the calderon problem with partial data in this paper we improve an earlier result by bukhgeim and uhlmann 1 by showing that in dimension n 3 the knowledge of the cauchy data for the schr nodinger equation measured on possibly very small subsets of the boundary determines uniquely the potential we follow the general strategy of 1 but use a richer set of solutions to the dirichlet problem this implies a similar result for the problem of electrical impedance tomography which consists in determining the conductivity of a body by making voltage and current measurements at the boundary

generalized ergodic problems existence and uniqueness structures of solutions abstract we study a generalized ergodic problem e which is a hamilton jacobi equation of contact type in the flat n dimensional torus we first obtain existence of solutions to this problem under quite general assumptions various examples are presented and analyzed to show that e does not have unique solutions in general we then study uniqueness structures of solutions to e in the convex setting by using the nonlinear adjoint method

constructing infinitely many smooth structures on 3 mathbb cp 2 n overline mathbb cp 2 using seiberg witten theory and rational blow down procedures of r fintushel and r j stern we construct infinitely many irreducible smooth structures both symplectic and non symplectic on the four manifold 3 mathbb cp 2 n overline mathbb cp 2 for each integer n lying in the interval 10 leq n leq 13

symplectic tori in rational elliptic surfaces let e 1 p denote the rational elliptic surface with a single multiple fiber fp of multiplicity p we construct an infinite family of homologous non isotopic symplectic tori representing the primitive 2 dimensional homology class fp in e 1 p when p 1 as a consequence we get infinitely many non isotopic symplectic tori in the fiber class of the rational elliptic surface open image in new window we also show how these tori can be non isotopically embedded as homologous symplectic submanifolds in other symplectic 4 manifolds

exotic smooth structures on small 4 manifolds let m be either mathbb cp 2 3 overline mathbb cp 2 or 3 mathbb cp 2 5 overline mathbb cp 2 we construct the first example of a simply connected irreducible symplectic 4 manifold that is homeomorphic but not diffeomorphic to m

exotic smooth structures on small 4 manifolds with odd signatures let m be 2n 1 mathbb cp 2 2n overline mathbb cp 2 for any integer n 1 we construct an irreducible symplectic 4 manifold homeomorphic to m and also an infinite family of pairwise non diffeomorphic irreducible non symplectic 4 manifolds homeomorphic to m we also construct such exotic smooth structures when m is mathbb cp 2 4 overline mathbb cp 2 or 3 mathbb cp 2 k overline mathbb cp 2 for k 6 8 10

knots links and 4 manifolds in this paper we investigate the relationship between isotopy classes of knots and links in s and the di eomorphism types of homeomorphic smooth 4 manifolds as a corollary of this initial investigation we begin to uncover the surprisingly rich structure of di eomorphism types of manifolds homeomorphic to the k3 surface in order to state our theorems we need to view the seiberg witten invariant of a smooth 4 manifold as a multivariable laurent polynomial to do this recall that the seiberg witten invariant of a smooth closed oriented 4 manifold x with b2 x 1 is an integer valued function which is de ned on the set of spinc structures over x cf w km ko1 t1 in case h1 x z has no 2 torsion which will be the situation in this paper there is a natural identi cation of the spinc structures of x with the characteristic elements of h2 x z in this case we view the seiberg witten invariant as

a weighted identity for stochastic partial differential operators and its applications abstract in this paper a weighted identity for some stochastic partial differential operators with complex principal parts is established this identity presents a unified approach in establishing carleman type estimates for some deterministic stochastic partial differential equations based on this identity one can deduce some known global carleman estimates for stochastic parabolic equations stochastic schrodinger equations stochastic transport equations and their deterministic counterparts meanwhile as its applications we derive two different carleman estimates for linear forward stochastic complex ginzburg landau equations they can be used to study the controllability observability and inverse problems for some stochastic complex ginzburg landau equations respectively

an inverse problem for the dynamical lame system with two sets of boundary data we prove uniqueness and a holder type stability of reconstruction of all three time independent elastic parameters in the dynamical isotropic system of elasticity from two special sets of boundary measurements in proofs we use carleman type estimates in sobolev spaces of negative order 2003 wiley periodicals inc

three region inequalities for the second order elliptic equation with discontinuous coefficients and size estimate abstract in this paper we would like to derive a quantitative uniqueness estimate the three region inequality for the second order elliptic equation with jump discontinuous coefficients the derivation of the inequality relies on the carleman estimate proved in our previous work 5 we then apply the three region inequality to study the size estimate problem with one boundary measurement

