semiclassical analysis of low lying eigenvalues ii tunneling we discuss the leading asymptotics of eigenvalue splittings of 2 x2v in the limit as x o and where v is a non negative potential with several zeros for example if eo x e1 x are the two lowest eigenvalues in a situation where v has precisely two zeros a and b related by a symmetry then limx x lin e1 x eo x is given as the distance from a to b in a certain riemann metric

intrinsic ultracontractivity and eigenfunction estimates for schrodinger operators let d be a domain in rn n 2 and let h h0 v where h0 is a divergence form uniformly elliptic operator with dirichlet boundary conditions and v is in the kato class intrinsic ultracontractivity is proved for the semigroup of h when d is holder domain of order 0 or a uniformly holder domain of order for 0 2 for every 2 there exists a uniformly holder domain of order for which the dirichlet laplacian is not intrinsically ultracontractive for a large class of domains it is shown that the heat kernels for h0 and h decay at the same rate at the boundary applications are given to the lifetime of conditioned brownian motion some of our results seem to be new even for smooth domains

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

on the matrix equation ak j i we concern ourselves with the problem of solving the 0 1 matrix equation ak j i in this paper where j is the matrix of all one s i the identity matrix and a an unknown 0 1 matrix in particular our effort brings about a complete solution of ak j2k 1 i2k 1 this generalizes a theorem of lam and van lint in the course of our solution we provide a local characterization of the webs i e the powers of the cycles our results mainly rely on the analysis of the intersection pattern of a collection of some specific sets namely the row sets of a matrix some results on partitionable graphs are also introduced to tackle this problem our work suggests an approach to investigate ak j i by studying a number theoretical question and a conjecture of ravindra on partitionable graphs several open problems are also presented

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

intrinsic ultracontractivity of a schrodinger semigroup in rn abstract we give a possibly sharp sufficient condition on the electric potential q r n 0 in the schrodinger operator a q x on l 2 r n that guarantees that the schrodinger heat semigroup e a t t 0 on l 2 r n generated by a is intrinsically ultracontractive moreover if q x q x is radially symmetric we show that our condition on q is also necessary i e truly sharp it reads r 0 q r 1 2 d r for some r 0 0 our proofs make essential use of techniques based on a logarithmic sobolev inequality rosen s inequality proved via a new fenchel young inequality and a very precise asymptotic formula due to hartman and wintner

a characterization of operator algebras abstract an operator algebra is a uniformly closed algebra of bounded operators on a hilbert space in this paper we give a characterization of unital operator algebras in terms of their matricial norm structure more precisely if a is an l matricially normed space and also an algebra with a completely contractive multiplication and an identity of norm 1 then there is a completely isometric isomorphism of a onto a unital operator algebra indeed the multiplication on a need not be assumed to be associative for this conclusion to follow examples are given to show that the condition on the identity is necessary it follows from the above that the quotient of an operator algebra by a closed two sided ideal with the natural matricial structure is again an operator algebra up to complete isometric isomorphism

on minimal parabolic functions and time homogeneous parabolic h transforms burdzy was supported in part by nsf grant dms 9700721 saliosbury was supported in part by a grant from nserc a portion of this research took place during his stay at the fields institute

spectral properties of metastable markov semigroups abstract it is shown that if a symmetric markov semigroup e ht on the hilbert space l 2 x is hypercontractive then the approximate degeneracy of the ground state has several consequences concerning other parts of the spectrum of h and concerning the unitary group e iht in particular in the presence of a space inversion symmetry all the eigenvalues occur in pairs with gaps comparable to the gap between the bottom two eigenvalues

potential theory of truncated stable processes for any alpha in 0 2 a truncated symmetric stable process is a symmetric levy process in mathbb r d with a levy density given by c x d alpha 1 x 1 for some constant c in this paper we study the potential theory of truncated symmetric stable processes in detail we prove a harnack inequality for nonnegative harmonic functions of these processes we also establish a boundary harnack principle for nonnegative functions which are harmonic with respect to these processes in bounded convex domains we give an example of a non convex domain for which the boundary harnack principle fails

