homogenization of some degenerate pseudoparabolic variational inequalities abstract multiscale analysis of a degenerate pseudoparabolic variational inequality modelling the two phase flow with dynamical capillary pressure in a perforated domain is the main topic of this work regularisation and penalty operator methods are applied to show the existence of a solution of the nonlinear degenerate pseudoparabolic variational inequality defined in a domain with microscopic perforations as well as to derive a priori estimates for solutions of the microscopic problem the main challenge is the derivation of a priori estimates for solutions of the variational inequality uniformly with respect to the regularisation parameter and to the small parameter defining the scale of the microstructure the method of two scale convergence is used to derive the corresponding macroscopic obstacle problem

the navier stokes equations in a space of bounded functions we establish a blow up rate of the navier stokes equations subject to the non slip boundary condition for a certain class of domains including bounded and exterior domains

the l p regularity problem on lipschitz domains this paper contains two results on the l p regularity problem on lipschitz domains for second order elliptic systems and 1 2 the solvability of the l p regularity problem is equivalent to a weak reverse holder condition with exponent p

neumann problem for p laplace equation in metric spaces using a variational approach existence boundedness and boundary regularity we employ a variational approach to study the neumann boundary value problem for the p laplacian on bounded smooth enough domains in the metric setting and show that solutions exist and are bounded the boundary data considered are borel measurable bounded functions we also study boundary continuity properties of the solutions one of the key tools utilized is the trace theorem for newton sobolev functions and another is an analog of the de giorgi inequality adapted to the neumann problem

green s matrices and boundary estimates in parabolic homogenization abstract for a family of second order parabolic systems in divergence form with rapidly oscillating periodic and time dependent coefficients we establish the pointwise estimates of the green s matrices we investigate uniform boundary lipschitz estimates and boundary holder estimates as well as non tangential maximum function estimates in c 1 or c 1 cylinders consequently we also obtain the agmon miranda maximum principle

the orbifold chow ring of toric deligne mumford stacks the orbifold chow ring of a deligne mumford stack defined by abramovich graber and vistoli 2 is the algebraic version of the orbifold cohomology ring in troduced by w chen and ruan 7 8 by design this ring incorporates numerical invariants such as the orbifold euler characteristic and the orbifold hodge num bers of the underlying variety the product structure is induced by the degree zero part of the quantum product in particular it involves gromov witten invariants inspired by string theory and results in batyrev 3 and yasuda 28 one expects that in nice situations the orbifold chow ring coincides with the chow ring of a resolution of singularities fantechi and g ttsche 14 and uribe 25 verify this conjecture when the orbifold is symn 5 where 5 is a smooth projective surface with ks 0 and the resolution is hilbn the initial motivation for this project was to compare the orbifold chow ring of a simplicial toric variety with the chow ring of a cr pant resolution to achieve this goal we first develop the theory of toric deligne mumford stacks modeled on simplicial toric varieties a toric deligne mumford stack corresponds to a combinatorial object called a stacky fan as a first approximation this object is a simplicial fan with a distinguished lattice point on each ray in the fan more precisely a stacky fan s is a triple consisting of a finitely generated abelian group n a simplicial fan e in q z n with n rays and a map zn n where the image of the standard basis in zn generates the rays in e a rational simplicial fan e produces a canonical stacky fan s n e where n is the distinguished lattice and is the map defined by the minimal lattice points on the rays hence there is a natural toric deligne mumford stack associated to every simplicial toric variety a stacky fan encodes a group action on a quasi affine variety and the toric deligne mumford stack is the quotient if e corresponds to a smooth toric variety x e and s is the canonical stacky fan associated to e then we simply have x z we show that many of the basic concepts such as open and closed toric substacks line bundles and maps between toric deligne mumford stacks correspond to combinatorial notions we expect that many more results about toric varieties lift to the realm of stacks and hope that toric deligne mumford stacks will serve as a useful testing ground for general theories

lagrangian floer superpotentials and crepant resolutions for toric orbifolds we investigate the relationship between the lagrangian floer superpotentials for a toric orbifold and its toric crepant resolutions more specifically we study an open string version of the crepant resolution conjecture crc which states that the lagrangian floer superpotential of a gorenstein toric orbifold mathcal x and that of its toric crepant resolution y coincide after analytic continuation of quantum parameters and a change of variables relating this conjecture with the closed crc we find that the change of variable formula which appears in closed crc can be explained by relations between open orbifold gromov witten invariants we also discover a geometric explanation in terms of virtual counting of stable orbi discs for the specialization of quantum parameters to roots of unity which appears in ruan s original crc gromov witten theory of spin curves and orbifolds contemp math amer math soc providence ri pp 117 126 2006 we prove the open crc for the weighted projective spaces mathcal x mathbb p 1 ldots 1 n using an equality between open and closed orbifold gromov witten invariants along the way we also prove an open mirror theorem for these toric orbifolds

