the mixed problem in lipschitz domains with general decompositions of the boundary this paper continues the study of the mixed problem for the laplacian we consider a bounded lipschitz domain rn n 2 with boundary that is decomposed as d n d and n disjoint we let denote the boundary of d relative to and impose conditions on the dimension and shape of and the sets n and d under these geometric criteria we show that there exists p0 1 depending on the domain such that for p in the interval 1 p0 the mixed problem with neumann data in the space lp n and dirichlet data in the sobolev space w 1 p d has a unique solution with the non tangential maximal function of the gradient of the solution in lp we also obtain results for p 1 when the dirichlet and neumann data comes from hardy spaces and a result when the boundary data comes from weighted sobolev spaces

some potential theory for reflecting brownian motion in holder and lipschitz domains bounds are found on the transition densities and green functions for brownian motion with normal reflection in holder and lipschitz domains for lipschitz domains reflecting brownian motion and boundary local time are constructed a harnack inequality valid up to the boundary is proved a probabilistic solution to the neumann problem is given and the kuramochi boundary is identified

weighted gradient estimates for elliptic problems with neumann boundary conditions in lipschitz and semi convex domains abstract let n 2 and be a bounded lipschitz domain in r n in this article the authors investigate global weighted norm estimates for the gradient of solutions to neumann boundary value problems of second order elliptic equations of divergence form with real valued bounded measurable coefficients in more precisely for any given p 2 two necessary and sufficient conditions for w 1 p estimates of solutions to neumann boundary value problems respectively in terms of a weak reverse holder inequality with exponent p or weighted w 1 q estimates of solutions with q 2 p and some muckenhoupt weights are obtained as applications for any given p 1 and a p r n the class of muckenhoupt weights the authors establish weighted w 1 p estimates for solutions to neumann boundary value problems of second order elliptic equations of divergence form with small bmo coefficients on bounded semi convex domains as further applications the global gradient estimates are obtained respectively in weighted lorentz spaces lorentz morrey spaces weighted orlicz spaces and variable lebesgue spaces

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

boundary harnack inequality for markov processes with jumps we prove a boundary harnack inequality for jump type markov processes on metric measure state spaces under comparability estimates of the jump kernel and urysohn type property of the domain of the generator of the process the result holds for positive harmonic functions in arbitrary open sets it applies e g to many subordinate brownian motions levy processes with and without continuous part stable like and censored stable processes jump processes on fractals and rather general schrodinger drift and jump perturbations of such processes

on the quasi regularity of non sectorial dirichlet forms by processes having the same polar sets abstract we obtain a criterion for the quasi regularity of generalized non sectorial dirichlet forms which extends the result of p j fitzsimmons on the quasi regularity of sectorial semi dirichlet forms given the right markov process associated to a semi dirichlet form we present sufficient conditions for a second right process to be a standard one having the same state space the above mentioned quasi regularity criterion is then an application the conditions are expressed in terms of the associated capacities nests of compacts polar sets and quasi continuity the second application is on the quasi regularity of the generalized dirichlet forms obtained by perturbing a semi dirichlet form with kernels

boundary value problems in morrey spaces for elliptic systems on lipschitz domains let q be a bounded lipschitz domain in rn n 3 let be a second order elliptic system with constant coefficients satisfying the legendre hadamard condition we consider the dirichlet problem u 0 in q u f on 4 where e 0 depends on q and 0 a 2 for n 3 we obtain existence and uniqueness results with nontangential maximal function estimate u 2 a q c f 2 a 0q if c satisfies the strong elliptic condition and 0 a min n 1 2 e we show that the neumann type problem u 0 in q g g 2 a dq on 9q vu wfp idci has a unique solution here h2 x dq is an atomic space with the property h2 x dq l2 a 9q the invertibility of layer potentials on l2 x dq and h2 x dq is also obtained finally we study the dirichlet problem for the biharmonic equation we establish a similar estimate in l2 x for the biharmonic equation in which case the range 0 a 2 is sharp for n 4 or 5