a short proof of boundary harnack principle abstract we give a direct analytic proof of the classical boundary harnack principle for solutions to linear uniformly elliptic equations in either divergence or non divergence form

nonuniqueness of the positive dirichlet problem for parabolic equations in cylinders abstract we investigate uniqueness of nonnegative solutions to parabolic equations in cylinders of r n 1 with zero dirichlet boundary condition where bases of cylinders are unbounded domains we give necessary and or sufficient conditions for the uniqueness in terms of geometric properties of the bases

boundary behaviour of dirichlet eigenfunctions of second order elliptic operators we investigate the rate of decay of solutions of a second order elliptic equation tf f in an arbitrary domain r n which satisfy zero boundary conditions and then apply the results to a study of the behaviour of the associated heat semigroup in various weighted l 2 spaces our results concern the behaviour at infinity and at points on the boundary of especially irregular ones like corners and cusps the decay rates we obtain are expressed in terms of weighted l 2 estimates which are governed by the sharpness of the cusps and the properties of the coefficients of also the meaning of the zero boundary condition at a cusp is addressed

sharp estimates for dirichlet eigenfunctions in horn shaped regions we prove sharp exponential decay for the dirichlet eigenfunctions in horn shaped regions in the plane the estimate is obtained using a method of carleman which is widely used in the study of harmonic measure such estimates can be applied to study the intrinsic ultracontractivity properties of the heat semigroup for such regions

boundary harnack principle for δ δ 2 for d 1 and 2 0 2 consider the family of pseudo differential operators f b 2 b 2 0 1 g on r d that evolves continuously fromto 2 in this paper we establish a uniform boundary harnack principle bhp with explicit boundary decay rate for nonnegative functions which are harmonic with respect to b 2 or equivalently the sum of a brownian motion and an independent symmetric stable process with constant multipleb 1 inc 1 1 open sets here a uniform bhp means that the comparing constant in the bhp is independent of b 2 0 1 along the way a uniform carleson type estimate is established for nonnegative functions which are harmonic with respect to b 2 in lipschitz open sets our method employs a combination of probabilistic and analytic techniques

boundary harnack principle for brownian motions with measure valued drifts in bounded lipschitz domains let mu mu 1 dots mu d be such that each mu i is a signed measure on r d belonging to the kato class k d 1 a brownian motion in r d with drift mu is a diffusion process in r d whose generator can be informally written as frac12 delta mu cdot nabla when each mu i is given by u i x dx for some function u i a brownian motion with drift mu is a diffusion in r d with generator frac12 delta u cdot nabla in kim and song ill j math 50 3 635 688 2006 some properties of brownian motions with measure value drifts in bounded smooth domains were discussed in this paper we prove a scale invariant boundary harnack principle for the positive harmonic functions of brownian motions with measure value drifts in bounded lipschitz domains we also show that the martin boundary and the minimal martin boundary with respect to brownian motions with measure valued drifts coincide with the euclidean boundary for bounded lipschitz domains the results of this paper are also true for diffusions with measure valued drifts that is when delta is replaced by a uniformly elliptic divergence form operator sum i j 1 d partial i a ij partial j with c 1 coefficients or a uniformly elliptic non divergence form operator sum i j 1 d a ij partial i partial j with c 1 coefficients

g circulant solutions to the 0 1 matrix equation am jn this paper discusses the structure of g circulant solutions to a m jn where a is an unknown 0 1 matrix and jn is a matrix of order n with all entries equal to 1 wang has made a conjecture on the form of all such solutions partially verifying his conjecture we discover a close relationship among the hall polynomial a x the shifting parameter g and the order n of any 0 1 g circulant solution a to a m jn as a consequence all the g circulant solutions to a m jn are completely determined in the case that n is a prime power moreover in the case that the constant line sum r of a is square free all g circulant solutions to a m jn are proved to be permutation similar to the adjacency matrix of the de bruijn digraph b r m motivated by the current status of this subject we identify all 0 1 g circulant solutions to a m jn whose hall polynomials have some specific properties and we further determine the possible values that the shifting parameter g of such solutions may take the uniqueness of these solutions up to isomorphism is also investigated our paper is concluded with some open problems in particular we give the concept of standard factorization and conjecture that all factorizations of x n 1 x 1 into a product of 0 1 polynomials must be standard and thus point out the close similarity between wang s conjecture and a conjecture appearing in the study of perfect graph 2002 elsevier science inc all rights reserved

