conditional gauge and potential theory for the schrodinger operator on etend le theoreme de jauge conditionnelle a des operateurs plus generaux et a des domaines moins reguliers on obtient des resultats de theorie du potentiel pour l equation de schrodinger

henselian implies large in this note we show that the quotient field of a domain which is henselian with respect to a nontrivial ideal is a large field and give some applications of this fact using a specialization theorem for ramified covers of the line over generalized krull fields

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 equivalence of certain heat kernel and green function bounds abstract we show that certain upper and lower bounds on the green function and heat kernel of a second order elliptic operator in a bounded region are equivalent and imply a number of apparently stronger bounds we also show that these bounds are equivalent to the harnack inequality except for peculiar regions and are therefore almost always valid

prime ends for domains in metric spaces in this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions we compare our prime ends to those of caratheodory and nakki modulus ends and

harmonic measure and quantitative connectivity geometric characterization of the l p solvability of the dirichlet problem it is well known that quantitative scale invariant absolute continuity more precisely the weak a infty property of harmonic measure with respect to surface measure on the boundary of an open set omega subset mathbb r n 1 with ahlfors david regular boundary is equivalent to the solvability of the dirichlet problem in omega with data in l p partial omega for some p infty in this paper we give a geometric characterization of the weak a infty property of harmonic measure and hence of solvability of the l p dirichlet problem for some finite p this characterization is obtained under background hypotheses an interior corkscrew condition along with ahlfors david regularity of the boundary that are natural and in a certain sense optimal we provide counter examples in the absence of either of them or even one of the two upper or lower ahlfors david bounds moreover the examples show that the upper and lower ahlfors david bounds are each quantitatively sharp

properties of harmonic measures in the dirichlet problem for nilpotent lie groups of heisenberg type in groups of heisenberg type we introduce a large class of domains which we call adp admissible for the dirichlet problem and we prove that on the boundary of such domains harmonic measure ordinary surface measure and the perimeter measure are mutually absolutely continuous we also establish the solvability of the dirichlet problem when the boundary datum belongs to l p 1 p with respect to the ordinary surface measure here the harmonic measure is that relative to a sub laplacian associated with a basis of the first layer of the lie algebra a domain is called adp if it is a nontangentially accessible domain and it satisfies an intrinsic outer ball condition

a carleson type inequality for fully nonlinear elliptic equations with non lipschitz drift term abstract this paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term we elaborate on the non homogeneous generalized harnack inequality proved by the second author in 26 to prove a generalized carleson estimate we also prove boundary holder continuity and a boundary harnack type inequality

estimates for p harmonic functions vanishing on a flat we study p harmonic functions in a domain crn near an m dimensional plane an m flat m where 0 m n 1 in particular let u be a positive p harmonic function with n p vanishing on a porti

canonical analysis of two convex polyhedral cones and applications canonical analysis of two convex polyhedral cones consists in looking for two vectors one in each cone whose square cosine is a maximum this paper presents new results about the properties of the optimal solution to this problem and also discusses in detail the convergence of an alternating least squares algorithm the set of scalings of an ordinal variable is a convex polyhedral cone which thus plays an important role in optimal scaling methods for the analysis of ordinal data monotone analysis of variance and correspondence analysis subject to an ordinal constraint on one of the factors are both canonical analyses of a convex polyhedral cone and a subspace optimal multiple regression of a dependent ordinal variable on a set of independent ordinal variables is a canonical analysis of two convex polyhedral cones as long as the signs of the regression coefficients are given we discuss these three situations and illustrate them by examples

the dirichlet problem for p harmonic functions with respect to the mazurkiewicz boundary and new capacities in this paper we develop the perron method for solving the dirichlet problem for the analog of the p laplacian i e for p harmonic functions with mazurkiewicz boundary values the setting conside

counting arithmetic lattices and surfaces we give estimates on the number al h x of conjugacy classes of arithmetic lattices of covolume at most x in a simple lie group h in particular we obtain a first concrete estimate on the number of arithmetic 3 manifolds of volume at most x our main result is for the classical case h psl 2 where we show that limx log al h x x log x 1 2 the proofs use several different techniques geometric bounding the number of generators of as a function of its covolume number theoretic bounding the number of maximal such and sharp estimates on the character values of the symmetric groups to bound the subgroup growth of