on comparability of integral forms we prove comparability of certain homogeneous anisotropic integral forms as a consequence we obtain a hardy type inequality generalising that for the fractional laplacian we give an application to anisotropic censored stable processes in lipschitz domains

excessive kernels and revuz measures we consider a proper submarkovian resolvent of kernels on a lusin measurable space and a given excessive measure with every quasi bounded excessive function we associate an excessive kernel and the corresponding revuz measure every finite measure charging no polar set is such a revuz measure provided the hypothesis b of hunt holds under a weak duality hypothesis we prove the revuz formula and characterize the quasi boundedness and the regularity in terms of revuz measures we improve results of azema 2 and getoor and sharpe 20 for the natural additive functionals of a borel right process

the l p robin problem for laplace equations in lipschitz and semi convex domains abstract let n 3 and be a bounded lipschitz domain in r n assume that p 2 and the function b l is non negative where denotes the boundary of denote by the outward unit normal to in this article the authors give two necessary and sufficient conditions for the unique solvability of the robin problem for the laplace equation u 0 in with boundary data u b u f l p respectively in terms of a weak reverse holder inequality with exponent p or the unique solvability of the robin problem with boundary data in some weighted l 2 space as applications the authors obtain the unique solvability of the robin problem for the laplace equation in the bounded semi convex domain with boundary data in weighted l p for any given p 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

the probabilistic solution of the third boundary value problem for second order elliptic equations using standard reflected brownian motion srbm and martingales we define in the spirit of stroock and varadhan see s v the probabilistic solution of the boundary value problem begin gathered frac 1 2 delta u qu 0 in d hfill frac partial u partial n cu f on partial d hfill end gathered whered is a bounded domain withc3 boundary andn is the inward unit oormal vector on d the assumptions forq c andf are quite general

partially linear monotone methods with automatic variable selection and monotonicity direction discovery in many statistical regression and prediction problems it is reasonable to assume monotone relationships between certain predictor variables and the outcome genomic effects on phenotypes are for instance often assumed to be monotone however in some settings it may be reasonable to assume a partially linear model where some of the covariates can be assumed to have a linear effect one example is a prediction model using both high dimensional gene expression data and low dimensional clinical data or when combining continuous and categorical covariates we study methods for fitting the partially linear monotone model where some covariates are assumed to have a linear effect on the response and some are assumed to have a monotone potentially nonlinear effect most existing methods in the literature for fitting such models are subject to the limitation that they have to be provided the monotonicity directions a priori for the different monotone effects we here present methods for fitting partially linear monotone models which perform both automatic variable selection and monotonicity direction discovery the proposed methods perform comparably to or better than existing methods in terms of estimation prediction and variable selection performance in simulation experiments in both classical and high dimensional data settings

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

skorokhod decomposition of reflected diffusions on bounded lipschitz domains with singular non reflection part let overline g subset mathbb r d be a compact set with interior g let l1 g dx 0 dx a e on g and m dx let a aij be symmetric and globally uniformly strictly elliptic on g let be such that mathcal e r f g frac 1 2 sum i j 1 d int g a ij partial i f partial j g dm f g in c infty overline g is closable in l2 g m with closure r d r the latter is fulfilled if satisfies the hamza type condition or i l1loc g dx 1 i d conservative non symmetric diffusion processes xt related to the extension of a generalized dirichlet form mathcal e r f g sum i 1 d int g rho 1 overline b i partial i f g dm f g in d mathcal e r b where rho 1 overline b 1 overline b d in l 2 g mathbb r d m satisfies sum i 1 d int g overline b i partial i f dx 0 quad rm for all f in c infty overline g are constructed and analyzed if g is a bounded lipschitz domain h1 1 g and aij d r a skorokhod decomposition for xt is given this happens through a local time that is uniquely associated to the smooth measure 1 tr 0 d where tr denotes the trace and the surface measure on g

