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

perturbations of elliptic operators in 1 sided chord arc domains part i small and large perturbation for symmetric operators the first and third authors acknowledge financial support from the spanish ministry of economy and competitiveness through the severo ochoa programme for centres of excellence in r d sev 2015 0554 they also acknowledge 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 second author was supported by nsf grant dms 1664047

the martin boundary in non lipschitz domains the martin boundary with respect to the laplacian and with respect to uniformly elliptic operators in divergence form can be identified with the euclidean boundary in c domains where x bx loglog 1 x logloglog 1 x b small a counterexample shows that this result is very nearly sharp

sharp estimates for the nontangential maximal function and the lusin area function in lipschitz domains let m be a harmonic function on a domain of the form d x y x 6 ft y ft y 4 x where 0 r r is a lipschitz function the authors show a good a inequality between au the lusin area function of u and nu the nontangential maximal function of u this leads to an lp inequality of the form au p 0 we also consider the functions au p 0 t and nu p 0 t and show that a corollary of the good a inequality is a law of the iterated logarithm involving these two functions as 0 if n 1 and 4 has a small lipschitz constant the above results are shown valid with the roles of nu and au interchanged in this paper we will show some good a inequalities between the nontangen tial maximal function and the lusin area function on lipschitz domains in addition we give some applications of these in particular we prove a law of the iterated logarithm for the two functions which gives a precise bound on the ratios of their growth as they approach the boundary of the domain this law of the iterated logarithm is the analogue of the result proved in 1 with the roles of the maximal function and the area function interchanged we call f r r a lipschitz function if there exists a constant m such that x cp y we will consider two types of lipschitz domains an unbounded type and a bounded type the unbounded type will be of the form d x y x gr y gr y 4 x where is a lipschitz function the bounded domains we will consider are the starlike lipschitz domains these are domains d c rn with the following two properties first we require that the boundary of d is locally lipschitz that is for each p g dd there is a coordinate system z t z g r t g r and a lipschitz function such that for some neighborhood v of p v r d z t t p z we require

loops and branches of coexistence states in a lotka volterra competition model abstract a two species lotka volterra competition diffusion model with spatially inhomogeneous reaction terms is investigated the two species are assumed to be identical except for their interspecific competition coefficients viewing their common diffusion rate as a parameter we describe the bifurcation diagram of the steady states including stability in terms of two real functions of we also show that the bifurcation diagram can be rather complicated namely given any two positive integers l and b the interspecific competition coefficients can be chosen such that there exist at least l bifurcating branches of positive stable steady states which connect two semi trivial steady states of the same type they vanish at the same component and at least b other bifurcating branches of positive stable steady states that connect semi trivial steady states of different types

multiple positive solutions for p laplacian m point boundary value problems on time scales let t be a time scale such that 0 email protected t a i 0 for i 1 m 2 let x i satisfy 0 x 1 x 2 x m 2 r t and i 1 m 2a i 1 we consider the following p laplacian m point boundary value problem on time scales f p u d t a t f t u t 0 email protected 0 t u 0 0 f p u d t email protected i 1m 2a email protected p u d x i where email protected c l d 0 t 0 and email protected c l d 0 t x 0 0 some new results are obtained for the existence of at least twin or triple positive solutions of the above problem by applying avery henderson and leggett williams fixed point theorems respectively in particular our criteria extend and improve some known results

eigenvalue problem for p laplacian three point boundary value problems on time scales abstract let t be a time scale such that 0 t t 0 and 0 t we consider the following p laplacian three point boundary problem on time scales p u t h t f u t 0 t 0 t u 0 u 0 u u t 0 where p 1 0 h c ld 0 t 0 and f c 0 0 some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established in doing so the usual restriction that f 0 lim u 0 f u p u and f lim u f u p u exist is removed an example is also given to illustrate the main results

