary bounded domains r n this method which has a direct functional analysis approach does not approximate the eigenvalues of the laplacian as those of a finite linear operator it is based on the uniform convergence away from nodal surfaces and can produce a simple and fast algorithm for computing the eigenvalues with minimal computational requirements instead of using the ubiquitous rayleigh quotient of finite linear algebra also an alternative expression for the rayleigh quotient in the associated infinite dimensional sobolev space which avoids the integration of gradients is introduced and shown to be more efficient the method can also be used in order to produce the spectral decomposition of any given function u l 2

sharp estimates for dirichlet eigenfunctions in horn shaped regions we prove sharp exponential decay for the dirichlet eigenfunctions in horn shaped regions in the plane the estimate is obtained using a method of carleman which is widely used in the study of harmonic measure such estimates can be applied to study the intrinsic ultracontractivity properties of the heat semigroup for such regions

boundary behaviour of dirichlet eigenfunctions of second order elliptic operators we investigate the rate of decay of solutions of a second order elliptic equation tf f in an arbitrary domain r n which satisfy zero boundary conditions and then apply the results to a study of the behaviour of the associated heat semigroup in various weighted l 2 spaces our results concern the behaviour at infinity and at points on the boundary of especially irregular ones like corners and cusps the decay rates we obtain are expressed in terms of weighted l 2 estimates which are governed by the sharpness of the cusps and the properties of the coefficients of also the meaning of the zero boundary condition at a cusp is addressed

spectral stability of the p laplacian we study the dependence of the eigenvalues of the pp laplacian upon domain perturbation we prove lipschitz type estimates for the deviation of the eigenvalues following a domain perturbation such estimates are expressed in terms of suitable measures of vicinity between open sets such as the atlas distance and the lower hausdorff pompeiu deviation in the case of open sets with holder continuous boundaries our results improve a result known for the first eigenvalue

bounds on natural frequencies of composite circular plates integral equation methods abstract the problem of finding upper and lower bounds to the natural frequencies of free vibration of a circular plate with stepped radial density has been investigated such problems involving discontinuous coefficients in the governing differential equation are formulated with an integral equation by using a green function and the basic theory of linear integral equations it is shown that eigenvalue estimation techniques based on the integral equation formulation are more effective than those based on the traditional differential equation formulation by using the galerkin method the integral equation formulation is shown to provide more accurate upper bounds than the differential equation formulation for both the clamped and simply supported plates additionally it is shown that the corresponding green functions can be used to supply improvable lower bounds

quantitative stability for the first dirichlet eigenvalue in reifenberg flat domains in rn abstract in this paper we prove that if and are close enough for the complementary hausdorff distance and their boundaries satisfy some geometrical and topological conditions then 1 1 c n where 1 resp 1 is the first dirichlet eigenvalue of the laplacian in resp and is the lebesgue measure of the symmetric difference here the constant 1 could be taken arbitrary close to 1 but strictly less and c is a constant depending on a lot of parameters including dimension n and some geometric properties of the domains

vibrations of non homogeneous rectangular membranes abstract an exact solution for the title problem is obtained for the case where the two subdomains of the non homogeneous membrane are of rectangular shape it is shown that the fundamental frequency coefficient obtained by means a of the classical kantorovish method and b the optimized kantorovich method are in very good agreement with the result predicted by the exact method

semiflexible polymer in the half plane and statistics of the integral of a brownian curve a continuum model of a polymer with non zero bending energy fluctuating without overhangs in the half plane is considered the exact partition function is obtained from the marshall watson solution of the klein kramers equation for brownian motion in the half space the partition function contains information on probabilities associated with the integral of a brownian curve and reproduces sinai s t 5 4 result for the asymptotic first passage time density the t 5 2 dependence of a different passage probability implies a first order polymer adsorption transition for short range pinning potentials

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

spectral stability of dirichlet second order uniformly elliptic operators we prove sharp stability results for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous dirichlet boundary conditions upon domain perturbation

use of lagrange multipliers with polynomial series for dynamic analysis of constrained plates part i polynomial series abstract a theory for prediction of the dynamic response of a constrained plate is presented here the boundaries of the plate may be partially fixed its dynamic loading is due to elastically mounted vibrating machines and its constraints include beam like stiffeners the theory yields the eigenvalues and modal shapes of the plate and stiffeners which comprise the system the solution given in part i is based on galerkin s method combined with use of special polynomial series presented by kantorovich and krylov these eigenvalues are used in part ii 1 for response analysis of the complete system and the eigenvalues of the complete system will be obtained by the application of lagrange equations and multipliers the various coefficients used in the process are presented in the appendices to the work comparisons with published results show good agreement

