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

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

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

convergence of spectra of uniformly fattened open book structures abstract we consider a stratified compact surface m in r 3 which we call an open book structure whose singular points form simple closed smooth curves and its neighborhood m which we call a fattened open book structure assuming a transversality condition at the singularities and absence of zero dimensional strata i e corners we show that the discrete spectrum of the neumann laplacian in m converges when 0 to the spectrum of a differential operator on m similar results have been obtained before for the case of fattened graphs i e m being one dimensional in the case of a smooth surfaces m the problem has been studied well however having singularities along strata of lower dimensions significantly complicates considerations as in the quantum graph case such considerations are triggered by various applications

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

higher integrability result for nonlinear elliptic systems with conormal boundary conditions abstract we prove a boundary higher integrability result for nonlinear elliptic systems with conormal boundary conditions in locally uniform domains to do it we derive a boundary version of gehring giaquinta modica lemma and sobolev poincare type inequality in locally uniform domains our result plays a key role for handling the coefficients when obtaining caldero n zygmund type estimates with conormal boundary conditions in nonsmooth domains

energy entropy inequalities for classical lattice systems a new characterization of equilibrium states for classical lattice systems is given in terms of correlation inequalities their physical meaning is found to express thermodynamic stability we demonstrate the applicability of the inequalities in specific models

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

the kato square root problem for mixed boundary conditions we consider the negative laplacian subject to mixed boundary conditions on a bounded domain we prove under very general geometric assumptions that slightly above the critical exponent 12 its fractional power domains still coincide with suitable sobolev spaces of optimal regularity in combination with a reduction theorem recently obtained by the authors this solves the kato square root problem for elliptic second order operators and systems in divergence form under the same geometric assumptions thereby we answer a question posed by j l lions in 1962 30

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

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

lower order positivity and holomorphic operators abstract questions concerning holomorphic extensions of operator valued functions in domains d or complex manifolds of c n are studied conditions for such an extension are formulated in terms the of positive definiteness of order 3 of certain hermitian kernels on d d the results of this study constitute a generalization and an extension of previous results of hindmarsh and fitzgerald and horn and recent results of burbea

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

degree independent sobolev extension on locally uniform domains abstract we consider the problem of constructing extensions l k p l k p r n where l k p is the sobolev space of functions with k derivatives in l p and r n is a domain in the case of lipschitz calderon gave a family of extension operators depending on k while stein later produced a single k independent operator for the more general class of locally uniform domains which includes examples with highly non rectifiable boundaries a k dependent family of operators was constructed by jones in this work we produce a k independent operator for all spaces l k p on a locally uniform domain

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

statistical mechanics of elastic moduli we consider continuous systems of particles in the framework of classical statistical mechanics and derive a general expression for the static elastic moduli tensor in terms of correlation functions we find sufficient conditions for the vanishing of the shear modulus relationships between these conditions and others insuring translational or rotational invariance are discussed

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

mathematical basis for a general theory of laplacian transport towards irregular interfaces the theory of laplacian transport towards and across irregular surfaces is reformulated in terms of the dirichlet to neumann operator and its spectral characteristics this permits us to obtain an exact equivalent circuit for the impedance of a working interface of arbitrary shape the important result is that only very few eigenmodes of this operator do govern the entire response of a macroscopic system this property drastically simplifies the understanding of irregular or prefractal interfaces the results can be applied in electrochemistry physiology and chemical engineering fields where exchange processes across surfaces with complex geometry are ubiquitous

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

lattice systems with a continuous symmetry iii low temperature asymptotic expansion for the plane rotator model we prove that the expansion in powers of the temperature t of the correlation functions and the free energy of the plane rotator model on a d dimensional lattice is asymptotic to all orders in t the leading term in the expansion is the spin wave approximation and the higher powers are obtained by the usual perturbation series we also prove the inverse power decay of the pair correlation at low temperatures for d 3

on the absence of spontaneous symmetry breaking and of crystalline ordering in two dimensional systems we develop a unified approach based on araki s relative entropy concept to proving absence of spontaneous breaking of continuous internal symmetries and translation invariance in two dimensional statistical mechanical systems more precisely we show that under rather general assumptions on the interactions all equilibrium states of a two dimensional system have all the symmetries compact internal and spatial of the dynamics except possibly rotation invariance rotation invariance remains unbroken if connected correlations decay more rapidly than the inverse square distance we also prove that two dimensional systems with a non compact internal symmetry group like anharmonic crystals typically do not have gibbs states