lagrangian floer theory on compact toric manifolds i we introduced the notion of weakly unobstructed lagrangian submanifolds and constructed their potential function po purely in terms of a model data in fooo3 in this article we carry out explicit calculations involving po on toric manifolds and study the relationship between this class of lagrangian submanifolds with the earlier work of givental g1 which advocates that the quantum cohomology ring is isomorphic to the jacobian ring of a certain function called the landau ginzburg superpotential combining this study with the results from fooo3 we also apply the study to various examples to illustrate its implications to symplectic topology of lagrangian fibers of toric manifolds in particular we relate it to the hamiltonian displacement property of lagrangian fibers and to entov polterovich s symplectic quasi states

square function non tangential maximal function estimates and the dirichlet problem for non symmetric elliptic operators a we consider divergence form elliptic operators l div a x defined in the half space rn 1 n 2 where the coefficient matrix a x is bounded measurable uniformly elliptic t independent and not necessarily symmetric we establish square function non tangential maximal function estimates for solutions of the homogeneous equation lu 0 and we then combine these estimates with the method of approximability to show that l harmonic measure is absolutely continuous with respect to surface measure i e n dimensional lebesgue measure on the boundary in a scale invariant sense more precisely that it belongs to the class a with respect to surface measure equivalently that the dirichlet problem is solvable with data in lp for some p previously these results had been known only in the case n 1

w1 p estimates for elliptic problems with neumann boundary conditions in lipschitz domains abstract we study w 1 p estimates in lipschitz domains for second order elliptic equations and systems of divergence form with real valued bounded measurable coefficients for any fixed p 2 we prove that a weak reverse holder inequality implies the w 1 p estimates for solutions with neumann boundary conditions as a result we are able to show that if the coefficient matrix of elliptic equation is symmetric and in vmo r n the w 1 p estimate holds for 3 2 e p 3 e if n 3 and for 4 3 e p 4 e if n 2 and for elliptic systems the w 1 p estimate is true for 2 n n 1 e p 2 n n 1 e

directional homogenization of elliptic equations in non divergence form abstract the paper concerns regularities of solutions of elliptic equations in non divergence form with directional homogenization the main interest is to investigate the differences of regularities of datums that are needed and of solutions that are obtained between directions with homogenization and without homogenization we establish uniform interior l p 1 p estimates for d 2 u e and uniform interior c 0 1 estimates for d d x u e where x are the non homogenization directions our results are optimal

uniform lipschitz estimates in bumpy half spaces this paper is devoted to the proof of uniform holder and lipschitz estimates close to oscillating boundaries for divergence form elliptic systems with periodically oscillating coefficients our main point is that no structure is assumed on the oscillations of the boundary in particular those oscillations are neither periodic nor quasiperiodic nor stationary ergodic we investigate the consequences of our estimates on the large scales of green and poisson kernels our work opens the door to the use of potential theoretic methods in problems concerned with oscillating boundaries which is an area of active research

quantitative estimates in reiterated homogenization abstract this paper investigates quantitative estimates in the homogenization of second order elliptic systems with periodic coefficients that oscillate on multiple separated scales we establish large scale interior and boundary lipschitz estimates down to the finest microscopic scale via iteration and rescaling arguments we also obtain a convergence rate in the l 2 space by the reiterated homogenization method

weak maximum principle for biharmonic equations in quasiconvex lipschitz domains abstract in dimension two or three the weak maximum principle for biharmonic equation is valid in any bounded lipschitz domains in higher dimensions greater than three it was only known that the weak maximum principle holds in convex domains or c 1 domains and may fail in general lipschitz domains in this paper we prove the weak maximum principle in higher dimensions in quasiconvex lipschitz domains which is a sharp condition in some sense and recovers both convex and c 1 domains