holder domains and poincare domains a domain d c rd of finite volume is said to be a p poincare domain if there is a constant mp d so that fu udii dx d then d is a p poincare domain this answers a question of axler and shields regarding the image of the unit disk under a holder continuous conformal mapping we also consider geometric conditions which imply that the imbedding of the sobolev space w p d lp d is compact and prove that this is the case for a holder domain d

spectra and elementary cycles of the digraphs with unique paths of fixed length abstract a digraph g whose adjacency matrix a satisfies a k j n i n where j n is the n n matrix of all ones is called a digraph with unique paths of fixed length k or simply a upfl k digraph we prove that all the upfl k digraphs of the same order are co spectral and have the same number of elementary cycles of length l for each l k we also provide some techniques helpful for computing the spectrum and the numbers of short elementary cycles of a upfl digraph including the determination of the numbers of reentrant paths of every fixed length in a upfl digraph at the end of the paper we point out an interesting relation between the number of elementary cycles of the upfl digraphs and the number of circular sequences with equal length and period our theorems generalize corresponding results of lam and van lint

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

lifetimes of conditioned diffusions we investigate when an upper bound on expected lifetimes of conditioned diffusions associated with elliptic operators in divergence and non divergence form can be found the critical value of the parameter is found for each of the following classes of domains l p domains p n 1 uniformly regular twistedl p domains p n 1 and twisted holder domains 1 3 a related parabolic boundary harnack principle is proved

sharp estimates for the green function in lipschitz domains we give sharp global estimates for the green function and martin kernel in lipschitz domains we apply the estimates to obtain short proofs of some classical results

heat kernel eigenfunctions and conditioned brownian motion in planar domains abstract some recent results concerning uniform convergence of the shape of the heat kernel to that of the first eigenfunction and the lifetime of h processes in bounded lipschitz domains in r n are extended to all planar domains of finite area

recovery based error estimator for stabilized finite element method for the stationary navier stokes problem a recovery based error estimator is proposed and analyzed for stabilized p 1 p 0 continuous linear velocity constant pressure finite element approximations to the stationary navier stokes problem we establish the reliability and efficiency of the error estimator a crucial part of this work is the estimation for the nonlinear term of the navier stokes problem it turns out such a term can be bounded by the recovery based error estimator numerical results are provided to illustrate the performance of the error estimator

gaussian upper bounds for the heat kernels of some second order operators on riemannian manifolds abstract we describe a method of obtaining gaussian upper bounds on heat kernels which unifies and improves recent results for hypoelliptic operators in divergence form and for laplace beltrami operators on complete riemannian manifolds

uniform bounds for solutions to quasilinear parabolic equations abstract we consider a class of quasilinear parabolic equations whose model is the heat equation corresponding to the p laplacian operator u pu di 1 i u p 2 iu with p 2 d on a domain d r d of finite measure we prove that u t x c d t u0 r for all t 0 x d and for all initial data u0 lr d provided r is not smaller than a suitable r0 where are positive constants explicitly computed in terms of d p r the nonlinear cases associated with the case p 2 display exactly the same contractivity properties which hold for the linear heat equation we also show that the nonlinear evolution considered is contractive on any lq space for any q 2

uniformly integrable operators and large deviations for markov processes abstract in this paper we introduce and investigate the notion of uniformly integrable operators on l p e its relations to classical compactness and hypercontractivity are exhibited several consequences of this notion are established such as perron frobenius type theorems independence on p of the spectral radius in l p p continuity of spectral radius and especially the existence of spectral gap in the irreducible case we also present some infinitesimal criteria ensuring the uniform integrability of a positive semigroup or of its resolvent for a essentially irreducible markov process we show that the uniform integrability in l p of some type of resolvent associated with the transition semigroup implies the large deviation principle of level 3 with some rate function given by a modified donsker varadhan entropy functional we also prove that the uniform integrability condition becomes even necessary in the symmetric case finally we present several applications of our results to feynman kac semigroups to the thermodynamical limits of grand ensembles and to non symmetric markov processes given by girsanov s formula