a bilinear estimate for biharmonic functions in lipschitz domains we show that a bilinear estimate for biharmonic functions in a lipschitz domain is equivalent to the solvability of the dirichlet problem for the biharmonic equation in as a result we prove that for any given bounded lipschitz domain in mathbb r d and 1 q the solvability of the l q dirichlet problem for 2 u 0 in with boundary data in wa 1 q is equivalent to that of the l p regularity problem for 2 u 0 in with boundary data in wa 2 p where frac 1 p frac 1 q 1 this duality relation together with known results on the dirichlet problem allows us to solve the l p regularity problem for d 4 and p in certain ranges

markov processes on the lipschitz boundary for the neumann and robin problems abstract we investigate the markov process on the boundary of a bounded lipschitz domain associated to the neumann and robin boundary value problems we first construct l p semigroups of sub markovian contractions on the boundary generated by the boundary conditions and we show that they are induced by the transition functions of the forthcoming processes as in the smooth boundary case the process on the boundary is obtained by the time change with the inverse of a continuous additive functional of the reflected brownian motion the robin problem is treated with a kato type l p perturbation method using the revuz correspondence an exceptional polar set occurs on the boundary we make the link with the dirichlet forms approach

regularity and well posedness of some quasi linear elliptic and parabolic problems with nonlinear general wentzell boundary conditions on nonsmooth domains abstract we show that on a bounded domain r n with lipschitz continuous boundary if 2 n n 2 p n 1 then weak solutions of the quasi linear elliptic equation p u u p 2 u 1 x u f in with the general wentzell boundary conditions p u u p 2 u u p 2 u 2 x u g weakly on are globally holder continuous on here p and p denote the p laplace operator and the p laplace beltrami operator on the boundary respectively the functions 1 x for x and 2 x for x are continuous on r and satisfy a certain growth condition we also obtain that a realization of the operator p u 1 x u in c with the boundary conditions p u 1 x u p u u p 2 u 2 x u 0 on generates a strongly continuous semigroup of nonlinear operators on c

layer potentials and boundary value problems for elliptic systems in lipschitz domains various layer potential operators are constructed for general elliptic systems of partial differential equations with constant coefficients these operators are used to study the boundary value problems for these systems for a bounded lipschitz domain in rn n 3 it is shown that if the coefficients of the system satisfy the legendre hadamard condition and a symmetry condition then there exists a unique solution u of the system that solves the dirichlet problem for given data g in lp for p in a neighborhood of 2 also the oblique derivative problem for the system is studied furthermore some regularity results for the dirichlet problem are obtained

notes on exterior lp boundary value problems for nonhomogeneous elliptic operators abstract in terms of non smooth domains and non homogeneous elliptic operators this paper is devoted to study non tangential maximal function estimates for exterior l p regular problems we do not repeat the layer potential arguments which is a possible way instead appealing to a radial maximal function we can employ the results on homogeneous operators to make the proof much easier nevertheless this method relies on an exterior w 1 p estimate which is established in the course of the proof

the multidirectional neumann problem in ℝ4 the neumann problem as formulated in lipschitz domains with l p boundary data is solved for harmonic functions in any compact polyhedral domain of 4 that has a connected 3 manifold boundary energy estimates on the boundary are derived from new polyhedral rellich formulas together with a whitney type decomposition of the polyhedron into similar lipschitz domains the classical layer potentials are thereby shown to be semi fredholm to settle the onto question a method of continuity is devised that uses the classical 3 manifold theory of e e moise in order to untwist the polyhedral boundary into a lipschitz boundary it is shown that this untwisting can be extended to include the interior of the domain in local neighborhoods of the boundary in this way the flattening arguments of b e j dahlberg and c e kenig for the h 1 at neumann problem can be extended to polyhedral domains in 4 a compact polyhedral domain in 6 of m l curtis and e c zeeman based on a construction of m h a newman shows that the untwisting and flattening techniques used here are unavailable in general for higher dimensional boundary value problems in polyhedra