dirichlet problems on varying domains abstract the aim of the paper is to characterise sequences of domains for which solutions to an elliptic equation with dirichlet boundary conditions converge to a solution of the corresponding problem on a limit domain necessary and sufficient conditions are discussed for strong and uniform convergence for the corresponding resolvent operators examples are given to illustrate that most results are optimal

shape sensitivity analysis of the hardy constant abstract we consider the hardy constant associated with a domain in the n dimensional euclidean space and we study its variation upon perturbation of the domain we prove a frechet differentiability result and establish a hadamard type formula for the corresponding derivatives we also prove a stability result for the minimizers of the hardy quotient finally we prove stability estimates in terms of the lebesgue measure of the symmetric difference of domains

singularity of eigenfunctions at the junction of shrinking tubes part ii abstract consider two domains connected by a thin tube it can be shown that generically the mass of a given eigenfunction of the dirichlet laplacian concentrates in only one of them the restriction to the other domain when suitably normalized develops a singularity at the junction of the tube as the channel section tends to zero our main result states that under a nondegeneracy condition the normalized limiting profile has a singularity of order n 1 where n is the space dimension we give a precise description of the asymptotic behavior of eigenfunctions at the singular junction which provides us with some important information about its sign near the tunnel entrance more precisely the solution is shown to be one sign in a neighborhood of the singular junction in other words we prove that the nodal set does not enter inside the channel

distance to ck hypersurfaces abstract it is shown by elementary means that a ck hypersurface m of positive reach in r n 1 has the property that the signed distance function to it is ck k 1 this extends and complements work of federer gilbarg and trudinger and serrin

transverse vibrations of composite membranes of arbitrary boundary shape this paper deals with an approximate method based on the technique of conformal mapping and a variational formulation which allows for a straightforward solution of the title problem for a certain type of composite membranes numerical results are presented for membranes of regular polygonal shape whose inner circular core possesses a density 1 while the remaining is characterized by 2 the fundamental eigenvalues are also verified by the fourier expansion collocation method and by a finite element algorithm

spectral theory for perturbed krein laplacians in nonsmooth domains abstract we study spectral properties for h k the krein von neumann extension of the perturbed laplacian v defined on c 0 where v is measurable bounded and nonnegative in a bounded open set r n belonging to a class of nonsmooth domains which contains all convex domains along with all domains of class c 1 r r 1 2 in particular in the aforementioned context we establish the weyl asymptotic formula j n k j 2 n v n n 2 o n 1 2 2 as where v n n 2 n 2 1 denotes the volume of the unit ball in r n and k j j n are the non zero eigenvalues of h k listed in increasing order according to their multiplicities we prove this formula by showing that the perturbed krein laplacian i e the krein von neumann extension of v defined on c 0 is spectrally equivalent to the buckling of a clamped plate problem and using an abstract result of kozlov from the mid 1980s our work builds on that of grubb in the early 1980s who has considered similar issues for elliptic operators in smooth domains and shows that the question posed by alonso and simon in 1980 pertaining to the validity of the above weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting we also study certain exterior type domains r n k n 3 with k r n compact and vanishing bessel capacity b 2 2 k 0 to prove equality of friedrichs and krein laplacians in l 2 d n x that is c 0 has a unique nonnegative self adjoint extension in l 2 d n x

n body schrodinger operators with finitely many bound states we consider a class of second order elliptic operators which includes atomic type n body operators for n 2 our concern is the problem of predicting the existence of only a finite number of bound states corresponding to eigenvalues below the essential spectrum we obtain a criterion which is natural for the problem and easy to apply as is demonstrated with various examples while the criterion applies to general second order elliptic operators sharp results are obtained when the hamiltonian of an atom with an infinitely heavy nucleus of charge z and n electrons of charge 1 and mass 1 2 is considered

the singularly perturbed domain and the characterization for the eigenfunctions with neumann boundary condition on considere un domaine mobile 0 parametre qui degenere partiellement quand 0 et on donne une caracterisation elaboree des comportements des fonctions propres du laplacien avec la condition limite de neumann

domain dependence of eigenvalues of elliptic type operators the dependence on the domain for the dirichlet eigenvalues of elliptic operators considered in bounded domains is studied the proximity of domains is measured by a norm of the difference of two orthogonal projectors corresponding to the reference domain and the perturbed one this allows to compare eigenvalues corresponding to domains that have non smooth boundaries and different topology the main result is an asymptotic formula in which the remainder is evaluated in terms of this quantity applications of this result are given the results are new for the laplace operator

