critical properties of the classical xy and classical heisenberg models a renormalization group study abstract by using two approaches of renormalization group rg mean field rg mfrg and effective field rg efrg we study the critical properties of the simple cubic lattice classical xy and classical heisenberg models the methods are illustrated by employing its simplest approximation version in which small clusters with one n 1 and two n 2 spins are used the thermal and magnetic critical exponents y t and y h and the critical parameter k c are numerically obtained and are compared with more accurate methods monte carlo series expansion and e expansion the results presented in this work are in excellent agreement with these sophisticated methods we have also shown that the exponent y h does not depend on the symmetry n of the hamiltonian hence the criteria of universality for this exponent is only a function of the dimension d

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

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

analytic dependence of riemann mappings for bounded domains and minimal surfaces in this paper we prove that the functional which takes a closed lavrentiev curve to the corresponding riemann mapping is locally lipl on the set of all closed lavrentiev curves this set is a subset of bmo t it is however not open in bmo t we also prove that the previous functional is analytic for certain classes of closed lavrentiev curves including the class of curves which have some symmetry with respect to the unit circle these classes of curves are submanifolds of bmo t finally we consider the functional which takes a lavrentiev curve closed or not in n dimensional euclidean space to the corresponding minimal surface and we study the differentiability and analyticity of this functional on certain function spaces 1993 john wiley sons inc

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

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

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

weil petersson teichmuller space iii dependence of riemann mappings for weil petersson curves the classical riemann mapping theorem implies that there exists a so called riemann mapping which takes the upper half plane onto the left domain bounded by a jordan curve in the extended complex plane the primary purpose of the paper is to study the basic problem how does a riemann mapping depend on the corresponding jordan curve we are mainly concerned with those jordan curves in the weil petersson class namely the corresponding riemann mappings can be quasiconformally extended to the whole plane with beltrami coefficients being square integrable under the poincare metric after giving a geometric characterization of a weil petersson curve we endow the space of all normalized weil petersson curves with a new real hilbert manifold structure in a geometric manner and show that this new structure is topologically equivalent to the standard complex hilbert manifold structure which implies that an appropriately chosen riemann mapping depends continuously on a weil petersson curve and vice versa this can be considered as the first result about the continuous dependence of riemann mappings on non smooth jordan curves

on fatou bieberbach domains a fatou bieberbach domain is a domain in c2 which is a biholomorphic image of c2 and is not all of c2 the existence of such domains has been known for a long time there are several papers dealing with fatou bieberbach domains bf bs de e fs gl k n1 n2 rr1 rr2 s nevertheless they remain quite misterious objects and there are many open questions about them one of them is the following rr1 q 11 p 79 if is a fatoubieberbach domain and l is a complex line is it possible that a l is connected b l has finitely many components c l is a circular disc let be the open unit disc in c and let p suppose one wants to obtain a fatou bieberbach domain whose intersection with the z axis z 0 z c is approximately the unit disc one would try to find such that

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

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

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

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

well posedness for the navier stokes equations where u is the velocity and p is the pressure it is well known that the navierstokes equations are locally well posed for smooth enough initial data as long as one imposes appropriate boundary conditions on the pressure at for instance it is easy to see see 9 for much more general results that if s n 2 then for any h initial data there exists a unique c 0 t h r local solution with a pressure p c 0 t h r in the sequel we consider solutions for less regular initial data this has to be understood in the sense that the map from the initial data to the solution extends continuously to rougher function spaces the question we are interested in is the global well posedness for small data and local well posedness for large data with respect to a certain space of

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

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

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

coherent anomaly method for classical heisenberg model the power series coherent anomaly method is applied to study the critical properties of a classical heisenberg model the values of true critical temperature tc are obtained using these results the estimation of critical exponent for the zero field static susceptibility has been made the results for tc are in good agreement with those obtained from the ratio method and the pade approximant analysis of the direct susceptibility series but the results for are found to be different it is seen that for bcc and fcc lattices is approximately equal to 43 while for the sc lattice 2 43 in disagreement with the mean experimental value of 43 with the proposal of a possible correction due to confluent singularities for sc model we obtain the following expression for susceptibility a 1 tc 43 1 b 1 tc with xc xcxc xc jkbtc kb being the boltzmann constant j the nearest neighbour exchange constant tcc the critical temperature b and a are numerical constants the confluent correction has been found to be 0 42 for the sc lattice and non existent in bcc and fcc lattices