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

the boundary behavior of heat kernels of dirichlet laplacians we prove the following complete and qualitatively sharp description of heat kernels g of dirichlet laplacians on bounded c1 1 domains d there exist positive constants c1 c2 and t 0 depending on d such that for x dist x d x y t 1c1tn 2e c2 x y 2 t g x t y 0 x y t 11c1tn 2e x y 2 c2t for all x y d and 0 t are known bounds when d is unbounded are also given

an approach to solving ak j i abstract we conjecture that under the permutation similar equivalence relation there are exactly k solutions a to the matrix equation a k j d k 1 i d k 1 where is euler s totient function d 1 is an integer k 0 is an odd integer j is the matrix of all ones i is the identity matrix and a is an unknown 0 1 matrix we present an approach to verify this conjecture it establishes a connection between the work of solving the matrix equation a k j i and the problems of both determining the structure of near k factor factorizations of cyclic groups and characterizing cycle powers we also collect some results about the latter two problems in order to give more insight into this approach

eigenfunctions of the koch snowflake domain we obtain non tangential boundary estimates for the dirichlet eigenfunctions n and their gradients n for a class of planar domains with fractal boundaries this class includes the quasidiscs and in particular snowflake type domains when applied to the case when is the koch snowflake domain one of our main results states that 1 tends to or 0 as approaches certain types of boundary points where 1 0 denotes the ground state eigenfunction of the dirichlet laplacian on more precisely let ob resp ac denote the set of boundary points which are vertices of obtuse resp acute angles in an inner polygonal approximation of the snowflake curve then given eob resp e ac we show that 1 resp 0 as tends to in within a cone based at moreover we show that blowup of 1 also occurs at all boundary points in a cantor type set these results have physical relevance to the damping of waves by fractal coastlines as pointed out by sapovalet al in their experiments on the koch drum

a harnack type inequality for non symmetric markov semigroups abstract for a strong feller and irreducible markov semigroup on a locally compact polish space the harnack type inequality 1 1 holds if and only if the semigroup has a unique invariant probability measure and is ultracontractive moreover new sufficient conditions for this inequality to hold as well as upper bound estimates of the underlying constant are presented for diffusion semigroups on riemannian manifolds

heat kernel bounds for second order elliptic operators on riemannian manifolds 1 introduction we investigate the relationship between ultracontractive bounds on heat kernels weighted sobolev inequalities and logarithmic sobolev inequalities in a more general context than previously we find conditions which imply these bounds for a wide class of second order elliptic operators on manifolds we apply our result to laplacebeltrami operators on manifolds with cusps and to operators on a manifold whose coefficients become degenerate or infinite on the boundary of the manifold in an earlier paper with b simon 11 we introduced the notion of intrinsic ultracontractivity for schrodinger operators on rn and for the dirichlet laplacian on certain regions q in rn kusuoka and stroock 13 independently discovered the same phenomena for a somewhat different class of operators using a probabilistic intuition at the end of their paper they remarked that their ideas could be extended to manifolds with

the liouville property and a conjecture of de giorgi we consider bounded entire solutions of the nonlinear pde u u u 3 0i n r d and prove that under certain monotonicity conditions these solutions must be constant on hyperplanes the proof uses a liouville theorem for harmonic functions associated with a nonuniformly elliptic divergence form operator c 2000 john wiley sons inc