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

higher order m point boundary value problems on time scales in this paper we investigate the existence of positive solutions for nonlinear even order m point boundary value problems on time scales by means of fixed point theorems

multiple positive solutions for nonlinear dynamical systems on a measure chain in this paper we consider the following dynamical system on a measure chain u 1 t f1 t u1 t u2 t 0 t a b u 2 t f2 t u1 t u2 t 0 t a b with the sturm liouville boundary value conditions ui a u i a 0 ui b ui b 0 for i 1 2 some results are obtained for the existence of three positive solutions of the above problem by using leggett williams fixed point theorem

twin positive solutions of boundary value problems for finite difference equations with p laplacian operator abstract in this paper a new twin fixed point theorem is applied to obtain the existence of at least two positive solutions for the boundary value problem p u k a k f u k 0 k 0 n u n 1 0 u 0 b 0 u 0 0 in addition two corollaries and an example are given to illustrate the main results

twin solutions of boundary value problems for ordinary differential equations and finite difference equations a new twin fixed point theorem is applied first to obtain the existence of at least two positive solutions for the right focal boundary value problem y f ity 0 0 t 1 y 0 y 1 0 it is applied later to obtain the existence of at least two positive solutions for the analogous discrete right focal boundary value problem 2u k g u k 0 k 0 n u 0 u n 1 0

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

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

behavior near the boundary of positive solutions of second order parabolic equations ii a boundary backward harnack inequality is proved for positive solutions of second order parabolic equations in non divergence form in a bounded cylinder q q x 0 t which vanish on o q oq x 0 t where q is a bounded lipschitz domain in r this inequality is applied to the proof of the holder continuity of the quotient of two positive solutions vanishirlg on a portion of o q

positive solutions of singular p laplacian dynamic equations with sign changing nonlinearity abstract let t be a time scale such that 0 t t by the schauder fixed point theorem and the upper and lower solution method we present some existence criteria of the positive solution of m point singular p laplacian dynamic equation p u t q t f t u t 0 t 0 t t with boundary conditions u 0 0 i 1 m 1 i u i u t 0 m 2 where p s s p 2 s with p 1 i r r is continuous for i 1 2 m 1 and nonincreasing if m 3 0 1 2 m 2 m 1 t the nonlinear term may be singular in its dependent variable and is allowed to change sign our results are new even for the corresponding differential t r and difference equations t z as an application an example is given to illustrate our result

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

boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions in this work we introduce the use of powerful tools from geometric measure theory gmt to study problems related to the size and structure of sets of mutual absolute continuity for the harmonic measure omega of a domain omega omega subset mathbb r n and the harmonic measure omega of omega omega int omega c in dimension n ge 3

existence of positive solutions to second order time scale systems in this paper the author consider the following dynamic system on a measure chain u i d d t f i t u 1 s t u 2 s t u n s t 0 email protected a b satisfying sturm liouville boundary conditions a iu i a email protected iu i d a 0 c iu i s b email protected iu i d s b 0 i 1 2 n some new results are obtained for the existence of single twin and triple positive solutions of the above system by using nonlinear alternative of leray schauder type krasnosel skii s fixed point theorem in a cone and leggett williams fixed point theorem

a martin boundary in the plane let e be an open connected subset of euclidean space with a green function and let a be harmonic measure on the martin boundary i of e we will show that except for a a x a null set of x y g i 2 x is an entrance point for brownian motion conditioned to leave e at y r s martin gave examples in dimension 3 or higher for which there exist minimal accessible martin boundary points x 7 y for which this condition fails we will give a similar example in dimension 2

positive solutions for nonlinear three point boundary value problems on time scales abstract let t be a time scale such that 0 t t consider the following three point boundary value problem on time scales u t a t f t u t 0 t 0 t u 0 u 0 0 u u t where 0 0 0 t 0 t and d t 1 0 by using fixed point theorems in cones some new and general results are obtained for the existence of single and multiple positive solutions of the above problem in particular our criteria generalize and improve some known results

existence results for singular three point boundary value problems on time scales abstract we prove the existence of a positive solution for the three point boundary value problem on time scale t given by y f x y 0 x 0 1 t y 0 0 y p y 2 1 where p 0 1 t is fixed and f x y is singular at y 0 and possibly at x 0 y we do so by applying a fixed point theorem due to gatica oliker and waltman j differential equations 79 1989 62 for mappings that are decreasing with respect to a cone we also prove the analogous existence results for the related dynamic equations y f x y 0 y f x y 0 and y f x y 0 satisfying similar three point boundary conditions