homogenization of elliptic boundary value problems in lipschitz domains in this paper we study the lp boundary value problems for mathcal l u 0 in mathbb r d 1 where mathcal l rm div a nabla is a second order elliptic operator with real and symmetric coefficients assume that a is periodic in xd 1 and satisfies some minimal smoothness condition in the xd 1 variable we show that the lp neumann and regularity problems are uniquely solvable for 1 p 2 we also present a new proof of dahlberg s theorem on the lp dirichlet problem for 2 p dahlberg s original unpublished proof is given in the appendix as the periodic and smoothness conditions are imposed only on the xd 1 variable these results extend directly from mathbb r d 1 to regions above lipschitz graphs consequently by localization techniques we obtain uniform lp estimates for the dirichlet neumann and regularity problems on bounded lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization the results on the neumann and regularity problems are new even for smooth domains

the quantum orbifold cohomology of weighted projective spaces we calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces we generalize givental s heuristic argument which relates small quantum cohomology to s1 equivariant floer cohomology of loop space to weighted projective spaces and use this to conjecture an explicit formula for the small j function a generating function for certain genus zero gromov witten invariants we prove this conjecture using a method due to bertram this provides the first non trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known in addition we obtain formulas for the small j functions of weighted projective complete intersections satisfying a combinatorial condition this condition naturally singles out the class of orbifolds with terminal singularities

commutator estimates for the dirichlet to neumann map of stokes systems in lipschitz domains abstract in the paper we establish commutator estimates for the dirichlet to neumann map of stokes systems in lipschitz domains the approach is based on dahlberg s bilinear estimates and the results may be regarded as an extension of 8 21 to stokes systems

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

on second order periodic elliptic operators in divergence form we consider second order strongly elliptic operators with complex coecients in divergence form on r d we assume that the coecients are all periodic with a common period if the coecients are continu ous we derive gaussian bounds with the correct small and large time asymptotic behaviour on the heat kernel and all its holder deriva tives moreover we show that the first order riesz transforms are bounded on the lp spaces with p 2 h1 1i secondly if the coe cients are holder continuous we prove that the first order derivatives of the kernel satisfy good gaussian bounds then we establish that the second order derivatives exist and satisfy good bounds if and only if the coecients are divergence free or if and only if the second order riesz transforms are bounded finally if the third order derivatives exist with good bounds then the coecients must be constant

localised necessary conditions for singularity formation in the navier stokes equations with curved boundary abstract we generalize two results in the navier stokes regularity theory whose proofs rely on zooming in on a presumed singularity to the local setting near a curved portion of the boundary suppose that u is a boundary suitable weak solution with singularity z x t where x then under weak background assumptions the l 3 norm of u tends to infinity in every ball centered at x lim t t u t l 3 b x r r 0 additionally u generates a non trivial mild bounded ancient solution in r 3 or r 3 through a rescaling procedure that zooms in on the singularity our proofs rely on a truncation procedure for boundary suitable weak solutions the former result is based on energy estimates for l 3 initial data and a liouville theorem for the latter result we apply perturbation theory for l initial data based on linear estimates due to k abe and y giga

optimized constraints for the linearized geoacoustic inverse problem a geoacoustic inversion scheme to estimate the depth dependent sound speed characteristics of the shallow water waveguide is presented the approach is based on the linearized perturbative technique developed by rajan et al j acoust soc am 82 998 1017 1987 this method is applied by assuming a background starting model for the environment that includes both the water column and the seabed typically the water column properties are assumed to be known and held fixed in the inversion successful application of the perturbative inverse technique lies in handling issues of stability and uniqueness associated with solving a discrete ill posed problem conventionally such problems are regularized a procedure which results in a smooth solution past applications of this inverse technique have been restricted to cases for which the water column sound speed profile was known and sound speed in the seabed could be approximated by a smooth profile in this work constraints that are better suited to specific aspects of the geoacoustic inverse problem are applied these techniques expand on the original application of the perturbative inverse technique by including the water column sound speed profile in the solution and by allowing for discontinuities in the seabed sound speed profile

an integral structure in quantum cohomology and mirror symmetry for toric orbifolds abstract we introduce an integral structure in orbifold quantum cohomology associated to the k group and the class in the case of compact toric orbifolds we show that this integral structure matches with the natural integral structure for the landau ginzburg model under mirror symmetry by assuming the existence of an integral structure we give a natural explanation for the specialization to a root of unity in y ruan s crepant resolution conjecture yongbin ruan the cohomology ring of crepant resolutions of orbifolds in contemp math vol 403 amer math soc providence ri 2006 pp 117 126