calderon zygmund estimates for stochastic homogenization abstract we prove quenched l p type estimates for the gradient of a solution of a quasilinear elliptic equation with random coefficients

on integrated l 1 convergence rate of an isotonic regression estimator for multivariate observations we consider a general monotone regression estimation where we allow for independent and dependent regressors we propose a modification of the classical isotonic least squares estimator and establish its rate of convergence for the integrated l 1 loss function the methodology captures the shape of the data without assuming additivity or a parametric form for the regression function furthermore the degree of smoothing is chosen automatically and no auxiliary tuning is required for the theoretical analysis some simulations and two real data illustrations complement the study of the proposed estimator

the dirichlet problem for higher order equations in composition form the present paper commences the study of higher order differential equations in composition form specifically we consider the equation lu divb adiva u 0 where a and b are elliptic matrices with complex valued bounded measurable coefficients and a is an accretive function elliptic operators of this type naturally arise for instance via a pull back of the bilaplacian 2 from a lipschitz domain to the upper half space more generally this form is preserved under a lipschitz change of variables contrary to the case of divergence form fourth order differential equations we establish well posedness of the dirichlet problem for the equation lu 0 with boundary data in l2 and with optimal estimates in terms of nontangential maximal functions and square functions

pseudospectral method for non isotropic heat transfer using mixed hermite legendre interpolation abstract in this paper we develop a new mixed pseudospectral method for heat transfer by using generalized hermite functions and legendre polynomials in unbounded domains fundamental to spectral methods for various unbounded domain problems we establish some basic results on the mixed generalized hermite legendre interpolation as an example a new mixed generalized hermite legendre pseudospectral scheme is provided for non isotropic heat transfer its convergence is proved numerical results demonstrate the spectral accuracy of this approach

effective condition number bounds for convex regularization we derive bounds relating renegar s condition number to quantities that govern the statistical performance of convex regularization in settings that include the ell 1 analysis setting using results from conic integral geometry we show that the bounds can be made to depend only on a random projection or restriction of the analysis operator to a lower dimensional space and can still be effective if these operators are ill conditioned as an application we get new bounds for the undersampling phase transition of composite convex regularizers key tools in the analysis are slepian s inequality and the kinematic formula from integral geometry

weighted estimates in l 2 for laplace s equation on lipschitz domains let c r d d 3 be a bounded lipschitz domain for laplace s equation au 0 in q we study the dirichlet and neumann problems with boundary data in the weighted space l 2 leader where q q q 0 q 0 is a fixed point on and da denotes the surface measure on we prove that there exists e e 0 2 such that the dirichlet problem is uniquely solvable if 1 d d 3 e and the neumann problem is uniquely solvable if 3 d e a d 1 if q is a c 1 domain one may take e 2 the regularity for the dirichlet problem with data in the weighted sobolev space l 2 1 d is also considered finally we establish the weighted l 2 estimates with general ap weights for the dirichlet and regularity problems

error estimation of hermite spectral method for nonlinear partial differential equations hermite approximation is investigated some inverse inequalities imbedding inequalities and approximation results are obtained a hermite spectral scheme is constructed for burgers equation the stability and convergence of the proposed scheme are proved strictly the techniques used in this paper are also applicable to other nonlinear problems in unbounded domains

mixed legendre hermite spectral method for heat transfer in an infinite plate nonisotropic heat transfer in an infinite plate is considered a weak formulation is derived which is appropriate for its numerical simulation the mixed legendre hermite spectral method is proposed for this problem its convergence is proved numerical results show the efficiency of this new approach

