properly posed sets of nodes for multivariate lagrange interpolation in c s in this paper we apply techniques from the theory of ideals and varieties in algebraic geometry to study the geometric structure of a properly posed set of nodes or ppsn for short for multivariate lagrange interpolation along an algebraic hypersurface we provide a hyperplane superposition process to construct the ppsn for interpolation along an algebraic hypersurface and as a result we offer a clear understanding of the geometric structure of the ppsn for multivariate lagrange interpolation in cs

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

extrapolation of the dirichlet problem for elliptic equations with complex coefficients abstract in this paper we prove an extrapolation result for complex coefficient divergence form operators that satisfy a strong ellipticity condition known as p ellipticity specifically let be a chord arc domain in r n and the operator l i a i j x j b i x i be elliptic with b i x k x 1 for a small k let p 0 sup p 1 a is p elliptic we establish that if the l q dirichlet problem is solvable for l for some 1 q p 0 n 1 n 2 then the l p dirichlet problem is solvable for all p in the range q p 0 n 1 n 2 in particular if the matrix a is real or n 2 the l p dirichlet problem is solvable for p in the range q

the lp neumann problem for the heat equation in non cylindrical domains we consider the lp neumann problem for the heat equation when 1 p 2 in a time varying domain using a perturbation technique and previous l2 neumann results the authors obtain a generalization of earlier work in caloric cylinders

boundary layers on sobolev besov spaces and poisson s equation for the laplacian in lipschitz domains we study inhomogeneous boundary value problems for the laplacian in arbitrary lipschitz domains with data in sobolev besov spaces as such this is a natural continuation of work in jerison and kenig j funct anal 1995 16 219 where the inhomogeneous dirichlet problem is treated via harmonic measure techniques the novelty of our approach resides in the systematic use of boundary integral methods in this regard the key results are establishing the invertibility of the classical layer potential operators on scales of sobolev besov spaces on lipschitz boundaries for optimal ranges of indices applications tolp based helmholtz type decompositions of vector fields in lipschitz domains are also presented

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

word maps isotopy and entropy on trouve des diffeomorphismes de basse entropie dans chaque classe d isotopie sur s 3 s 3

greenʼs function for second order parabolic systems with neumann boundary condition abstract we study the neumann green s function for second order parabolic systems in divergence form with time dependent measurable coefficients in a cylindrical domain q where r n is an open connected set such that a multiplicative sobolev embedding inequality holds there such a domain includes for example a bounded sobolev extension domain a special lipschitz domain and an unbounded domain with compact lipschitz boundary we construct the neumann green s function in q under the assumption that weak solutions of the systems satisfy an interior holder continuity estimate we also establish global gaussian bounds for neumann green s function under an additional assumption that weak solutions with zero neumann data satisfy a local boundedness estimate in the scalar case such a local boundedness estimate is a consequence of de giorgi moser nash theory holds for equations with bounded measurable coefficients in sobolev extension domains while in the vectorial case one may need to impose further regularity assumptions on the coefficients of the system as well as on the domain to obtain such an estimate we present a unified approach valid for both the scalar and vectorial cases and discuss some applications of our results including the construction of neumann functions for second order elliptic systems with measurable coefficients in two dimensional domains

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

uniqueness results for inverse robin problems with bounded coefficient in this paper we address the uniqueness issue in the classical robin inverse problem on a lipschitz domain omega subset rr n with l infty robin coefficient l 2 neumann data and isotropic conductivity of class w 1 r omega r n we show that uniqueness of the robin coefficient on a subpart of the boundary given cauchy data on the complementary part does hold in dimension n 2 but needs not hold in higher dimension we also raise on open issue on harmonic gradients which is of interest in this context

existence and uniqueness for a semilinear elliptic problem on lipschitz domains in riemannian manifolds ii abstract for a general class of semilinear equations in a lipschitz domain where m is a smooth compact riemannian manifold we establish the existence and uniqueness of solutions to dirichlet and neumann boundary problems the main contribution of this paper is that we consider rough boundary data that are typically just functions results of this type for the linear equation are typically obtained using singular integral techniques

the dirichlet problem in non smooth domains dirichlet problem for an elliptic operator 2s laj jaj with smooth coefficients aij in domains d in rm m 3 with non smooth boundaries the domains we treat are given locally in some cm coordinate system by the graph of a continuous function qs with vq5 e li for some p 1 p oo such domains are called li domains see section 1 for the precise definitions note that l domains are usually called lipschitz domains if d is an li domain and x e d we study the elliptic measure a7 associated with s and d at x thus for f a continuous function on ad the formula u x hf x f doh gives the perron wiener brelot generalized soluid