quantum cohomology and toric minimal model programs abstract we give a quantum version of the danilov jurkiewicz presentation of the cohomology of a compact toric orbifold with projective coarse moduli space more precisely we construct a canonical isomorphism from a formal version of the batyrev ring from 4 to the quantum orbifold cohomology at a canonical bulk deformation this isomorphism generalizes results of givental 23 iritani 34 and fukaya oh ohta ono 21 for toric manifolds and coates lee corti tseng 11 for weighted projective spaces the proof uses a quantum version of kirwan surjectivity theorem 2 6 below and an equality of dimensions theorem 4 19 below deduced using a toric minimal model program tmmp we show that there is a natural decomposition of the quantum cohomology where summands correspond to singularities in the tmmp each of which gives rise to a collection of hamiltonian non displaceable lagrangian tori

layer potential methods for elliptic homogenization problems in this paper we use the method of layer potentials to study l 2 boundary value problems in a bounded lipschitz domain for a family of second order elliptic systems with rapidly oscillating periodic coefficients defining l e div a e 1 x under the assumption that a x is elliptic symmetric periodic and holder continuous we establish the solvability of the l 2 dirichlet regularity and neumann problems for l e u e 0 in with optimal estimates uniform in e 0

on lp estimates in homogenization of elliptic equations of maxwell s type abstract for a family of second order elliptic systems of maxwell s type with rapidly oscillating periodic coefficients in a c 1 domain we establish uniform estimates of solutions u e and u e in l p for 1 p the proof relies on the uniform w 1 p and lipschitz estimates for solutions of scalar elliptic equations with periodic coefficients

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

uniform regularity estimates in homogenization theory of elliptic systems with lower order terms on the neumann boundary problem abstract in this paper we mainly employed the idea of the previous paper 34 to study the sharp uniform w 1 p estimates with 1 p for more general elliptic systems with the neumann boundary condition on a bounded c 1 domain arising in homogenization theory based on the skills developed by z shen in 27 and by t suslina in 31 32 we also established the l 2 convergence rates on a bounded c 1 1 domain and a lipschitz domain respectively here we found a rough version of the first order correctors see 1 12 which can unify the proof in 27 and 32 it allows us to skip the corresponding convergence results on r d that are the preconditions in 31 32 our results can be regarded as an extension of 23 developed by c kenig f lin z shen as well as of 32 investigated by t suslina

boundary estimates in homogenization of systems of elasticity on lipschitz domains l p theory abstract the primary purpose of this paper is to investigate a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients arising in the theory of homogenization in lipschitz domains as a consequence for d 4 we prove that the l p neumann and l p dirichlet boundary value problems for systems of second order linear elasticity are uniquely solvable for 2 d 1 d 1 p 2 and 2 p 2 d 1 d 3 respectively

regularity of solutions of linear second order elliptic and parabolic boundary value problems on lipschitz domains for a linear strictly elliptic second order differential operator in divergence form with bounded measurable coefficients on a lipschitz domain we show that solutions of the corresponding elliptic problem with robin and thus in particular with neumann boundary conditions are holder continuous up to the boundary for sufficiently lp regular right hand sides from this we deduce that the parabolic problem with robin or wentzell robin boundary conditions is well posed on c

perturbative inversion methods for obtaining bottom geoacoustic parameters in shallow water three perturbative inverse methods for obtaining bottom geoacoustic parameters as a function of depth in shallow water are described the required input data are the trapped mode eigenvalues for one or more frequencies the group velocity dispersion curves for one or more modes or the cw pressure field versus range in each case a fredholm integral equation arises that can be solved using linear inverse theory and for which resolution and variance estimates of the solution can be readily made attention is focused primarily on the modal eigenvalue inverse problem for which the theory for determining the compressional wave speed compressional wave attenuation and density is developed in detail properties of this technique are studied using synthetic data and include investigations of the dependence of the results on acoustic frequency number of modes excited and partial a priori knowledge of the bottom the method is demonstrated on experimental data obtained in nantucket sound at 140 and 220 hz d

gromov witten theory of deligne mumford stacks given a smooth complex deligne mumford stack cal x with a projective coarse moduli space we introduce gromov witten invariants of cal x and prove some of their basic properties including the wdvv equation

the lp boundary value problems on lipschitz domains abstract let be a bounded lipschitz domain in r n we develop a new approach to the invertibility on l p of the layer potentials associated with elliptic equations and systems in as a consequence for n 4 and 2 n 1 n 1 e p 2 where e 0 depends on we obtain the solvability of the l p neumann type boundary value problems for second order elliptic systems the analogous results for the biharmonic equation are also established