sharp bounds for ground state eigenfunctions on domains with horns and cusps abstract we prove a lower bound on the ground state eigenfunction of the dirichlet laplacian for horn shaped and cusped domains the novelty of the method is that for horn shaped domains it implies a new result for dimension two and more importantly is valid for higher dimensions moreover the same technique applies just as easily to cusps finally we construct an example to show that the estimates are sharp

large time behavior of the lp norm of schrodinger semigroups abstract letv 0 v c0 rv with v 3 be such that h 1 2 v 0 but for any e 0 1 2 1 e v is not positive we determine the exact rate of divergence of the norm of e th as a map from l to l a number of related problems are discussed

no triple point of planar brownian motion is accessible burdzy s research supported in part by nsf grant dms 9322689 and ams centennial research fellowship werner s research supported by an eec hcm leibniz fellowship

an invariance principle for branching diffusions in bounded domains we study branching diffusions in a bounded domain d of mathbb r d in which particles are killed upon hitting the boundary partial d it is known that any such process undergoes a phase transition when the branching rate beta exceeds a critical value a multiple of the first eigenvalue of the generator of the diffusion we investigate the system at criticality and show that the associated genealogical tree when the process is conditioned to survive for a long time converges to aldous continuum random tree under appropriate rescaling the result holds under only a mild assumption on the domain and is valid for all branching mechanisms with finite variance and a general class of diffusions

on the matrix equation al al k jn abstract this paper studies the matrix equation a l a l k j n where l k are nonnegative integers j n is the n n matrix of all l s and a is an unknown 0 1 matrix we shall provide a solution for every odd k and every n which is feasible i e n d l d l k for some nonnegative integer d and show that the equation has no solution in other cases with some trivial cases excluded we also show that for any solution a to this equation there must be a 0 1 matrix c satisfying i c k j d k 1 and a the associated digraph of a is the l th iterated line digraph of c in particular the well known kautz digraph k d l 1 can be characterized as a where a satisfies a l a l 1 j n for n d l d l 1

finite difference approximation of space fractional diffusion problems a mathematical analysis is presented to establish the convergence of the matrix transformation or matrix transfer method for the finite difference approximation of space fractional diffusion problems combined this with an implicit euler time discretization the optimal order convergence is proved with respect to the discrete l2 and the maximum norm the analysis is performed on general two and three dimensional domains with homogeneous boundary conditions the corresponding error estimates are illustrated with some numerical experiments

on the lp independence of the spectrum of the hodge laplacian on non compact manifolds abstract the central aim of this paper is the study of the spectrum of the hodge laplacian on differential forms of any order k in l p the underlying space is a c smooth open manifold m n with ricci curvature bounded below and uniformly subexponential volume growth it will be demonstrated that on such manifolds the l p spectrum of the hodge laplacian on differential k forms is independent of p for 1 p whenever the weitzenbock tensor on k forms is also bounded below it follows as a corollary that the isolated eigenvalues of finite multiplicity are l p independent the proof relies on the existence of a gaussian upper bound for the heat kernel of the hodge laplacian by considering the l p spectra on the hyperbolic space h n 1 we conclude that the subexponential volume growth condition is necessary in the case of one forms as an application we will show that the spectrum of the laplacian on one forms has no gaps on certain manifolds with a pole or that are in a warped product form this will be done under less strict curvature restrictions than what has been known so far and it was achieved by finding the l 1 spectrum of the laplacian

maximum and anti maximum principlesand eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations in this paper we discuss some new results concerning perturbation theory for second order elliptic partial differential equations related to positivity properties of such equations we continue the study of some different notions of small perturbations and discuss their relations to comparisons of green s functions refined maximum and anti maximum principles ground state and the decay of eigenfunctions in particular we show that if v is a positive function which is a semismall perturbation of a subcritical schrodinger operator h defined on a domain omega subset mathbb r d and phi k k geq 0 are the dirichlet eigenfunctions of the equation hu lambda vu then for any k geq 0 the function phi k phi 0 is bounded and has a continuous extension up to the martin boundary of the pair omega h where phi 0 is the ground state of h with a principal eigenvalue lambda 0