a criterion for the absolute continuity of the harmonic measure associated with an elliptic operator for some a o and for all x e q and 4 e r n we also assume a a for such operators the dirichlet problem is solvable in q if and only if it is solvable for the laplace operator according to a theorem of littman stampacchia and weinberger 1 this means that if q c r n is a sufficiently nice bounded region the unit ball b is an example and f is a given continuous function on the boundary of q then there exists a unique function u continuous on q so that l u 0 in q and u f on aq let us assume for convenience that the origin belongs to q then the mapping f c f aq u o is a positive linear functional so there exists a unique nonnegative measure w on aq such that for every f e aq

algebraic entropy and the action of mapping class groups on character varieties abstract we extend the definition of algebraic entropy to endomorphisms of affine varieties we then calculate the algebraic entropy of the action of elements of mapping class groups on various character varieties and show that it is equal to a quantity we call the spectral radius a generalization of the dilatation of a pseudo anosov mapping class our calculations are compatible with all known calculations of the topological entropy of this action

analyticity of layer potentials and l 2 solvability of boundary value problems for divergence form elliptic equations with complex l infty coefficients we consider divergence form elliptic operators of the form l dv a x nabla defined in r n 1 x t in r n times r n geq 2 where the l infty coefficient matrix a is n 1

on the mapping class groups of r sp sp for p 3 7 for m r sharp r s p times s p p 3 7 we calculate pi 0 rm diff m r theta 2p 1 and mathcal e m r respectively the group of isotopy classes of orientation preserving diffeomorphisms of mr modulo isotopy classes with representatives which are the identity outside a 2p disc and the group of homotopy classes of orientation preserving homotopy equivalences of mr

green s matrices of second order elliptic systems with measurable coefficients in two dimensional domains we study green s matrices for divergence form second order strongly elliptic systems with bounded measurable coefficients in two dimensional domains we establish existence uniqueness and pointwise estimates of green s matrices

one problem of the navier type for the stokes system in planar domains we study the problem u f u f u g u g in u gu g h h on for a bounded simply connected lipschitz domain in the plane for f 0f 0 g 0g 0 g lp g lp h lq h lq we study a solution in the sense of a nontangential limit for f ws 1 q r2 f ws 1 q r2 g ws q g ws q g wt 1 p p g wt 1 p p h ws 1 q q h ws 1 q q with t s 1t s 1 we prove the existence of a unique solution u wt p r2 ws q u wt p r2 ws q for f bs 1q r r2 g bsq r g bt 1 pp h bs 1 qq r with t s 1t s 1 we prove the existence of a unique solution u btp r2 bsq r for f ck 1 r2 g ck h ck h ck g ck 1 g ck 1 we prove the existence of a unique solution u ck 1 r2 ck

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

polynomial interpolation of minimal degree minimal degree interpolation spaces with respect to a finite set of points are subspaces of multivariate polynomials of least possible degree for which lagrange interpolation with respect to the given points is uniquely solvable and degree reducing this is a generalization of the concept of least interpolation introduced by de boor and ron this paper investigates the behavior of lagrange interpolation with respect to these spaces giving a newton interpolation method and a remainder formula for the error of interpolation moreover a special minimal degree interpolation space will be introduced which is particularly beneficial from the numerical point of view

lp resolvent estimates for constant coefficient elliptic systems on lipschitz domains abstract in this paper we establish the l p resolvent estimates on a lipschitz domain in r d for constant coefficient elliptic systems with homogeneous neumann boundary conditions where 1 p for d 3 and 2 d d 3 p 2 d d 3 for d 4 with some positive constant we also give the global l p estimates for the derivatives of solutions to the previous systems where 2 p 2 d d 1 for d 3 finally we extend our main results to the case of some variable coefficient elliptic systems on a bounded lipschitz domain

the application of cayley bacharach theorem to bivariate lagrange interpolation in this paper we give a new proof of the famous cayley bacharach theorem by means of interpolation and deduce a general method of constructing properly posed set of nodes for bivariate lagrange interpolation as a result we generalize the main results in liang on the interpolations and approximations in several variables jilin university 1965 liang and lu approximation theory ix vanderbilt university press 1988 and liang et al analysis combinatorics and computing nova science publishers inc new york 2002 to the more extensive situations

sharp estimates for the neumann functions and applications to quantitative photo acoustic imaging in inhomogeneous media we obtain sharp lp and holder estimates for the neumann function of the operator ik on a bounded domain we also obtain quantitative description of its singularity we then apply these estimates to quantitative photo acoustic imaging in inhomogeneous media the problem is to reconstruct the optical absorption coefficient of a diametrically small anomaly from the absorbed energy density