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

the l p regularity problem for the stokes system on lipschitz domains we study the lp regularity problem for the stationary stokes system on lipschitz domains for any p 2 we show that a weak reverse holder inequality with exponent p is both necessary and sufficient for the solvability of the regularity problem with data in w1 p lnp we also obtain the w1 p estimate u lp q lp f lp in a bounded lipschitz domain rd for solutions to the dirichlet problem u q div f div u 0 in and u 0 on where 1p 12 12d e

traces extensions and co normal derivatives for elliptic systems on lipschitz domains this is the post print version of the article the official published version can be accessed from the link below copyright 2011 elsevier

on the robin problem in fractal domains research partially supported by national science foundation nsf grants dms 0244737 and dms 0303310

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

hardy spaces of the conjugate beltrami equation we study hardy spaces of solutions to the conjugate beltrami equation with lipschitz coefficient on dini smooth simply connected planar domains in the range of exponents 1 infty we analyse their boundary behaviour and certain density properties of their traces we derive on the way an analog of the fatou theorem for the dirichlet and neumann problems associated with the equation mbox div sigma nabla u 0 with l p boundary data

measure valued branching processes associated with neumann nonlinear semiflows abstract we construct a measure valued branching markov process associated with a nonlinear boundary value problem where the boundary condition has a nonlinear pseudo monotone branching mechanism term which includes the case u u m with 0 m 1 the process is then used in the probabilistic representation of the solution of the parabolic problem associated with a nonlinear neumann boundary value problem in this way the classical association of the superprocesses to the dirichlet boundary value problems also holds for the nonlinear neumann boundary value problems it turns out that the obtained branching process behaves on the measures carried by the given open set like the linear continuous semiflow induced by the reflected brownian motion while the branching occurs on the measures having non zero traces on the boundary of the open set with the behavior of the superprocess having as spatial motion the process on the boundary associated to the reflected brownian motion

regularity of solutions to the polyharmonic equation in general domains the present paper establishes boundedness of m frac n 2 frac 1 2 derivatives for the solutions to the polyharmonic equation of order 2m in arbitrary bounded open sets of mathbb r n 2 n 2m 1 without any restrictions on the geometry of the underlying domain it is shown that this result is sharp and cannot be improved in general domains moreover it is accompanied by sharp estimates on the polyharmonic green function

approximation of some diffusion evolution equations in unbounded domains by hermite functions spectral and pseudospectral approximations of the heat equation are analyzed the solution is represented in a suitable basis constructed with hermite polynomials stability and convergence estimates are given and numerical tests are discussed

solution regularity and co normal derivatives for elliptic systems with non smooth coefficients on lipschitz domains abstract elliptic pde systems of the second order with coefficients from l or holder lipschitz spaces are considered in the paper continuity of the operators in corresponding sobolev spaces is stated and the internal local solution regularity theorems are generalized to the non smooth coefficient case for functions from the sobolev space h s 1 2 s 3 2 definitions of non unique generalized and unique canonical co normal derivatives are considered which are related to possible extensions of a partial differential operator and the pde right hand side from the domain to its boundary it is proved that the canonical co normal derivatives coincide with the classical ones when both exist a generalization of the boundary value problem settings which makes them insensitive to the co normal derivative inherent non uniqueness is given

regularity of the free boundary for the one phase hele shaw problem abstract in this paper we prove that in a local neighborhood lipschitz continuous free boundary of a solution of the one phase hele shaw problem is indeed smooth if the solution is lipschitz continuous and non degenerate in the neighborhood

generalized hermite spectral method and its applications to problems in unbounded domains in this paper we develop a spectral method based on generalized hermite functions with weight chi x equiv1 we also establish some basic results on generalized hermite orthogonal approximations which play an important role in spectral methods as examples the generalized ginzburg landau equation in a population problem and an elliptic equation with a harmonic potential are considered related spectral schemes are proposed and their convergence is proved numerical results demonstrate the spectral accuracy of this approach

sub markovian resolvents under weak duality hypothesis we study the weak duality between two sub markovian resolvents of kernels on a lusin topological space with respect to a given measure m our frame covers the probabilistic context of two borel markov processes in weak duality the main results are related to the coincidence of the m semipolar and m cosemipolar sets the revuz correspondence between the measures charging no cofinely open m polar set and the potential kernels associated with the homogeneous random measures of the process the equivalence between smoothness and cosmoothness a smooth measure is the revuz measure of a continuous additive functional we extend and improve results of r m blumenthal r k getoor j azema d revuz j b walsh r k getoor m j sharpe