the lp resolvents of elliptic operators with uniformly continuous coefficients abstract we consider the strongly elliptic operator a of order 2 m in the divergence form with bounded measurable coefficients and assume that the coefficients of top order are uniformly continuous for 1 p a is a bounded linear operator from the l p sobolev space h m p into h m p we will prove that a 1 exists in h m p for some and estimate its operator norm

an approach to symmetrization via polarization we prove that the steiner symmetrization of a function can be approximated in l p r n by a sequence of very simple rearrangements which are called polarizations this result is exploited to develop elementary proofs of many inequalities including the isoperimetric inequality in euclidean space in this way we also obtain new symmetry results for solutions of some varia tional problems furthermore we compare the solutions of two boundary value problems one of them having a polarized geometry and we show some point wise inequalities between the solutions this leads to new proofs of well known functional inequalities which compare the solutions of two elliptic or parabolic problems one of them having a steiner symmetrized geometry the method also allows us to investigate the case of equality in the inequalities roughly speaking we prove that the equality sign is valid only if the original problem has the symmetrized geometry

on the rate of convergence to equilibrium for the linear boltzmann equation with soft potentials abstract in this work we present several quantitative results of convergence to equilibrium for the linear boltzmann operator with soft potentials under grad s angular cut off assumption this is done by an adaptation of the famous entropy method and its variants resulting in explicit algebraic or even stretched exponential rates of convergence to equilibrium under appropriate assumptions the novelty in our approach is that it involves functional inequalities relating the entropy to its production rate which have independent applications to equations with mixed linear and non linear terms we also briefly discuss some properties of the equation in the non cut off case and conjecture what we believe to be the right rate of convergence in that case

averaging lemmas without time fourier transform and application to discretized kinetic equations we prove classical averaging lemmas in the l2 framework with the help of the fourier transform in variables x and v but not t this method is then used to study discretized problems arising out of the numerical analysis of kinetic equations

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

asymptotic behavior of an initial boundary value problem for the vlasov poisson fokker planck system the asymptotic behavior for the vlasov poisson fokker planck system in bounded domains is analyzed in this paper boundary conditions defined by a scattering kernel are considered it is proven that the distribution of particles tends for large time to a maxwellian determined by the solution of the poisson boltzmann equation with dirichlet boundary condition in the proof of the main result the conservation law of mass and the balance of energy and entropy identities are rigorously derived an important argument in the proof is to use a lyapunov type functional related to these physical quantities

the landau equation with the specular reflection boundary condition the existence and stability of the landau equation 1936 in a general bounded domain with a physical boundary condition is a long outstanding open problem this work proves the global stability of the landau equation with the coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the maxwellian equilibrium states the highlight of this work also comes from the low regularity assumptions made for the initial distribution this work generalizes the recent global stability result for the landau equation in a periodic box kim et al in peking math j 2020 our methods consist of the generalization of the wellposedness theory for the fokker planck equation hwang et al siam j math anal 50 2 2194 2232 2018 hwang et al arch ration mech anal 214 1 183 233 2014 and the extension of the boundary value problem to a whole space problem as well as the use of a recent extension of de giorgi nash moser theory for the kinetic fokker planck equations golse et al ann sc norm super pisa cl sci 19 1 253 295 2019 and the morrey estimates bramanti et al j math anal appl 200 2 332 354 1996 to further control the velocity derivatives which ensures the uniqueness our methods provide a new understanding of the grazing collisions in the landau theory for an initial boundary value problem