on w1 p estimates for elliptic equations in divergence form first we will apply the method to study w 1 p regularity for a nonlinear elliptic operator in divergence form we would like to point out that in the particular case of a linear elliptic equation this method gives an alternative proof to the classical one which uses the general theory of singular integrals the utility of the method that we describe below is that hypotheses a and b are obtained directly by studying the deviation of the coefficients of a from the coefficients of a0 and this is usually not a difficult task a more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give w 1 p estimates with a weak hypothesis of regularity in the coefficients for instance we are able to study the following cases

applications of fourier analysis in homogenization of dirichlet problem i pointwise estimates in this paper we prove convergence results for homogenization problem for solutions of partial differential system with rapidly oscillating dirichlet data our method is based on analysis of oscill

the neumann problem for elliptic equations with non smooth coefficients in this paper we study the neumann and regularity problems for divergence form symmetric elliptic operators l div av where a aifix is a matrix of bounded measurable functions and ellipticity is the existence of a 0 such that 2 114 2 ais ir 21412 for all i r we assume that l is defined in the unit ball b but the methods of proof are sufficiently general to allow b to be a bounded starlike lipschitz domain the neumann problem lu 0 in b avun q f

dielectric breakdown optimal bounds we study dielectric breakdown for composites made of two isotropic phases we show that sachs s bound is optimal this simple example is used to illustrate a variational principle which departs from the traditional one we also derive the usual variational principle by elementary means without appealing to the technology of convex duality

homogenization of nonlinear equations with mixed boundary conditions in this paper we will study the convergence rates of solutions for homogenization of quasilinear elliptic equations with the mixed dirichlet robin boundary conditions thanks to the smoothing operator as well as homogenization tools we could handle the different boundary conditions in a uniform fashion as a consequence we establish the sharp rates of convergence in h1 and l2 which may be regarded as an extension from the classical linear equations dirichlet or neumann problems to a nonlinear case with the mixed boundary settings in this paper we will study the convergence rates of solutions for homogenization of quasilinear elliptic equations with the mixed dirichlet robin boundary conditions thanks to the smoothing operator as well as homogenization tools we could handle the different boundary conditions in a uniform fashion as a consequence we establish the sharp rates of convergence in h1 and l2 which may be regarded as an extension from the classical linear equations dirichlet or neumann problems to a nonlinear case with the mixed boundary settings

inverse optical tomography through pde constrained optimization l infty fluorescent optical tomography fot is a new bio medical imaging method with wider industrial applications it is currently intensely researched since it is very precise and with no side effects for humans as it uses non ionising red and infrared light mathematically fot can be modelled as an inverse parameter identification problem associated with a coupled elliptic system with robin boundary conditions herein we utilise novel methods of calculus of variations in l to lay the mathematical foundations of fot which we pose as a pde constrained minimisation problem in lp and l

necessary and sufficient conditions for the solvability of the l p dirichlet problem on lipschitz domains we study the homogeneous elliptic systems of order 2 ell with real constant coefficients on lipschitz domains in mathbb r n n ge 4 for any fixed p 2 we show that a reverse holder condition with exponent p is necessary and sufficient for the solvability of the dirichlet problem with boundary data in l p we also obtain a simple sufficient condition as a consequence we establish the solvability of the l p dirichlet problem for n ge 4 and 2 epsilon p frac 2 n 1 n 3 epsilon the range of p is known to be sharp if ell ge 2 and 4 le n le 2 ell 1 for the polyharmonic equation the sharp range of p is also found in the case n 6 7 if ell 2 and n 2 ell 2 if ell ge 3

compactness methods in the theory of homogenization ii equations in non divergence form we prove c0 c1 and c1 1 a priori estimates for solutions of boundary value problems for elliptic operators with periodic coefficients of the form j 1ai j x 2 xi xj the constants in these estimates are independent of the small parameter and hence our results imply strengthened versions of the classical averaging theorems these results generalize to a wide class of linear operators in non divergence form including equations with lower order terms and nonuniformly oscillating coefficients as well as to certain nonlinear problems which we discuss in the last section

properness for scaled gauged maps we prove properness of moduli stacks of gauged maps satisfying a stability conditition introduced by mundet 42 schmitt 48 and ziltener 59 the proof combines a git construction of schmitt 48 properness for twisted stable maps by abramovich vistoli 1 a variation of a boundedness argument due to ciocan fontanine kim maulik 13 and a removal of singularities for bundles on surfaces in colliot thelene sansuc 14

