eigenvalue convergence on perturbed lipschitz domains for elliptic systems with mixed general decompositions of the boundary abstract we consider eigenvalues of an elliptic operator l u x j a i j u x i where u u 1 u m t is a vector valued function and the coefficients a i j are m m matrices whose elements a i j are bounded and symmetric we perturb our domain 0 by adding a set of small measure t e to form the domain e we prescribe mixed boundary conditions on quite general decompositions of the boundary and look at the behavior of the eigenvalues of e as t e shrinks to zero we look at systems which satisfy either a strong ellipticity condition a legendre hadamard condition or in particular the system of linear elasticity

the mixed problem in lipschitz domains with general decompositions of the boundary this paper continues the study of the mixed problem for the laplacian we consider a bounded lipschitz domain rn n 2 with boundary that is decomposed as d n d and n disjoint we let denote the boundary of d relative to and impose conditions on the dimension and shape of and the sets n and d under these geometric criteria we show that there exists p0 1 depending on the domain such that for p in the interval 1 p0 the mixed problem with neumann data in the space lp n and dirichlet data in the sobolev space w 1 p d has a unique solution with the non tangential maximal function of the gradient of the solution in lp we also obtain results for p 1 when the dirichlet and neumann data comes from hardy spaces and a result when the boundary data comes from weighted sobolev spaces

two sided heat kernel estimates for censored stable like processes in this paper we study the precise behavior of the transition density func tions of censored resurrected stable like processes in c 1 1 open sets in r d where d 1 and 1 2 we first show that the semigroup of the censored stable like process in any bounded lipschitz open set is intrinsically ultracontractive we then establish sharp two sided estimates for the transition density functions of a large class of censored stable like processes in c 1 1 open sets we further obtain sharp two sided estimates for the green functions of these censored stable like processes in bounded c 1 1 open sets

integration by parts formula for regional fractional laplacian we obtain the integration by parts formula for the regional fractional laplacian which are generators of symmetric stable processes on a subset of mathbb r n 0 2 in this formula a local operator appears on the boundary connected with the regional fractional laplacian on domain hence this formula can be understood as the green formula for the regional fractional laplacian the similar integration by parts formula is also given for more general nonlocal operators the reflected stable like processes are studied

semilinear fractional elliptic equations with gradient nonlinearity involving measures we study the existence of solutions to the fractional elliptic equation e1 delta alpha u epsilon g nabla u nu in a bounded regular domain omega of r n n ge2 subject to the condition e2 u 0 in omega c where epsilon 1 or 1 delta alpha denotes the fractional laplacian with alpha in 1 2 1 nu is a radon measure and g r mapsto r is a continuous function we prove the existence of weak solutions for problem e1 e2 when g is subcritical furthermore the asymptotic behavior and uniqueness of solutions are described when nu is dirac mass g s s p p geq 1 and epsilon 1

the dirichlet elliptic problem involving regional fractional laplacian in this paper we study the solutions of elliptic equations involving regional fractional laplacian e u f in a bounded regular domain in rn n 2 with c2 boundary subject to dirichlet boundary g on where 12 1 and the operator denotes the regional fractional laplacian we prove that when g 0 problem e admits a unique weak solution under the hypotheses that f l2 f l1 dx and f m where x dist x 2 1 and m is a space of all radon measures satisfying d finally we provide an integration by parts formula for the classical solution of e with boundary data g

mountain pass solutions for non local elliptic operators abstract the purpose of this paper is to study the existence of solutions for equations driven by a non local integrodifferential operator with homogeneous dirichlet boundary conditions these equations have a variational structure and we find a non trivial solution for them using the mountain pass theorem to make the nonlinear methods work some careful analysis of the fractional spaces involved is necessary we prove this result for a general integrodifferential operator of fractional type and as a particular case we derive an existence theorem for the fractional laplacian finding non trivial solutions of the equation s u f x u in u 0 in r n as far as we know all these results are new

green function estimate for censored stable processes sharp two sided estimates for green functions of censored stable process y in a bounded c 1 1 open set d are obtained where 1 2 it is shown that the martin boundary and minimal martin boundary of y can all be identified with the euclidean boundary of d sharp two sided estimates for the martin kernel of y are also derived

the lp boundary value problems on lipschitz domains abstract let be a bounded lipschitz domain in r n we develop a new approach to the invertibility on l p of the layer potentials associated with elliptic equations and systems in as a consequence for n 4 and 2 n 1 n 1 e p 2 where e 0 depends on we obtain the solvability of the l p neumann type boundary value problems for second order elliptic systems the analogous results for the biharmonic equation are also established

on w1 p estimates for elliptic equations in divergence form first we will apply the method to study w 1 p regularity for a nonlinear elliptic operator in divergence form we would like to point out that in the particular case of a linear elliptic equation this method gives an alternative proof to the classical one which uses the general theory of singular integrals the utility of the method that we describe below is that hypotheses a and b are obtained directly by studying the deviation of the coefficients of a from the coefficients of a0 and this is usually not a difficult task a more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give w 1 p estimates with a weak hypothesis of regularity in the coefficients for instance we are able to study the following cases