a free boundary problem for the parabolic poisson kernel abstract we study parabolic chord arc domains introduced by hofmann lewis and nystrom 14 and prove a free boundary regularity result below the continuous threshold more precisely we show that a reifenberg flat parabolic chord arc domain whose poisson kernel has logarithm in vmo must in fact be a vanishing chord arc domain i e satisfies a vanishing carleson measure condition this generalizes to the parabolic setting a result of kenig and toro 26 and answers in the affirmative a question left open in the aforementioned paper of hofmann et al a key step in this proof is a classification of flat blowups for the parabolic problem

relative fatou s theorem for δ α 2 harmonic functions in bounded κ fat open sets recently it was shown in p kim fatou s theorem for censored stable processes stochastic process appl 108 1 2003 63 92 that fatou s theorem for transient censored stable processes in a bounded c1 1 open set is true here we give a probabilistic proof of relative fatou s theorem for 2 harmonic functions equivalently for symmetric stable processes in bounded fat open set where 0 2 that is if u is positive 2 harmonic function in a bounded fat open set d and h is singular positive 2 harmonic function in d then nontangential limits of u h exist almost everywhere with respect to the martin representing measure of h this extends the result of bogdan and dyda k bogdan b dyda relative fatou theorem for harmonic functions of rotation invariant stable processes in smooth domain studia math 157 1 2003 83 96 it is also shown that under the gaugeability assumption relative fatou s theorem is true for operators obtained from the generator of the killed stable process in bounded fat open set d through nonlocal feynman kac transforms as an application relative fatou s theorem for relativistic stable processes is also true if d is bounded c1 1 open set

optimal doubling reifenberg flatness and operators of p laplace type in this paper we consider operators of p laplace type of the form a x u 0 concerning a we assume for p 1 fixed an appropriate ellipticity type condition holder continuity in x and that a x p 1 a x whenever x r n and r n 0 let r n be a bounded domain let d be a compact subset of we say that u up d is the a capacitary

estimates for solutions to equations of p laplace type in ahlfors regular nta domains this thesis consists of six scientific papers an introduction and a summary all six papers concern the boundary behavior of non negative solutions to partial differential equations paper i concerns solutions to certain p laplace type operators with variable coefficients suppose that u is a non negative solution that vanishes on a part of an ahlfors regular nta domain we prove among other things that the gradient du of u has non tangential limits almost everywhere on the boundary piece and that log du is a bmo function on the boundary furthermore for ahlfors regular nta domains that are uniformly n r0 approximable by lipschitz graph domains we prove a boundary harnack inequality provided that is small enough paper ii concerns solutions to a p laplace type operator with lower order terms in reifenberg flat domains we prove that the ratio of two non negative solutions vanishing on a part of the boundary is holder continuous provided that is small enough furthermore we solve the martin boundary problem provided is small enough in paper iii we prove that the boundary type riesz measure associated to an a capacitary function in a reifenberg flat domain with vanishing constant is asymptotically optimal doubling paper iv concerns the boundary behavior of solutions to certain parabolic equations of p laplace type in lipschitz cylinders among other things we prove an intrinsic carleson type estimate for the degenerate case and a weak intrinsic carleson type estimate in the singular supercritical case in paper v we are concerned with equations of p laplace type structured on hormander vector fields we prove that the boundary type riesz measure associated to a non negative solution that vanishes on a part of an x nta domain is doubling on paper vi concerns a one phase free boundary problem for linear elliptic equations of non divergence type assume that we know that the positivity set is an nta domain and that the free boundary is a graph furthermore assume that our solution is monotone in the graph direction and that the coefficients of the equation are constant in the graph direction we prove that the graph giving the free boundary is lipschitz continuous

assessing a set of additive utility functions for multicriteria decision making the uta method abstract the purpose of the method presented in this paper is to assess additive utility functions which aggregate multiple criteria in a composite criterion using the information given by a subjective ranking on a set of stimuli or actions weak order comparison judgments and the multicriteria evaluations of these actions it is an ordinal regression method using linear programming to estimate the parameters of the utility function stability and sensitivity analysis leads to the assessment of a set of utility functions by means of post optimality analysis techniques in linear programming finally a simple illustrative example is presented and some extensions of the method are proposed