convergence rates and interior estimates in homogenization of higher order elliptic systems abstract this paper is concerned with the quantitative homogenization of 2m order elliptic systems with bounded measurable rapidly oscillating periodic coefficients we establish the sharp o e convergence rate in w m 1 p 0 with p 0 2 d d 1 in a bounded lipschitz domain in r d as well as the uniform large scale interior c m 1 1 estimate with additional smoothness assumptions the uniform interior c m 1 1 w m p and c m 1 estimates are also obtained as applications of the regularity estimates we establish asymptotic expansions for fundamental solutions

the neumann problem in lp on lipschitz and convex domains abstract given p 2 and a bounded lipschitz domain we obtain a necessary and sufficient condition for the solvability of the neumann problem for laplace s equation u 0 in with boundary data in l p as a result we show that the l p neumann problem on convex domains in r d is solvable for 1 p if d 2 for 1 p 4 if d 3 and for 1 p 3 e if d 4

continuous alignment of vorticity direction prevents the blow up of the navier stokes flow under the no slip boundary condition abstract this paper is concerned with a regularity criterion based on vorticity direction for navier stokes equations in a three dimensional bounded domain under the no slip boundary condition it asserts that if the vorticity direction is uniformly continuous in space uniformly in time there is no type i blow up a similar result has been proved for a half space by y maekawa and the first and the last authors 2014 the result of this paper is its natural but non trivial extension based on l theory of the stokes and the navier stokes equations recently developed by k abe and the first author

boundary korn inequality and neumann problems in homogenization of systems of elasticity this paper is concerned with a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients arising in the theory of homogenization we establish uniform optimal regularity estimates for solutions of neumann problems in a bounded lipschitz domain with l2 boundary data the proof relies on a boundary korn inequality for solutions of systems of linear elasticity and uses a large scale rellich estimate obtained in shen anal pde arxiv 1505 00694v2

hardy spaces and the neumann problem in l p for laplace s equation in lipschitz domains on donne des resultats optimaux pour la resolubilite du probleme de neumann dans des domaines de lipschitz a donnees dans l p on obtient des resultats du point extreme correspondant pour des espaces de hardy

quantifying uncertainty in geoacoustic inversion i a fast gibbs sampler approach this paper develops a new approach to estimating seabed geoacoustic properties and their uncertainties based on a bayesian formulation of matched field inversion in bayesian inversion the solution is characterized by its posterior probability density ppd which combines prior information about the model with information from an observed data set to interpret the multi dimensional ppd requires calculation of its moments such as the mean covariance and marginal distributions which provide parameter estimates and uncertainties computation of these moments involves estimating multi dimensional integrals of the ppd which is typically carried out using a sampling procedure important goals for an effective bayesian algorithm are to obtain efficient unbiased sampling of these moments and to verify convergence of the sample this is accomplished here using a gibbs sampler gs approach based on the metropolis algorithm which also forms the basis for simulated annealing sa although gs can be computationally slow in its basic form just as modifications to sa have produced much faster optimization algorithms the gs is modified here to produce an efficient algorithm referred to as the fast gibbs sampler fgs an automated convergence criterion is employed based on monitoring the difference between two independent fgs samples collected in parallel comparison of fgs gs and monte carlo integration for noisy synthetic benchmark test cases indicates that fgs provides rigorous estimates of ppd moments while requiring orders of magnitude less computation time

quantifying uncertainty in geoacoustic inversion ii application to broadband shallow water data this paper applies the new method of fast gibbs sampling fgs to estimate the uncertainties of seabed geoacoustic parameters in a broadband shallow water acoustic survey with the goal of interpreting the survey results and validating the method for experimental data fgs applies a bayesian approach to geoacoustic inversion based on sampling the posterior probability density to estimate marginal probability distributions and parameter covariances this requires knowledge of the statistical distribution of the data errors including both measurement and theory errors which is generally not available invoking the simplifying assumption of independent identically distributed gaussian errors allows a maximum likelihood estimate of the data variance and leads to a practical inversion algorithm however it is necessary to validate these assumptions i e to verify that the parameter uncertainties obtained represent meaningful estimates to this end fgs is applied to a geoacoustic experiment carried out at a site off the west coast of italy where previous acoustic and geophysical studies have been performed the parameter uncertainties estimated via fgs are validated by comparison with i the variability in the results of inverting multiple independent data sets collected during the experiment ii the results of fgs inversion of synthetic test cases designed to simulate the experiment and data errors and iii the available geophysical ground truth comparisons are carried out for a number of different source bandwidths ranges and levels of prior information and indicate that fgs provides reliable and stable uncertainty estimates for the geoacoustic inverse problem