existence of positive solutions of nonlinear m point bvp for an increasing homeomorphism and positive homomorphism on time scales in this paper we will consider the following multipoint boundary value problem for the following second order dynamic equations on time scales f u d a t f t u t 0 t 0 t u 0 i 1m 2a iu x i f u d t i 1m 2b i f u d x i where f r r is an increasing homeomorphism and positive homomorphism and f 0 0 by using fixed point theorems we obtain an existence theorem of positive solutions for the above boundary value problem which includes and improve some related results in the relevant literature as an application an example to demonstrate our results is given

boundary tangential convergence on spaces of homogeneous type we study tangential convergence of convolutions with approximate identities of functions defined on a homogeneous type space and having a certain regularity our results contain those already known for the euclidean case and give new ones for stratified nilpotent lie groups and for solutions of the dirichlet problem on lipschitz domains

on green s functions and positive solutions for boundary value problems on time scales in this paper we offer a form of self adjoint differential equations on time scales so that the associated green s function is found symmetric in the usual sense for this purpose together with the delta derivative we employ the nabla derivative as well we introduce the concepts of lebesgue delta and nabla integrals on time scales next sign properties of the green s function are investigated and existence results for positive solutions of nonlinear boundary value problems are established upper and lower bounds for these positive solutions also are given

triple positive solutions for p laplacian m point boundary value problem on time scales in this paper we discuss the following p laplacian m point boundary value problem on time scales t f p u d t h t f t u t u d t 0 email protected 0 t t u 0 0 f p u d t email protected i 1m 2a email protected p u d x i where f p u u p 2u for p 1 some new existence criteria for at least three positive solutions are established by using a generalization of the leggett williams fixed point theorem due to bai and ge an example is also given to demonstrate the main result

the sub elliptic obstacle problem c1 a regularity of the free boundary in carnot groups of step two the sub elliptic obstacle problem arises in various branches of the applied sciences e g in mechanical engineering and robotics mathematical finance image reconstruction and neurophysiology in the recent paper donatella danielli nicola garofalo sandro salsa variational inequalities with lack of ellipticity i optimal interior regularity and non degeneracy of the free boundary indiana univ math j 52 2 2003 361 398 mr1976081 2004c 35424 it was proved that weak solutions to the sub elliptic obstacle problem in a carnot group belong to the folland stein optimal lipschitz class the analogue of the well known interior local regularity for the classical obstacle problem however the regularity of the free boundary remained a challenging open problem in this paper we prove that in carnot groups of step r 2 the free boundary is euclidean c1 a near points satisfying a certain thickness condition this constitutes the sub elliptic counterpart of a celebrated result due to caffarelli luis a caffarelli the regularity of free boundaries in higher dimensions acta math 139 3 4 1977 155 184 mr0454350 56 12601

some new results on the existence of positive solutions for the one dimensional p laplacian boundary value problems on time scales abstract this paper is devoted to using the leggett williams fixed point theorem to derive some sufficient conditions for the existence of triple positive solutions to a type of p laplacian boundary value problems on time scales the interesting point is that the nonlinear term f depends on the first order delta derivative explicitly

regular domains in homogeneous groups we study john uniform and non tangentially accessible domains in homogeneous groups of steps 2 and 3 we show that c 1 1 domains in groups of step 2 are non tangentially accessible and we give an explicit condition which ensures the john property in groups of step 3

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

multiple positive solutions for functional dynamic equations on time scales in this paper we study the following functional dynamic equation on time scales f u d t a t f u t u m t 0 email protected 0 t t u t email protected t email protected r 0 t u 0 b 0 u d 0 0 u d t 0 where f r r is an increasing homeomorphism and a positive homomorphism and f 0 0 by using the well known leggett williams fixed point theorem existence criteria for multiple positive solutions are established an example is also given to illustrate the main results

a criterion for the boundedness of singular integrals on hypersurfaces on donne des conditions geometriques sur une hypersurface de r n pour que certaines integrales singulieres sur cette hypersurface definissent des operateurs bornes sur l 2