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

well posedness in sobolev spaces of the full water wave problem in 2 d we consider the motion of the interface of 2 d irrotational incompressible inviscid water wave with air above water and surface tension zero we show that the interface is always not accelerating into the water region normal to itself as rapidly as the normal acceleration of gravity as long as the interface is nonself intersect we therefore obtain the existence and uniqueness of solutions of the full water wave problem locally in time for any initial interface which is nonself intersect including the case that the interface is of multiple heights

oka properties of ball complements let n 1 be an integer we prove that holomorphic maps from stein manifolds x of dimension n to the complement mathbb c n setminus l of a compact convex set l subset mathbb c n satisfy the basic oka property with approximation and interpolation if l is polynomially convex then the same holds when 2 dim x le n we also construct proper holomorphic maps immersions and embeddings x rightarrow mathbb c n with additional control of the range thereby extending classical results of remmert bishop and narasimhan

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

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

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

equivalence of the o n vector ferromagnetic and antiferromagnetic models abstract the effective field renormalization group efrg approach is used to find the neel temperature t n of the o n vector model with antiferromagnetic af interaction the efrg method is illustrated by employing approximations in which clusters with one n 1 and two n 2 spins are used the critical temperature t n is obtained as a function of component n and coordination z numbers for all values of n and z we show that t n t c where t c is the curie temperature for the ferromagnetic f case as a comparison the results of the quantum heisenberg model n 3 with f and af interactions are also presented and we find that t n t c which is different from the classical result t c t n

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

high temperature series analyses of the classical heisenberg and xy models abstract although there is now a good measure of agreement between monte carlo and high temperature series expansion estimates for ising n 1 models published results for the critical temperature from series expansions up to 12th order for the three dimensional classical heisenberg n 3 and xy n 2 models do not agree very well with recent high precision monte carlo estimates in order to clarify this discrepancy we have analyzed extended high temperature series expansions of the susceptibility the second correlation moment and the second field derivative of the susceptibility which have been derived a few years ago by luscher and weisz for general o n vector spin models on d dimensional hypercubic lattices up to 14th order in k j kbt by analyzing these series expansions in three dimensions with two different methods that allow for confluent correction terms we obtain good agreement with the standard field theory exponent estimates and with the critical temperatures estimates from the new high precision monte carlo simulations furthermore for the heisenberg model we also reanalyze existing series for the susceptibility on the bcc lattice up to 11th order and on the fcc lattice up to 12th order using the same methods

critical behaviour of the 3d xy model a monte carlo study we present the results of a study of the three dimensional xy model on a simple cubic lattice using the single cluster updating algorithm combined with improved estimators we have measured the susceptibility and the correlation length for various couplings in the high temperature phase on lattices of size up to l 112 at the transition temperature we studied the fourth order cumulant and other cumulant like quantities on lattices of size up to l 64 from our numerical data we obtain for the critical coupling kc 0 45420 2 and for the static critical exponents v 1 976 6 and v 0 662 7

a model for two dimensional orientational order abstract we use a generalization of kikuchi s cluster variational method cvm in the pair approximation to study anisotropic plane rotors on a square lattice as a simple model for two dimensional nematics the system was shown to exhibit nematic ordered states at low temperature disordering for all anisotropies through a continuous transition the orientational distribution functions order parameter internal energy and specific heat were calculated within the cvm and compared with the results of a recent computer simulation estimates for the non classical exponents were obtained using a mean field renormalization group analysis a spin wave argument was also used to investigate the role of long range fluctuations near the isotropic fixed point the accuracy of these different techniques is discussed within the context of our model