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

uniform estimates for transmission problems with high contrast in heat conduction and electromagnetism in this paper we prove uniform a priori estimates for transmission problems with constant coefficients on two subdomains with a special emphasis for the case when the ratio between these coefficients is large in the most part of the work the interface between the two subdomains is supposed to be lipschitz we first study a scalar transmission problem which is handled through a converging asymptotic series then we derive uniform a priori estimates for maxwell transmission problem set on a domain made up of a dielectric and a highly conducting material the technique is based on an appropriate decomposition of the electric field whose gradient part is estimated thanks to the first part as an application we develop an argument for the convergence of an asymptotic expansion as the conductivity tends to infinity

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

isotonic regression in general dimensions we study the least squares regression function estimator over the class of real valued functions on 0 1 d that are increasing in each coordinate for uniformly bounded signals and with a fixed cubic lattice design we establish that the estimator achieves the minimax rate of order n min 2 d 2 1 d in the empirical l 2 loss up to polylogarithmic factors further we prove a sharp oracle inequality which reveals in particular that when the true regression function is piecewise constant on k hyperrectangles the least squares estimator enjoys a faster adaptive rate of convergence of k n min 1 2 d again up to polylogarithmic factors previous results are confined to the case d leq2 finally we establish corresponding bounds which are new even in the case d 2 in the more challenging random design setting there are two surprising features of these results first they demonstrate that it is possible for a global empirical risk minimisation procedure to be rate optimal up to polylogarithmic factors even when the corresponding entropy integral for the function class diverges rapidly second they indicate that the adaptation rate for shape constrained estimators can be strictly worse than the parametric rate

combined hermite spectral finite difference method for the fokker planck equation the convergence of a class of combined spectral finite difference methods using hermite basis applied to the fokker planck equation is studied it is shown that the hermite based spectral methods are convergent with spectral accuracy in weighted sobolev space numerical results indicating the spectral convergence rate are presented a velocity scaling factor is used in the hermite basis and is shown to improve the accuracy and effectiveness of the hermite spectral approximation with no increase in workload some basic analysis for the selection of the scaling factors is also presented

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

bounded and lp weak solutions for nonlinear equations of measure valued branching processes abstract we study the nonlinear schrodinger type equation c u c k 1 q k u k f where c is a bounded positive function we may replace the laplace operator with the generator of a borel right markov process with state space e and consider a nonlinear term induced by a perturbing kernel we solve this type of equation in an l p weak sense involving positive definite extensions of the solution from e to the set of all finite positive measures on e the extensions to the set of measures are related to the classical connection from the works of m nagasawa n ikeda s watanabe and m l silverstein between this nonlinear equation and the discrete branching markov processes we also investigate the existence of the bounded solutions for the dirichlet problem associated to the above equation with bounded boundary data our approach is based on probabilistic and analytic potential theoretical methods used on both spaces e and the set of all finite configurations of e like the revuz formula for the continuous additive functionals and the perturbation with kernels of the transition functions and resolvents

estimates for green functions of stokes systems in two dimensional domains abstract we prove the existence and pointwise bounds of the green functions for stationary stokes systems with measurable coefficients in two dimensional domains we also establish pointwise bounds of the derivatives of the green functions under a regularity assumption on the l 1 mean oscillations of the coefficients

the dirichlet to neumann operator on rough domains abstract we consider a bounded connected open set r d whose boundary has a finite d 1 dimensional hausdorff measure then we define the dirichlet to neumann operator d 0 on l 2 by form methods the operator d 0 is self adjoint and generates a contractive c 0 semigroup s s t t 0 on l 2 we show that the asymptotic behaviour of s t as t is related to properties of the trace of functions in h 1 which may or may not have