a recursive method for computing interpolants in this paper we describe a recursive method for computing interpolants defined in a space spanned by a finite number of continuous functions in r d we apply this method to construct several interpolants such as spline interpolants tensor product interpolants and multivariate polynomial interpolants we also give a simple algorithm for solving a multivariate polynomial interpolation problem and constructing the minimal interpolation space for a given finite set of interpolation points

global pointwise estimates for green s matrix of second order elliptic systems abstract we establish global pointwise bounds for the green s matrix for divergence form second order elliptic systems in a domain under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate moreover we prove that such a local boundedness estimate for weak solutions of the system is equivalent to the usual global pointwise bound for the green s matrix in the scalar case such an estimate is a consequence of de giorgi moser nash theory and holds for equations with bounded measurable coefficients in arbitrary domains in the vectorial case one need to impose certain assumptions on the coefficients of the system as well as on domains to obtain such an estimate we present a unified approach valid for both the scalar and vectorial cases and discuss several applications of our result

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

the theory of weights and the dirichlet problem for elliptic equations in this work we consider questions from the area of partial differential equations and the related questions from harmonic analysis specifically we will study the dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients roughly speaking we consider the following question suppose that we are given two operators lo and li and that we have good estimates for the dirichlet problem for lo good in this context could mean several different things for instance it might mean that there exists p 1 p oo so that the dirichlet problem is solvable in a suitable sense with lp data what are the optimal conditions on the difference between the coefficients of lo and l1 which guarantee that the dirichlet problem for li also has good estimates this question has been extensively studied recently cf f j k r f and d4 our work uses in a crucial way the results in r f the main results in d4 and r f are proved using an ingenious differential inequality for a family of harmonic measures introduced in d4 another of our goals in this article is to give new direct proofs of the results in d4 and r f without the use of this differential inequality theorems 2 4 2 5 and 2 18 these direct approaches have already had important applications k p2 an interesting class of examples for the theorems quoted above arises when one pulls back the laplacian via a quasiconformal mapping of the plane into itself specializing our theorem to this case suggests a new characterization of the class of am weights introduced in connection with several problems in harmonic analysis by muckenhoupt mul and coifman c fefferman co f we shall present in theorem 2 3 a criterion in terms of carleson measures which is very close to log w e bmo and is necessary and sufficient for w e ad

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

the algebraic entropy of the special linear character automorphisms of a free group on two generators in this note we establish a connection between the dynamical de gree or algebraic entropy of a certain class of polynomial automorphisms of r 3 and the maximum topological entropy of the action when restricted to compact invariant subvarieties indeed when there is no cancellation of lead ing terms in the successive iterates of the polynomial automorphism the two quantities are equal in general however the algebraic entropy overestimates the topological entropy these polynomial automorphisms arise as extensions of mapping class actions of a punctured torus s on the relative su 2 character varieties of s embedded inr 3 it is known that the topological entropy of these mapping class actions is maximized on the relative character variety comprised of reducible characters those whose boundary holonomy is 2 here we cal culate the algebraic entropy of the induced polynomial automorphisms on the character varieties and show that it too solely depends on the topology of s

parabolic regularity and dirichlet boundary value problems abstract we study the relationship between the regularity and dirichlet boundary value problems for parabolic equations of the form l u div a u u t 0 in lip 1 1 2 time varying cylinders where the coefficient matrix a a i j x t is uniformly elliptic and bounded we show that if the regularity problem r p for the equation l u 0 is solvable for some 1 p then the dirichlet problem d p for the adjoint equation l v 0 is also solvable where p p p 1 this result is an analogue of the result established in the elliptic case by kenig and pipher 1993 in the parabolic settings in the special case of the heat equation in slightly smoother domains this has been established by hofmann and lewis 1996 and nystrom 2006 for scalar parabolic systems in comparison our result is abstract with no assumption on the coefficients beyond the ellipticity condition and is valid in more general class of domains

necessary and sufficient conditions for absolute continuity of elliptic harmonic measure our purpose in this paper is to give a necessary and sufficient condition on the modulus of continuity of the coefficients of an elliptic operator in divergence form in order that the corresponding poisson kernel for the dirichlet problem exist this condition on the coefficients is global continuity together with the additional property that the modulus of continuity along some nontangential direction at each boundary point be bounded uniformly in these points and directions by a function 7 t satisfying the dini type condition