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

insulating layers and robin problems on koch mixtures abstract this paper deals with a reinforcement problem for a plane domain whose boundary is a deterministic or random mixture of self similar koch curves we construct an e thin polygonal 2 dimensional fiber e n n n 0 e 1 around pre fractal approximating domains n and related suitable energy functionals the aim of this paper is to study the asymptotic behavior of the reinforced energy functionals while simultaneously the thickness of the fibers and the conductivity of the functionals on the fibers converges to 0 as 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

on the symmetry of the gibbs states in two dimensional lattice systems under fairly general conditions if a two dimensional classical lattice system has an internal symmetry groupg which is a compact connected lie group then all gibbs states areg invariant

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

positive definiteness and its applications to interpolation problems for holomorphic functions on etudie le caractere defini positif de certains noyaux en exploitant une identification naturelle de ces noyaux avec des operateurs sur un espace de hilbert approprie

euclidean invariance in statistical mechanics of classical continuous system it is shown that equilibrium states of classical particles with short range interactions are euclidean invariant whenever their correlation functions have a clustering which is integrable the relation between invariance and clustering is analysed for spatial rotations and internal rotational degrees of freedom the analysis is then extended to the case of long range interactions including the coulomb force and jellium systems

heat kernel for the elliptic system of linear elasticity with boundary conditions abstract we consider the elliptic system of linear elasticity with bounded measurable coefficients in a domain where the second korn inequality holds we construct heat kernel of the system subject to dirichlet neumann or mixed boundary condition under the assumption that weak solutions of the elliptic system are holder continuous in the interior moreover we show that if weak solutions of the mixed problem are holder continuous up to the boundary then the corresponding heat kernel has a gaussian bound in particular if the domain is a two dimensional lipschitz domain satisfying a corkscrew or non tangential accessibility condition on the set where we specify dirichlet boundary condition then we show that the heat kernel has a gaussian bound as an application we construct green s function for elliptic mixed problem in such a domain

lip j extension domains and uniform domains in this paper we prove the following two results 1 a domain d r n is a uniform domain if and only if d is a lip j extension domain 2 a jordan domain d r 2 is a quasidisk if and only if f lip j d whenever is analytic in d with f z d z d 1 in here j x 1 x 2 1 2 log l x 1 x 2 d x 1 d l x 1 x 2 d x 2 d for x 1 x 2 d

lattice systems with a continuous symmetry we prove that the expansion in powers of the temperature t of the correlation functions and the free energy of the plane rotator model on a d dimensional lattice is asymptotic to all orders in t the leading term in the expansion is the spin wave approximation and the higher powers are obtained by the usual perturbation series we also prove the inverse power decay of the pair correlation at low temperatures for d 3

zero frequency elastic moduli of uniform fluids a statistical mechanical treatment of equilibrium elasticity of a uniform fluid phase based on density functional theory is presented bulk expressions for the stress tensor and the zero frequency elastic moduli tensor involving the direct correlation function are found

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

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

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 new proof of the stillinger lovett complete shielding condition it is shown that any equilibrium state of classical charged particles with correlation having a spatial decay faster than 1 x v 2 in dimensionv 2 3 obeys the stillinger lovett second moment condition under the same clustering hypothesis arbitrary localized external charge distributions are completely shielded

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

green s functions for elliptic and parabolic systems with robin type boundary conditions abstract the aim of this paper is to investigate green s function for parabolic and elliptic systems satisfying a possibly nonlocal robin type boundary condition we construct green s function for parabolic systems with time dependent coefficients satisfying a possibly nonlocal robin type boundary condition assuming that weak solutions of the system are locally holder continuous in the interior of the domain and as a corollary we construct green s function for elliptic system with a robin type condition also we obtain gaussian bound for robin green s function under an additional assumption that weak solutions of robin problem are locally bounded up to the boundary we provide some examples satisfying such a local boundedness property and thus have gaussian bounds for their green s functions

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

non lipschitz uniform domain shape optimization in linear acoustics we introduce new parametrized classes of shape admissible domains in r n n 2 and prove that they are compact with respect to the convergence in the sense of characteristic functions the hausdorff sense the sense of compacts and the weak convergence of their boundary volumes the domains in these classes are bounded e domains with possibly fractal boundaries that can have parts of any non uniform hausdorff dimension greater or equal to n 1 and less than n we prove the existence of optimal shapes in such classes for maximum energy dissipation in the framework of linear acous tics a by product of our proof is the result that the class of bounded e domains with fixed e is stable under hausdorff convergence an additional and related result is the mosco convergence of robin type energy functionals on converging 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