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

unique solvability of the neumann problem with weighted boundary data on a bounded c1 domain abstract let d r n n 3 be a bounded c 1 domain we prove that if 1 p and w a p d then the following neumann problem u 0 in d u n q g on d has a unique up to a constant solution u with the boundary data g l p d w satisfying d g q d q 0 moreover u satisfies u l p d w c g l p d w for each 0 1 where u denotes the nontangential maximal function of u

sharp norm estimates of layer potentials and operators at high frequency abstract in this paper we investigate single and double layer potentials mapping boundary data to interior functions of a domain at high frequency 2 in the high frequency regime the key problem is the dependence of mapping norms on the parameter for single layer potentials we find that the l 2 l 2 norms decay in the rate of decay depends on the curvature of the norm is 3 4 in piecewise smooth domains and 5 6 if the boundary is positively curved the double layer potential however displays uniform l 2 l 2 bounds independent of curvature by various examples we show that all our estimates on layer potentials are sharp appendix a by galkowski gives bounds l 2 l 2 for the single and double layer operators at high frequency that are sharp modulo log in this case both the single and double layer operator bounds depend upon the curvature of the boundary

regularity for the one phase hele shaw problem from a lipschitz initial surface in this paper we show that if the lipschitz constant of the initial free boundary is small then for small positive time the solution is smooth and satisfies the hele shaw equation in the classical sense a key ingredient in the proof which is of independent interest is an estimate up to order of magnitude of the speed of the free boundary in terms of initial data

well posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks we prove a regularity result for the anisotropic linear elasticity equation p u rm div left boldmath mathsf c cdot nabla u right f with mixed displacement and traction boundary conditions on a curved polyhedral domain omega subset mathbb r 3 in weighted sobolev spaces mathcal k m 1 a 1 omega for which the weight is given by the distance to the set of edges in particular we show that there is no loss of mathcal k m a regularity our curved polyhedral domains are allowed to have cracks we establish a well posedness result when there are no neighboring traction boundary conditions and a 0 that depends on p on the boundary conditions and on the domain our results extend to other strongly elliptic systems and higher dimensions

homogeneous random measures and a weak order for the excessive measures of a markov process let x xt px be a right markov process and let m be an excessive measure for x associated with the pair x m is a stationary strong markov process yt qm with random times of birth and death with the same transition function as x and with m as one dimensional di stribution we use yt qm to study the cone of excessive measures for x a weak order is defined on this cone an excessive measure is weakly dominated by m if and only if there is a suitable homogeneous random measure tt such that yt q6 is obtained by birthing yt qm birth in t t dt occurring at rate s dt random measures such as s are studied through the use of palm measures we also develop aspects of the general theory of processes over yt qm including the moderate markov property of yt qm when the arrow of time is reversed applications to balayage and capacity are suggested

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

the neumann problem for higher order elliptic equations with symmetric coefficients in this paper we establish well posedness of the neumann problem with boundary data in l 2 or the sobolev space dot w 2 1 in the half space for linear elliptic differential operators with coefficients that are constant in the vertical direction and in addition are self adjoint this generalizes the well known well posedness result of the second order case and is based on a higher order and one sided version of the classic rellich identity and is the first known well posedness result for an elliptic divergence form higher order operator with rough variable coefficients and boundary data in a lebesgue or sobolev space

strongly supermedian kernels and revuz measures in the frame of borel right markov processes we investigate following an analytical point of view the revuz correspondence between classes of potential kernels and their associated measures improving upon the results of revuz azema getoor and sharpe fitzsimmons fitzsimmons and getoor and dellacherie maisonneuve and meyer in the probabilistic approach of the problem the kernels that occur are the potential operators of different types of homogeneous random measures we completely characterize the hypothesis b of hunt in terms of revuz measures