a strong contractivity property for semigroups generated by differential operators frequently nonconservative semigroups generated by partial differential operators in l2 p rk have the property that initial conditions which are large at ixl oo become immediately small at infinity for all t 0 this property is related to the rate of decay of eigenfunctions of the differential operator in this paper this phenomenon is investigated for a large class of differential operators of second and higher order new estimates on the rate of decay of the eigenfunctions are included which are related in special cases to those of agmon o introduction frequently in nonconservative abstract cauchy problems singular behavior is suppressed we deal with a phenomenon in this paper which is best described as the suppression of singularities at infinity suppose that the abstract cauchy problem is formulated in l2 p rk where p is a bounded positive measurable function such that p x approaches zero as ixl approaches infinity the phenomenon we study here is that when u 0 e l2 p rk with norm ii sii irk if x l2 p x dx 1 2 u t e l2 l p rk for all t 0 hence while u 0 snay be allowed to grow rapidly at infinity depending on p u t must decay rapidly at infinity for all t 0 that this behavior occurs at all is rather surprising in this paper it will be shown to occur frequently when the infinitesimal generator is an ordinary or partial differential operator with positive coefficients in the discrete spectrum case although we study this phenomenon as a mathematical one it is also related to such areas as heat conduction and nonconservative quantum mechanics for example in heat conduction it deals with the case where an initial temperature which is initially large at infinity becomes immediately small at infinity the physical interpretation for this is that the equation represents the temperature of a twodimensional thin slab with variable thermal conductivity surrounded by a medium at constant temperature with variable rates of conduction via newton s law of cooling to the surrounding medium under hypotheses of the theorems this conduction approaches perfect conduction at ixl x hence this instantaneous decrease in temperature might be expected nevertheless it seems reasonable that there is an initial growth rate at infinity determined by p beyond which we would not expect the phenomenon to occur a good deal of the paper is involved with the computation of p from the coefficients of the infinitesimal generator p turns out to decay very rapidly at infinity received by the editors july 16 1986 and in revised form march 11 1987 presented to the international conference on differential equations and mathematical physics birmingham alabama march 3 8 1986 sponsored by the nsf uab and ua 1980 mathematie s ks ubjeet 8la ssifie ltion 1985 revi sion primary 47f05 47d05 35p20 35k30 r 1988 americarl mathematical society 0002 9947 88 1 00 25 per page

sharp convergence rate of eigenvalues in a domain with a shrinking tube abstract in this paper we consider a class of singularly perturbed domains obtained by attaching a cylindrical tube to a fixed bounded region and letting its section shrink to zero we use an almgren type monotonicity formula to evaluate the sharp convergence rate of perturbed simple eigenvalues via courant fischer min max characterization and blow up analysis for scaled eigenfunctions

stability and approximations of symmetric diffusion semigroups and kernels abstract let pt t 0and pt t 0be two diffusion semigroups on r d d 2 associated with uniformly elliptic operatorsl a and l a with measurable coefficientsa aij and a aij respectively the corresponding diffusion kernels are denoted bypt x y and pt x y we derive a pointwise estimate on pt x y pt x y as well as anlp operator norm bound wherep 1 forpt ptin terms of the locall2 distance betweenaijandaij this implies in particular that pt x y pt x y converges to zero uniformly in x y r d r dand that thelp operator norm ofpt ptconverges to zero uniformly inp 1 whenaij aijgoes to zero in the locall2 norm for each 1 i j n

the lp resolvents of second order elliptic operators of divergence form under the dirichlet condition abstract let a be the 2mth order elliptic operator of divergence form with bounded measurable coefficients defined in a domain of r n for 1 h 0 m p to h m p it is known that when r n we can construct the resolvent a 1 and estimate its operator norm for some if the leading coefficients are uniformly continuous in this paper we try to extend this result to a general domain it is successful when m 1 if is the half space or a domain with c2 bounded boundary for m 1 it is shown that the problem is reduced to the case where is the half space and a is a homogeneous operator with constant coefficients we also give a perturbation theorem

a drop of ink falls from my pen it comes to earth i know not when the authors obtain for a brownian particle in a uniform force field the mean and asymptotic first passage times as functions of the particle s initial position and velocity with the recurrence times given as a special case they discuss the region of phase space for which the diffusion model of brownian motion provides an adequate approximation and conclude that there is no possibility of obtaining the recurrence times within that model they find that the nature of the boundary value problem is profoundly altered when the motion is treated as a process in phase rather than configuration space because the time development operator is then parabolic rather than elliptic they argue that such a change in the treatment of brownian motion places it within the sphere of transport theory rather than diffusion theory and that consequently results such as ours have relevance to the study of phenomena such as radiative transfer and neutron transport

bounds on natural frequencies of composite circular membranes integral equation methods this paper is concerned with finding upper and lower bounds for the natural frequencies of vibration of a circular membrane with stepped radial density such problems involving discontinuous coefficients in the differential equation may be treated by using classical variational methods however it is shown here that eigenvalue estimation techniques based on an integral equation formulation are more effective integral equation methods provide more accurate upper bounds for a comparable amount of effort and supply improvable lower bounds as well

spectral stability of general non negative self adjoint operators with applications to neumann type operators we prove a stability theorem for the eigenvalues of general non negative self adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous neumann boundary conditions upon domain perturbation

