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

on multilinear square function and its applications to multilinear littlewood paley operators with non convolution type kernels abstract let m 2 n 1 and x r n define the multilinear square function t by t f x 0 r n m k t x y 1 y m j 1 m f j y j d y 1 d y m 2 d t t 1 2 where the kernel k satisfies a class of integral smooth conditions which is much weaker than the standard calderon zygmund kernel conditions in this paper we first established the l p 1 w 1 l p m w m l p estimate of t when each p i 1 and weak type l p 1 w 1 l p m w m l p estimate of t when there is a p i 1 where i 1 m i p p i and each w i is a nonnegative function on r n as applications of the above results we obtained the boundedness of multilinear littlewood paley operators with non convolution type kernels including multilinear g function marcinkiewicz integral and g function

a criterion for the absolute continuity of the harmonic measure associated with an elliptic operator for some a o and for all x e q and 4 e r n we also assume a a for such operators the dirichlet problem is solvable in q if and only if it is solvable for the laplace operator according to a theorem of littman stampacchia and weinberger 1 this means that if q c r n is a sufficiently nice bounded region the unit ball b is an example and f is a given continuous function on the boundary of q then there exists a unique function u continuous on q so that l u 0 in q and u f on aq let us assume for convenience that the origin belongs to q then the mapping f c f aq u o is a positive linear functional so there exists a unique nonnegative measure w on aq such that for every f e aq

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

a magnetic resonance device designed via global optimization techniques in this paper we are concerned with the design of a small low cost low field multipolar magnet for magnetic resonance imaging with a high field uniformity by introducing appropriate variables the considered design problem is converted into a global optimization one this latter problem is solved by means of a new derivative free global optimization method which is a distributed multi start type algorithm controlled by means of a simulated annealing criterion in particular the proposed method employs as local search engine a derivative free procedure under reasonable assumptions we prove that this local algorithm is attracted by global minimum points additionally we show that the simulated annealing strategy is able to produce a suitable starting point in a finite number of steps with probability one

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

the solution of the kato problem for divergence form elliptic operators with gaussian heat kernel bounds which in turnhas applications to the perturbation theory for certain classes of hyperbolicequations see 16 we remark that 1 3 is equivalent to the oppositeinequality for the square root of the adjoint operator l in 13 14 kato conjectured that an abstract version of 1 3 might hold for regularly accretive operators see 14 18 for the details a counter example to this abstract conjecture was obtained by mcintosh 17 whothen reformulated the conjecture in the following form bearing in mind thatkato s interest in the problem had been motivated by the special case ofelliptic di erential operators conjecture 1 4 the estimate 1 3 holds for lde ned as in 1 2 for anyl

functional calculus of dirac operators and complex perturbations of neumann and dirichlet problems we prove that the neumann dirichlet and regularity problems for divergence form elliptic equations in the half space are well posed in l 2 for small complex l infty perturbations of a coefficient matrix which is either real symmetric of block form or constant all matrices are assumed to be independent of the transversal coordinate we solve the neumann dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order dirac type operator we establish quadratic estimates for this dirac operator which implies that the associated hardy projection operators are bounded and depend continuously on the coefficient matrix we also prove that certain transmission problems for k forms are well posed for small perturbations of block matrices

l2 solvability of boundary value problems for divergence form parabolic equations with complex coefficients abstract we consider parabolic operators of the form t l l div a x t in r n 2 x t x x n 1 t r n r r x n 1 0 n 1 we assume that a is a n 1 n 1 dimensional matrix which is bounded measurable uniformly elliptic and complex and we assume in addition that the entries of a are independent of the spatial coordinate x n 1 as well as of the time coordinate t for such operators we prove that the boundedness and invertibility of the corresponding layer potential operators are stable on l 2 r n 1 c l 2 r n 2 c under complex l perturbations of the coefficient matrix subsequently using this general result we establish solvability of the dirichlet neumann and regularity problems for t l by way of layer potentials and with data in l 2 assuming that the coefficient matrix is a small complex perturbation of either a constant matrix or of a real and symmetric matrix

hausdorff measure estimates and lipschitz regularity in inhomogeneous nonlinear free boundary problems in this paper we prove a hausdorff measure estimate for the free boundaries of subsolutions of fully nonlinear and quasilinear equations of the type f d 2u x geqq f x and rm div a x nabla u geqq mu where f in l q q n and is a signed radon measure with some appropriate growth condition gradient estimates for nonnegative harmonic functions with bounded normal derivatives along the boundary obtained by caffarelli and salsa geometric approach to free boundary problems 2005 are extended to the context of inhomogeneous problems involving fully nonlinear and p laplace equations as an application lipschitz regularity is obtained for one phase solutions of inhomogeneous nonlinear free boundary problems

sound diffraction around screens and wedges for arbitrary point source locations a development is presented for the green s function for a point source in the vicinity of a rigid wedge the diffraction contributions to the green s function for arbitrary source and listener location is expressed in a form which can be readily evaluated using the laguerre technique for numerical integration the present approach offers the advantages of efficient numerical evaluation and of relatively straightforward reduction to well known analytical approximations in limiting cases comparisons with previously obtained experimental and numerical results obtained by ambaud and bergassoli acustica 27 291 298 1972 are presented the comparison with the experimental results is excellent the advantages of the present numerical technique vis a vis that of ambaud and bergassoli are pointed out

multilinear littlewood paley estimates with applications to partial differential equations abstract we obtain a collection of multilinear littlewood paley estimates which we then apply to two problems in partial differential equations the first problem is the estimation of the square root of an elliptic operator in divergence form and the second is the estimation of solutions to the cauchy problem for nondivergence form parabolic equations

weighted maximal regularity estimates and solvability of non smooth elliptic systems i we develop new solvability methods for divergence form second order real and complex elliptic systems above lipschitz graphs with l 2 boundary data the coefficients a may depend on all

trace operator and the dirichlet problem for elliptic equations on arbitrary bounded open sets abstract we consider the dirichlet problem and the weak dirichlet problem on a general possibly nonregular bounded domain for elliptic linear equation with uniformly elliptic divergence form operator we investigate carefully the relationship between weak integral probabilistic soft and the perron wiener brelot solutions of the problem we also study the relationship between solutions of the dirichlet problem and weak dirichlet problem to this end we extend the usual notion of the trace operator and consider the dirichlet problem with martin boundary

the regularity problem for second order elliptic operators with complex valued bounded measurable coefficients the present paper establishes a certain duality between the dirichlet and regularity problems for elliptic operators with t independent complex bounded measurable coefficients t being the transversal direction to the boundary to be precise we show that the dirichlet boundary value problem is solvable in l p subject to the square function and non tangential maximal function estimates if and only if the corresponding regularity problem is solvable in l p moreover the solutions admit layer potential representations in particular we prove that for any elliptic operator with t independent real possibly non symmetric coefficients there exists a p 1 such that the regularity problem is well posed in l p

necessary and sufficient conditions for absolute continuity of elliptic harmonic measure our purpose in this paper is to give a necessary and sufficient condition on the modulus of continuity of the coefficients of an elliptic operator in divergence form in order that the corresponding poisson kernel for the dirichlet problem exist this condition on the coefficients is global continuity together with the additional property that the modulus of continuity along some nontangential direction at each boundary point be bounded uniformly in these points and directions by a function 7 t satisfying the dini type condition