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

boundedness of the gradient for a doubly nonlinear parabolic equation we prove the local boundedness of the gradient for positive solutions to a doubly nonlinear parabolic equation in the case when the standard lebesgue measure has been replaced by a doubling measure which supports a weak poincare inequality

on mean convergence of hermite fejer and hermite interpolation for erdos weights abstract we investigate convergence of hermite fejer and hermite interpolation polynomials in l p 0 for erdos weights

quantitative rectifiability and lipschitz mappings the classical notion of rectifiability of sets in r n is qualitative in nature and in this paper we are concerned with quantitative versions of it this issue arises in connection with l p estimates for singular integral operators on sets in r n we give a criterion for one reasonably natural quantitative rectifiability condition to hold and we use it to give a new proof of a theorem in d3 we also give some results on the geometric properties of a certain class of sets in r n which can be viewed as generalized hypersurfaces along the way we shall encounter some questions concerning the behavior of lipschitz functions with regard to approximation by affine functions in particular

extending sobolev functions with partially vanishing traces from locally e δ domains and applications to mixed boundary problems we prove that given any k n for each open set rn and any closed subset d of such that is locally an e domain near d there exists a linear and bounded extension operator ek d mapping for each p 1 the space wdk p into wdk p rn here with o denoting either or rn the space wdk p o is defined as the completion in the classical sobolev space wk p o of restrictions to o of functions from cc rn whose supports are disjoint from d in turn this result is used to develop a functional analytic theory for the class wdk p including intrinsic characterizations boundary traces and extensions results interpolation theorems among other things which is then employed in the treatment of mixed boundary value problems formulated in locally e domains finally we also prove extension results on the scales of besov and bessel potential spaces on e domains with partially vanishing traces on ahlfors regular sets and explore some of the implications of such extension results

a gradient bound for free boundary graphs we prove an analogue for a one phase free boundary problem of the classical gradient bound for solutions to the minimal surface equation it follows in particular that every energy minimizing free boundary that is a graph is also smooth the method we use also leads to a new proof of the classical minimal surface gradient bound 2010 wiley periodicals inc

traps for reflected brownian motion consider an open set d r d d 2 and a closed ball b d let e x tb denote the expectation of the hitting time of b for reflected brownian motion in d starting from x d we say that d is a trap domain if supx e x tb a domain d is not a trap domain if and only if the reflecting brownian motion in d is uniformly ergodic we fully characterize the simply connected planar trap domains using a geometric condition and give a number of less complete results for d 2

negatively curved manifolds elliptic operators and the martin boundary consider a riemannian manifold m of dimension n 2 and assume that m is complete simply connected and that its sectional curvatures are bounded between two negative constants in a recent paper 7 m anderson and r schoen have proved that the martin boundary of m with respect to the laplace beltrami operator a is naturally homeomorphic to soo m the sphere at infinity of m here we shall first establish a similar result for a wide class of elliptic operators on m giving at the same time a new proof of the theorem of anderson and schoen our proof relies heavily on several concepts of potential theory but on the other hand avoids the quite difficult calculations of 7 and perhaps makes the result more understandable roughly speaking we show that if y is a second order elliptic operator defined over m and has a uniformly nice behaviour on each ball of radius 1 in m then dq m the y martin boundary of m is s m provided that the following condition holds there is a positive y ci superharmonic function on m for some positive if instead we only assume the existence of the green function for y a minimal requirement when studying the y martin boundary then d m may be very different from so m even for a rotation invariant operator y on the unit ball of rn equipped

a finiteness theorem for zero cycles over p adic fields let r be a henselian discrete valuation ring let x be a regular projective flat scheme over spec r with generalized semistable reduction we prove a bijectivity theorem for etale cycle class maps of the chow group of 1 cycles on x as an application we prove a finiteness theorem for the chow group of 0 cycles on a projective smooth variety over a p adic field