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

analysis of an a posteriori error estimator for a variational inequality governed by the stokes equations abstract in this work a residual type a posteriori error estimator is presented for finite element approximations of a variational inequality arising in hydrodynamics which is governed by the stationary stokes equations with a nonlinear slip boundary condition of friction type reliability of the estimator is rigorously proved and efficient error control is analyzed both are based on an equality problem with a lagrangian multiplier and an algorithm of recovering the multiplier is also provided numerical results are reported to illustrate the good performance of the estimator in the adaptive solution of the variational inequality as well as the availability of the multiplier

hoffman polynomials of nonnegative irreducible matrices and strongly connected digraphs abstract for a nonnegative n n matrix a we find that there is a polynomial f x r x such that f a is a positive matrix of rank one if and only if a is irreducible furthermore we show that the lowest degree such polynomial f x with tr f a n is unique thus generalizing the well known definition of the hoffman polynomial of a strongly connected regular digraph for any irreducible nonnegative n n matrix a we are led to define its hoffman polynomial to be the polynomial f x of minimum degree satisfying that f a is positive and has rank 1 and trace n the hoffman polynomial of a strongly connected digraph is defined to be the hoffman polynomial of its adjacency matrix we collect in this paper some basic results and open problems related to the concept of hoffman polynomials

holder domains and the boundary harnack principle the constant c depends on l only through the constant cl defined in 1 below of course c also depends on d v and k the above result shows that the boundary harnack principle holds in all holder domains it was first proven for the range 1 2 1 by bass and burdzy 1990a theorems 3 5 and 4 5 it was subsequently extended by banuelos unpublished to the full range of provided the domain also satisfies a uniform capacity condition on the boundary in banuelos 1990 a class of domains called uniformly holder domains of order was introduced a boundary harnack principle may be obtained for such domains for all 0 1 by a variation of the proof of theorem 1 in order to make this note compact we refer

poincare s inequalities and talagrand s concentration phenomenon for the exponential distribution we present a simple proof based on modified logarithmic sobolev inequalities of talagrand s concentration inequality for the exponential distribution we actually observe that every measure satisfying a poincare inequality shares the same concentration phenomenon we also discuss exponential integrability under poincare inequalities and its consequence to sharp diameter upper bounds on spectral gaps

on diagonal estimates on schrodinger semigroup kernels and reduced heat kernels we prove various estimates for the kernels of semigroups generated by schrodinger operators with magnetic field and potential of polynomial growth we also investigate the reduced heat kernels

on modified logarithmic sobolev inequalities for bernoulli and poisson measures we show that for any positive functionfon the discrete cube 0 1 n ent np f pqe np1f df 2where npis the product measure of the bernoulli measure with probability of successp as well as related inequalities which may be shown to imply in the limit the classical gaussian logarithmic sobolev inequality as well as a logarithmic sobolev inequality for poisson measure we further investigate modified logarithmic sobolev inequalities to analyze integrability properties of lipschitz functions on discrete spaces in particular we obtain under modified logarithmic sobolev inequalities some concentration results for product measures that extend the classical exponential inequalities for sums of independent random variables

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

a priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains we investigate quantitative properties of the nonnegative solutions u t x geq 0 to the nonlinear fractional diffusion equation partial t u mathcal l u m 0 posed in a bounded domain x in omega subset mathbb r n with m 1 for t 0 as mathcal l we use one of the most common definitions of the fractional laplacian delta s 0 s 1 in a bounded domain with zero dirichlet boundary conditions we consider a general class of very weak solutions of the equation and obtain a priori estimates in the form of smoothing effects absolute upper bounds lower bounds and harnack inequalities we also investigate the boundary behaviour and we obtain sharp estimates from above and below in addition we obtain similar estimates for fractional semilinear elliptic equations either the standard laplacian case s 1 or the linear case m 1 are recovered as limits the method is quite general suitable to be applied to a number of similar problems

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