spectral stability underlp perturbation of the second order coefficients let h be a second order elliptic operator acting on a domain 0 r there has been a lot of work on the stability of the spectrum and the resolvent of h under various sorts of perturbations perturbations of the 0th order term are well studied see for example 7 there are also several results on perturbations of the higher order terms classical perturbation theory can be applied in the case of uniform or asymptotic perturbations see 5 p deift 4 has obtained results for measurable perturbations in the context of scattering theory more recent results 3 deal with boundary perturbations in this paper we study l perturbations of the second order terms we work on a bounded euclidean domain 0 r which we assume initially to have a c boundary the operators involved are uniformly elliptic with real measurable coefficients and satisfy dirichlet boundary conditions we first prove eigenvalue stability but our main aim is the stability of the resolvent in trace classes if one is only interested in the fact of convergence rather than controlling the rate then a simple approach based upon the monotone or dominated convergence theorem exists however quantitative control by such methods is not possible we emphasize the fact that we deal with operators with measurable coefficients if one restricts attention to the smooth coefficient case then standard methods of perturbation theory involve assuming uniform bounds on the first derivatives of the coefficients such bounds are not needed in our approach and are not always available in applications apart from being more general the measurable coefficients hypothesis is necessary for the study of the heat transport in a body with randomly distributed impurities in particular we study the asymptotic form of the heat diffusion article no 0011

upper and lower bounds for natural frequencies a property of the smoothed finite element methods node based smoothed finite element method ns fem using triangular type of elements has been found capable to produce upper bound solutions to the exact solutions for force driving static solid mechanics problems due to its monotonic soft behavior this paper aims to formulate an ns fem for lower bounds of the natural frequencies for free vibration problems to make the ns fem temporally stable an fem is devised by combining the compatible and smoothed strain fields in a partition of unity fashion controlled by 0 1 so that both the properties of stiff fem and the monotonically soft ns fem models can be properly combined for a desired purpose for temporally stabilizing ns fem is chosen small so that it acts like a regularization parameter making the ns fem stable but still with sufficient softness ensuring lower bounds for natural frequency solution our numerical studies demonstrate that 1 using a proper the spurious non zero energy modes can be removed and the ns fem becomes temporally stable 2 the stabilized ns fem becomes a general approach for solids to obtain lower bounds to the exact natural frequencies over the whole spectrum 3 fem can even be tuned for obtaining nearly exact natural frequencies

existence and uniqueness of a global smooth solution for the vlasov poisson fokker planck system in three dimensions abstract studying precisely the regularity of the force field created by the solution of the linear vlasov fokker planck equation we prove existence and uniqueness of a smooth solution to the vlasov poisson fokker planck system in three dimensions the attractive and repulsive cases are treated the method involves an analysis of the effect of the fokker planck kernel without any force field on non linear expressions coming from the vlasov term

on differentiability of eigenvalues of second order elliptic operators on non smooth domains abstract the regularity of eigenvalues of elliptic operators upon deformations of a given bounded domain is a classical problem in elliptic pdes which has been focused by many authors we establish a theorem on c r dependence of algebraically simple eigenvalues and eigenfunctions with respect to perturbations of c 1 class of non smooth domains and of c r class of coefficients of elliptic operators moreover we also compute the first variation of these eigenvalues in relation to both parameters for non smooth domains as a byproduct we extend hadamard s formula to second order elliptic operators for domains of c 2 class and other non smooth ones

vibration of inhomogeneous strings rods and membranes this paper is concerned with the vibration of inhomogeneous strings rods and membranes with continuously varying properties the work has two main purposes first we provide specific examples for which closed form exact solutions are obtained apart from their intrinsic interest such examples serve as benchmark problems against which the accuracy of approximate analytical or numerical methods may be assessed the second objective is to demonstrate the effectiveness of an integral equation based method for obtaining lower bounds for vibration frequencies

radiative transfer with long range interactions in the half space abstract we study the well posedness and regularity theory for the radiative transfer equation in the peaked regime posed in the half space an average lemma for the transport equation in the half space is established and used to generate interior regularity for solutions of the model the averaging also shows a fractional regularisation gain up to the boundary for the spatial derivatives

spectral stability of the neumann laplacian we prove the equivalence of hardy and sobolev type inequalities certain uniform bounds on the heat kernel and some spectral regularity properties of the neumann laplacian associated with an arbitrary region of finite measure in euclidean space we also prove that if one perturbs the boundary of the region within a uniform holder category then the eigenvalues of the neumann laplacian change by a small and explicitly estimated amount

on the hadamard formula for nonsmooth domains we consider the first eigenvalue of the dirichlet laplacian in three cases c1 1 domains lipschitz domains and bounded domains without any smoothness assumptions asymptotic formula for this eig

global existence for the vlasov poisson system in bounded domains in this paper we prove global existence for solutions of the vlasov poisson system in convex bounded domains with specular boundary conditions and with a prescribed outward electrical field at the boundary