vector valued hardy spaces in non smooth domains abstract we characterize the radon nikod m property of a banach space x in terms of the existence of non tangential limits of x valued harmonic functions u defined in a domain d r n n 2 with lipschitz boundary and belonging to maximal hardy spaces this extends the same result previously known for the unit disk of c we also prove an atomic decomposition of the borel x valued measures in d that arise as boundary limits of x valued harmonic functions whose non tangential maximal function is integrable with respect to harmonic measure of d

null sets of harmonic measure on nta domains lipschitz approximation revisited we show the david jerison construction of big pieces of lipschitz graphs inside a corkscrew domain does not require surface measure be upper ahlfors regular thus we can study absolute continuity of harmonic measure and surface measure on nta domains of locally finite perimeter using lipschitz approximations a partial analogue of the f and m riesz theorem for simply connected planar domains is obtained for nta domains in space as one consequence every wolff snowflake has infinite surface measure

boundary behavior of p harmonic functions in the heisenberg group in this paper we study the boundary behavior of nonnegative p harmonic functions in a bounded domain in the heisenberg group mathbb h n under suitable geometric assumptions on the ground domain omega subset mathbb h n our main contributions can be summarized as follows 1 in theorem 1 1 we obtain an estimate from above stating that any such function should vanish at most linearly like the sub riemannian distance from the boundary frac u g u a r g 0 leq c 1 frac d g partial omega r where a r g 0 in omega is a non tangential point relative to g 0 in partial omega 2 in theorem 1 2 we establish an estimate from below which states that away from the characteristic set of the order of vanishing is exactly linear i e frac u g u a r g 0 ge c frac d g partial omega r

area integral estimates for caloric functions we study the relationship between the area integral and the para bolic maximal function of solutions to the heat equation in domains whose boundary satisfies a 5 1 mixed lipschitz condition our main result states that the area integral and the parabolic maximal function are equivalent in lp p 0 p oo the measure p must satisfy muckenhoupt s loo condition with respect to caloric measure we also give a fatou theorem which shows that the existence of parabolic limits is a e with respect to caloric measure equivalent to the finiteness of the area integral

heegner divisors l functions and harmonic weak maass forms recent works mostly related to ramanujan s mock theta functions make use of the fact that harmonic weak maass forms can be combinatorial generating functions generalizing works of waldspurger kohnen and zagier we prove that such forms also serve as generating functions for central values and derivatives of quadratic twists of weight 2 modular l functions to obtain these results we construct differentials of the third kind with twisted heegner divisor by suitably generalizing the borcherds lift to harmonic weak maass forms the connection with periods fourier coefficients derivatives of l functions and points in the jacobian of modular curves is obtained by analyzing the properties of these differentials using works of scholl waldschmidt and gross and zagier

gleaning in the field of dual scaling some historical background and preliminary technical information are first presented and then a number of hidden but important methodological aspects of dual scaling are illustrated and discussed normed versus projected weights the amount of information accounted for by each solution a perfect solution to the problem of multidimensional unfolding multidimensional quantification space graphical display number of option problems option standardization versus item standardization and asymmetry of symmetric dual scaling contrary to the common perception that dual scaling and similar quantification methods are now mathematically transparent the present study demonstrates how much more needs to be clarified for routine use of the method to arrive at valid conclusions data analysis must be carried out in such a way that common sense intuition and sound logic will prevail

boundary harnack principle for symmetric stable processes in this paper we study potential theoretic properties of the symmetric stable processes 0 2 establishing the boundary harnack principle for ratios of harmonic functions on any open sets identifying the martin boundary with the euclidean boundary for open sets with a certain interior fatness property and extending earlier results on intrinsic ultracontractivity and the conditional gauge theorem to certain open sets

p harmonic functions in the heisenberg group boundary behaviour in domains well approximated by non characteristic hyperplanes in this paper we study for given p 1 p infty the boundary behaviour of non negative p harmonic functions in the heisenberg group mathbb h n i e we consider weak solutions to the non linear and potentially degenerate partial differential equation begin aligned sum i 1 2n x i xu p 2 x i u 0 end aligned where the vector fields x 1 ldots x 2n form a basis for the space of left invariant vector fields on mathbb h n in particular we introduce a set of domains omega subset mathbb h n which we refer to as domains well approximated by non characteristic hyperplanes and in omega we prove for 2 le p infty the boundary harnack inequality as well as the holder continuity for ratios of positive p harmonic functions vanishing on a portion of partial omega