the mixed problem in l p for some two dimensional lipschitz domains we consider the mixed problem left begin array ll delta u 0 quad rm in omega frac partial u partial nu f n quad rm on rm n u f d quad rm on d end array right in a class of lipschitz graph domains in two dimensions with lipschitz constant at most 1 we suppose the dirichlet data fd has one derivative in lp d of the boundary and the neumann data fn is in lp n we find a p0 1 so that for p in an interval 1 p0 we may find a unique solution to the mixed problem and the gradient of the solution lies in lp

on an inverse boundary problem arising in brain imaging abstract we consider the inverse problem of recovering both an unknown electric current and the surrounding electromagnetic parameters of a medium from boundary measurements this inverse problem arises in brain imaging we show that under generic conditions one can recover both the source and the electromagnetic parameters if these are piecewise constant and the source current is invariant in a fixed direction or a harmonic function respectively

layer potentials and boundary value problems for elliptic equations with complex l coefficients satisfying the small carleson measure norm condition abstract we consider divergence form elliptic equations l u a u 0 in the half space r n 1 x t r n 0 whose coefficient matrix a is complex elliptic bounded and measurable in addition we suppose that a satisfies some additional regularity in the direction transverse to the boundary namely that the discrepancy a x t a x 0 satisfies a carleson measure condition of fefferman kenig pipher type with small carleson norm under these conditions we establish a full range of boundedness results for double and single layer potentials in l p hardy sobolev bmo and holder spaces furthermore we prove solvability of the dirichlet problem for l with data in l p r n bmo r n and c r n and solvability of the neumann and regularity problems with data in the spaces l p r n h p r n and l 1 p r n h 1 p r n respectively with the appropriate restrictions on indices assuming invertibility of layer potentials for the t independent operator l 0 a 0

hardy and bmo spaces associated to divergence form elliptic operators consider a second order divergence form elliptic operator l with complex bounded measurable coefficients in general operators based on l such as the riesz transform or square function may lie beyond the scope of the calderon zygmund theory they need not be bounded in the classical hardy bmo and even some l p spaces in this work we develop a theory of hardy and bmo spaces associated to l which includes in particular a molecular decomposition maximal and square function characterizations duality of hardy and bmo spaces and a john nirenberg inequality

functional calculus of dirac operators and complex perturbations of neumann and dirichlet problems we prove that the neumann dirichlet and regularity problems for divergence form elliptic equations in the half space are well posed in l 2 for small complex l infty perturbations of a coefficient matrix which is either real symmetric of block form or constant all matrices are assumed to be independent of the transversal coordinate we solve the neumann dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order dirac type operator we establish quadratic estimates for this dirac operator which implies that the associated hardy projection operators are bounded and depend continuously on the coefficient matrix we also prove that certain transmission problems for k forms are well posed for small perturbations of block matrices

weighted maximal regularity estimates and solvability of non smooth elliptic systems i we develop new solvability methods for divergence form second order real and complex elliptic systems above lipschitz graphs with l 2 boundary data the coefficients a may depend on all

fundamental solutions for stationary stokes systems with measurable coefficients abstract we establish the existence and the pointwise bound of the fundamental solution for the stationary stokes system with measurable coefficients in the whole space r d d 3 under the assumption that weak solutions of the system are locally holder continuous we also discuss the existence and the pointwise bound of the green function for the stokes system with measurable coefficients on where is an unbounded domain such that the divergence equation is solvable such a domain includes for example half space and an exterior domain

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

potential theory on lipschitz domains in riemannian manifolds the case of dini metric tensors we study the applicability of the method of layer potentials in the treatment of boundary value problems for the laplace beltrami operator on lipschitz sub domains of riemannian manifolds in the case when the metric tensor g jk dx j desk has low regularity under the assumption that g jk x g jk y c x y where the modulus of continuity satisfies a dini type condition we prove the well posedness of the classical dirichlet and neumann problems with l p boundary data for sharp ranges of p s and with optimal nontangential maximal function estimates

on the jacobi group and the mapping class group of the paper contains a proof that the mapping class group of the manifold s 3 x s 3 is isomorphic to a central extension of the full jacobi group j by the group of 7 dimensional homotopy spheres using a presentation of the group j and the p invariant of the homotopy spheres we give a presentation of this mapping class group with generators and defining relations we also compute the cohomology of the group j and determine 2 cocycles that correspond to the mapping class group of s 3 s 3

dynamics on k3 surfaces salem numbers and siegel disks this paper presents the first examples of k3 surface automorphisms f x rightarrow x with siegel disks domains on which f acts by an irrational rotation the set of such examples is countable and the surface x must be non projective to carry a siegel disk these automorphisms are synthesized from salem numbers of degree 22 and trace 1 which play the role of the leading eigenvalue for f h 2 x the construction uses the torelli theorem the atiyah bott fixed point theorem and results from transcendence theory

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