the mixed problem for the lame system in two dimensions we consider the mixed problem for l the lame system of elasticity in a bounded lipschitz domain r2 we suppose that the boundary is written as the union of two disjoint sets d n we take traction data from the space lp n and dirichlet data from a sobolev space w1 p d and look for a solution u of lu 0 with the given boundary conditions in our main result we find a scale invariant condition on d and an exponent p0 1 so that for 1 p p0 we can find a unique solution of this boundary value problem with the non tangential maximal function of the gradient in lp we also establish the existence of a unique solution when the data is taken from hardy spaces and hardy sobolev spaces with p in p1 1 for some p1 1

some more universal scaling laws for critical mappings we study such nonlinear mappingsxn 1 f xn bcr of an intervali into itself for which the feigenbaum scaling laws hold i e for which bcr is an accumulation point of bifurcation points letx0 be a random variable with some absolutely continuous distribution ini we show in particular that i the geometric average distance ofxn from the nearest point of the attractor decreases liken 1 93387 ii the geometric average of xn x0 increases liken0 60 iii the geometric mean distance xn yn between the iterates of two close by pointsx0 y0 asymptotically tends towards a value x0 y0 0 77 these and other properties are also borne out from a simple probabilistic model which depicts the evolution as a random walklike process

the mixed problem in l p for some two dimensional lipschitz domains we consider the mixed problem left begin array ll delta u 0 quad rm in omega frac partial u partial nu f n quad rm on rm n u f d quad rm on d end array right in a class of lipschitz graph domains in two dimensions with lipschitz constant at most 1 we suppose the dirichlet data fd has one derivative in lp d of the boundary and the neumann data fn is in lp n we find a p0 1 so that for p in an interval 1 p0 we may find a unique solution to the mixed problem and the gradient of the solution lies in lp

reflected symmetric α stable processes and regional fractional laplacian in this paper we investigate the reflected symmetric stable processes and their generators we show that the generators are regional fractional laplacians on the closed region in the case of 1 2 their existence requires that open image in new window on the boundary among other things we obtain the integration by parts formula of the regional fractional laplacian and the semi martingale decomposition of the reflected symmetric stable processes

the mixed problem for the lame system in a class of lipschitz domains abstract we consider the mixed problem for the lame system l u 0 in u d f d on d u f n on n u l p in the class of bounded lipschitz creased domains here d and n partition and stands for the traction operator we suppose the dirichlet data f d has one derivative in l p d and the traction data f n is in l p n for p in a small interval containing 2 we find a unique solution to the mixed problem subject to the condition that the non tangential maximal function of the gradient of the solution is in l p

the poisson problem with mixed boundary conditions in sobolev and besov spaces in non smooth domains we introduce certain sobolev besov spaces which are particularly well adapted for measuring the smoothness of data and solutions of mixed boundary value problems in lipschitz domains in particular these are used to obtain sharp well posedness results for the poisson problem for the laplacian with mixed boundary conditions on bounded lipschitz domains which satisfy a suitable geometric condition introduced by r brown in 1994 in this context we obtain results which generalize those by d jerison and c kenig 1995 as well as e fabes o mendez and m mitrea 1998 applications to hodge theory and the regularity of green operators are also presented

well posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks we prove a regularity result for the anisotropic linear elasticity equation p u rm div left boldmath mathsf c cdot nabla u right f with mixed displacement and traction boundary conditions on a curved polyhedral domain omega subset mathbb r 3 in weighted sobolev spaces mathcal k m 1 a 1 omega for which the weight is given by the distance to the set of edges in particular we show that there is no loss of mathcal k m a regularity our curved polyhedral domains are allowed to have cracks we establish a well posedness result when there are no neighboring traction boundary conditions and a 0 that depends on p on the boundary conditions and on the domain our results extend to other strongly elliptic systems and higher dimensions

mixed boundary value problems for the stokes system we prove the well posedness of the mixed problem for the stokes system in a class of lipschitz domains in n n 3 the strategy is to reduce the original problem to a boundary integral equation and we establish certain new rellich type estimates which imply that the intervening boundary integral operator is semi fredholm we then prove that its index is zero by performing a homotopic deformation of it onto an operator related to the lame system which has recently been shown to be invertible

hardy spaces and the neumann problem in l p for laplace s equation in lipschitz domains on donne des resultats optimaux pour la resolubilite du probleme de neumann dans des domaines de lipschitz a donnees dans l p on obtient des resultats du point extreme correspondant pour des espaces de hardy

censored stable processes we present several constructions of a censored stable process in an open set d rn i e a symmetric stable process which is not allowed to jump outside d we address the question of whether the process will approach the boundary of d in a finite time we give sharp conditions for such approach in terms of the stability index and the thickness of the boundary as a corollary new results are obtained concerning besov spaces on non smooth domains including the critical exponent case we also study the decay rate of the corresponding harmonic functions which vanish on a part of the boundary we derive a boundary harnack principle in c1 1 open sets