regularity of lipschitz free boundaries in two phase problems for the p laplace operator in this paper we continue the study in ln6 concerning the regularity of the free boundary in a general two phase free boundary problem for the p laplace operator by proving regularity of the free boundary assuming that the free boundary is close to a lipschitz graph

a implies nta for a class of variable coefficient elliptic operators abstract we consider a certain class of second order variable coefficient divergence form elliptic operators in a uniform domain with ahlfors regular boundary and we show that the a property of the elliptic measure associated to any such operator and its transpose imply that the domain is in fact nta and hence chord arc the converse was already known and follows from work of kenig and pipher 30

sobolev w1p spaces on closed subsets of rn abstract for each p n we use local oscillations and doubling measures to give intrinsic characterizations of the restriction of the sobolev space w p 1 r n to an arbitrary closed subset of r n

free boundary regularity for harmonic measures and poisson kernels one of the basic aims of this paper is to study the relationship between the geometry of hypersurface like subsets of euclidean space and the properties of the measures they support in this context we show that certain doubling properties of a measure determine the geometry of its support a radon measure is said to be doubling with constant c if c times the measure of the ball of radius r centered on the support is greater than the measure of the ball of radius 2r and the same center we prove that if the doubling constant of a measure on r n 1 is close to the doubling constant of the ndimensional lebesgue measure then its support is well approximated by ndimensional a ne spaces provided that the support is relatively at to start with primarily we consider sets which are boundaries of domains in r n 1 the n dimensional hausdorfi measure may not be deflned on the boundary of a domain in r n 1 thus we turn our attention to the harmonic measure which is well behaved under minor assumptions see section 3 we obtain a new characterization of locally at domains in terms of the doubling properties of their harmonic measure see section 3 along these lines we investigate how the weak regularity of the poisson kernel of a domain determines the geometry of its boundary sections 5 and 6 pursue this goal as in alt and cafiarelli s work see ac c1 c2 and also jerison s j in both cases the goal is to prove that under the appropriate technical conditions at at points of the boundary the oscillation of the poisson kernel controls the oscillation of the unit normal vector the difierence between our work and the work in ac is that we measure the oscillation in an integral sense bmo estimates

conditional transformation of drift formula and potential theory for 1 2 b using conditional brownian motion and the transformation of drift formula of cameron martin girsarov maruyama we give integral conditions on a vector fieldb which imply the harmonic measures and green functions for 1 2 and 1 2 b on a bounded lipschitz domaind are equivalent by equivalent we mean there exist two sided inequalities with constants depending only onb andd this enables one to conclude the potential theory for 1 2 b ond and 1 2 ond are the same

boundary behavior of non negative solutions to degenerate sub elliptic equations let x x 1 x m be a system of c infinity vector fields in r n satisfying hormander s finite rank condition and let omega be a non tangentially accessible domain with respect to the carnot

boundary estimates for solutions to linear degenerate parabolic equations let omega subset mathbb r n be a bounded nta domain and let omega t omega times 0 t for some t 0 we study the boundary behaviour of non negativesolutions to the equation hu parti

boundary behavior of solutions to the parabolic p laplace equation ii abstract this paper is the second installment in a series of papers concerning the boundary behavior of solutions to the p parabolic equations in this paper we are interested in the short time behavior of the solutions which is in contrast with much of the literature where all results require a waiting time we prove a dichotomy about the decay rate of non negative solutions vanishing on the lateral boundary in a cylindrical c 1 1 domain furthermore we connect this dichotomy to the support of the boundary type riesz measure related to the p parabolic equation in nta domains which has consequences for the continuation of solutions

mean convergence of hermite fejer interpolation abstract weighted l p convergence of hermite fejer interpolation based on the zeros of generalized jacobi polynomials is investigated the main result of the paper gives necessary and sufficient conditions for such convergence for all continuous functions

the hadamard variational formula and the minkowski problem for p capacity abstract a hadamard variational formula for p capacity of convex bodies in r n is established when 1 p n the formula is applied to solve the minkowski problem for p capacity which involves a degenerate monge ampere type equation uniqueness for the minkowski problem for p capacity is established when 1 p n and existence and regularity when 1 p 2 these results are non linear extensions of the now classical solution of jerison of the minkowski problem for electrostatic capacity p 2

generalized 3g theorem and application to relativistic stable process on non smooth open sets let g x y and gd x y be the green functions of rotationally invariant symmetric stable process in rd and in an open set d respectively where 0 2 the inequality gd x y gd y z gd x z c g x y g y z is a very useful tool in studying local schrodinger operators when the above inequality is true with c c d 0 then we say that the 3g theorem holds in d in this paper we establish a generalized version of 3g theorem when d is a bounded fat open set which includes a bounded john domain the 3g we consider is of the form gd x y gd z w gd x w where y may be different from z when y z we recover the usual 3g the 3g form gd x y gd z w gd x w appears in non local schrodinger operator theory using our generalized 3g theorem we give a concrete class of functions belonging to the non local kato class introduced by chen and song on fat open sets as an application we discuss relativistic stable processes relativistic hamiltonian when 1 in fat open sets we identify the martin boundary and the minimal martin boundary with the euclidean boundary for relativistic stable processes in fat open sets furthermore we show that relative fatou type theorem is true for relativistic stable processes in fat open sets the main results of this paper hold for a large class of symmetric markov processes as are illustrated in the last section of this paper we also discuss the generalized 3g theorem for a large class of symmetric stable levy processes

the distribution of free path lengths in the periodic lorentz gas and related lattice point problems the periodic lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers and is one of the fundamental models for chaotic diffusion in the present paper we investigate the boltzmann grad limit where the radius of each scatterer tends to zero and prove the existence of a limiting distribution for the free path length we also discuss related problems such as the statistical distribution of directions of lattice points that are visible from a fixed position

quantitative analysis of qualitative data this paper presents an overview of an approach to the quantitative analysis of qualitative data with theoretical and methodological explanations of the two cornerstones of the approach alternating least squares and optimal scaling using these two principles my colleagues and i have extended a variety of analysis procedures originally proposed for quantitative interval or ratio data to qualitative nominal or ordinal data including additivity analysis and analysis of variance multiple and canonical regression principal components common factor and three mode factor analysis and multidimensional scaling the approach has two advantages a if a least squares procedure is known for analyzing quantitative data it can be extended to qualitative data and b the resulting algorithm will be convergent three completely worked through examples of the additivity analysis procedure and the steps involved in the regression procedures are presented

asymptotics for mixed dirichlet robin problems in irregular domains abstract we study the asymptotic behaviour of the solutions of two dimensional elliptic problems with robin boundary conditions on the prefractal curves approximating the koch curve type fractals

extending isotopies of planar continua in this paper we solve the following problem in the affirmative let z be a continuum in the plane c and suppose that h z x 0 1 c is an isotopy starting at the identity can h be extended to an isotopy of the plane we will provide a new characterization of an accessible point in a planar continuum z and use it to show that accessibility of a point is preserved during the isotopy we show next that the isotopy can be extended over small hyperbolic crosscuts which are shown to remain small under the isotopy the proof makes use of the notion of a metric external ray which mimics the notion of a conformal external ray but is easier to control during an isotopy it also relies on the existence of a partition of a hyperbolic simply connected domain u in the sphere into hyperbolically convex subsets which have limited distortion under conformal maps to the unit disk

einstein solvmanifolds are standard we study einstein manifolds admitting a transitive solvable lie group of isometries solvmanifolds it is conjectured that these exhaust the class of noncompact homogeneous einstein manifolds j heber has shown that under a simple algebraic condition he calls such a solvmanifold standard einstein solvmanifolds have many remarkable structural and uniqueness properties in this paper we prove that any einstein solvmanifold is standard by applying a stratification procedure adapted from one in geometric invariant theory due to f kirwan

boundary behavior of harmonic maps on non smooth domains and complete negatively curved manifolds abstract in this paper we develop a theory for harmonic maps which is analogous to the classical theory for bounded harmonic functions in harmonic analysis we solve the dirichlet problem in bounded lipschitz domains and manifolds with negative sectional curvatures generalize the wienner criterion and prove the fatou theorem for harmonic maps into convex balls

regularity and free boundary regularity for the p laplace operator in reifenberg flat and ahlfors regular domains regularity and free boundary regularity for the p laplace operator in reifenberg flat and ahlfors regular domains

doubling properties of second order parabolic equations we prove the doubling property of l caloric measure corresponding to the second order parabolic equation in the whole space and in lipschitz domains for parabolic equations in the divergence form a weaker form of the doubling property follows easily from a recent result the backward harnack inequality and known estimates of green s function our method works for both the divergence and nondivergence cases moreover the backward harnack inequality and estimates of green s function are not needed in the course of proof

uniform rectifiability carleson measure estimates and approximation of harmonic functions the first author was supported by nsf grant dms 1361701 the second author was supported in part by mineco grant mtm2010 16518 icmat severo ochoa project sev 2011 0087 he also acknowledges that the research leading to these results has received funding from the european research council under the european union s seventh framework programme fp7 2007 2013 erc agreement no 615112 hapdegmt the third author was supported by the alfred p sloan fellowship the nsf career award dms 1056004 the nsf inspire award dms 1344235 and the nsf materials research science and engineering center seed grant dmr 0212302

singularly perturbed elliptic systems and multi valued harmonic functions with free boundaries here we study the asymptotic limits of solutions of some singularly perturbed elliptic systems the limiting problems involve multiple valued harmonic functions or in general harmonic maps to singular spaces and free interfaces between supports of various components of the maps the main results of the paper are the uniform lipschitz regularity of solutions as well as the regularity of free interfaces

erratum for boundedness of families of canonically polarized manifolds a higher dimensional analogue of shafarevich s conjecture we show that the number of deformation types of canonically polarized manifolds over an arbitrary variety with proper singular locus is nite and that this number is uniformly bounded in any nite type family of base varieties as a corollary we show that a direct generalization of the geometric version of shafarevich s original conjecture holds for innitesimally rigid families of canonically polarized varieties

on time dependent domains for the degenerate p parabolic equation carleson estimate and holder continuity in this paper we propose a definition of parabolic nta for solutions to the degenerate p parabolic equation given this definition we prove the carleson estimate originally proved for this equation in avelin et al j eur math soc 2015 for cylindrical domains moreover we study a non optimal stronger outer corkscrew condition such that we obtain holder continuity up to the boundary for non negative solutions vanishing on a part of the boundary

a boundary harnack principle in twisted holder domains the boundary harnack principle for the ratio of positive harmonic functions is shown to hold in twisted holder domains of order a for a e 1 2 1 for each a e 0 1 2 there exists a twisted h6lder domain of order a for which the boundary harnack principle fails extensions are given to l harmonic

boundary behavior and the martin boundary problem for p harmonic functions in lipschitz domains boundary behavior and the martin boundary problem for p harmonic functions in lipschitz domains

a free boundary problem on three dimensional cones abstract we consider a free boundary problem on cones depending on a parameter c and study when the free boundary is allowed to pass through the vertex of the cone we show that when the cone is three dimensional and c is large enough the free boundary avoids the vertex we also show that when c is small enough but still positive the free boundary is allowed to pass through the vertex this establishes 3 as the critical dimension for which the free boundary may pass through the vertex of a right circular cone in view of the well known connection between area minimizing surfaces and the free boundary problem under consideration our result is analogous to a result of morgan that classifies when an area minimizing surface on a cone passes through the vertex

birth of homoclinic intersections a model for the central dynamics of partially hyperbolic systems creation d intersection homoclines un modele pour la dynamique centrale des systemes partiellement hyperboliques nous montrons une conjecture de j palis tout diffeomorphisme d une variete compacte peut etre approche en topologie c 1 par un diffeomorphisme morse smale ou par un diffeomorphisme ayant une intersection homocline transverse