symplectic rational blowdowns we prove that the rational blowdown a surgery on smooth 4 manifolds introduced by fintushel and stern can be performed in the symplectic category as a consequence interesting families of smooth 4 manifolds including the exotic k3 surfaces of gompf and mrowka admit symplectic structures a basic problem in symplectic topology is to understand what smooth manifolds admit a symplectic structure a closed non degenerate 2 form in this paper we focus on this question in dimension 4 currently the primary methods for constructing smooth irreducible 4 manifolds in such a way that one can distinguish them by donaldson or seiberg witten invariants are surgery constructions that use complex manifolds as building blocks these surgery methods are smooth logarithmic transforms rational blowdowns and connect sums along surfaces it is interesting to see when these surgeries can be performed in the symplectic category in this paper we prove that performing a rational blowdown of a symplectic manifold along symplectic surfaces yields a symplectic manifold this result establishes that certain exotic 4 manifolds including the exotic k3 surfaces of gompf and mrowka 9 are symplectic in any even dimension two symplectic manifolds can be summed along codimension 2 symplectic submanifolds to yield a symplectic manifold we refer to this symplectic operation which was proposed by gromov 11 as the symplectic sum gompf 8 used the symplectic sum to construct a plethora of interesting symplectic manifolds including the first examples of simply connected symplectic 4 manifolds that are not homotopic to any complex surface and some exotic k3 surfaces more recently fintushel and stern 6 have used the connect sum along smoothly embedded tori to produce a rich class of exotic 4 manifolds some homeomorphic to a k3 surface many of which cannot admit a symplectic structure the author is grateful for the support of an nsf post doctoral fellowship dsm9627749

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

many closed symplectic manifolds have infinite hofer zehnder capacity we exhibit many examples of closed symplectic manifolds on which there is an autonomous hamiltonian whose associated flow has no nonconstant periodic orbits the only previous explicit example in the literature was the torus t2n n 2 with an irrational symplectic structure the underlying smooth manifolds of our examples include for instance the k3 surface and also infinitely many smooth manifolds homeomorphic but not diffeomorphic to it infinitely many minimal four manifolds having any given finitely presented group as their fundamental group and simply connected minimal four manifolds realizing all but finitely many points in the first quadrant of the geography plane below the line corresponding to signature 3 the examples are constructed by performing symplectic sums along suitable tori and then perturbing the symplectic form in such a way that hypersurfaces near the neck in the symplectic sum have no closed characteristics we conjecture that any closed symplectic four manifold with b 1 admits symplectic forms with a similar property

carleman estimates for the one dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem abstract we study the observability and some of its consequences controllability identification of diffusion coefficients for one dimensional heat equations with discontinuous coefficients piecewise c 1 the observability for a linear equation is obtained by a carleman type estimate this kind of observability inequality yields controllability results for a semi linear equation as well as a stability result for the identification of the diffusion coefficient

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

simply connected symplectic 4 manifolds with b2 1 and c12 2 in this article we construct a new simply connected symplectic 4 manifold with b2 1 and c12 2 which is homeomorphic but not diffeomorphic to a rational surface by using rational blow down technique as a corollary we conclude that a rational surface mathbf c p 2 sharp 7 overline mathbf c p 2 admits an exotic smooth structure

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

three sphere inequality for second order elliptic equations with coefficients with jump discontinuity abstract this is a short note to complete the paper appeared in francini et al 2016 4 where a rough version of the classical well known hadamard three circle theorem for solution of an elliptic pde in divergence form has been proved precisely instead of circles the authors obtain a similar inequality in a more complicated geometry in this paper we clean the geometry and obtain a generalized version of the three circle inequality for elliptic equation with coefficients with discontinuity of jump type

symplectic surgeries along certain singularities and new lefschetz fibrations abstract we define a new 4 dimensional symplectic cut and paste operations arising from the generalized star relations t a 0 t a 1 t a 2 t a 2 g 1 2 g 1 t b 1 t b 2 g t b 3 also known as the trident relations in the mapping class group g 3 of an orientable surface of genus g 1 with 3 boundary components we also construct new families of lefschetz fibrations by applying the generalized star relations and the chain relations to the families of words t c 1 t c 2 t c 2 g 1 t c 2 g t c 2 g 1 2 t c 2 g t c 2 g 1 t c 2 t c 1 2 n 1 t c 1 t c 2 t c 2 g t c 2 g 1 2 g 2 n 1 and t c 1 t c 2 t c 2 g 1 t c 2 g 2 2 g 1 n 1 in the mapping class group g of the closed orientable surface of genus g 1 and n 1 furthermore we show that the total spaces of some of these lefschetz fibrations are irreducible exotic symplectic 4 manifolds using the degenerate cases of the generalized star relations we also realize all elliptic lefschetz fibrations and genus two lefschetz fibrations over s 2 with non separating vanishing cycles

carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations in a bounded domain of r n 1 n geq 2 we consider a second order elliptic operator a d x 0 2 nabla x cdot c x nabla x where the scalar coefficient c x is piecewise smooth yet discontinuous across a smooth interface s we prove a local carleman estimate for a in the nhd of any point of the interface the observation region can be chosen independently of the sign of the jump of the coefficient c at the considered point the derivation of this estimate relies on the separation of the problem into three microlocal regions and the calderon projector technique following the method of lebeau and robbiano cite lr 95 we then prove the null controllability for the linear parabolic initial problem with dirichlet boundary conditions associated to the operator d t nabla x cdot c x nabla x

viscosity solutions of general viscous hamilton jacobi equations we present comparison principles lipschitz estimates and study state constraints problems for degenerate second order hamilton jacobi equations

local and global carleman estimates for parabolic operators with coefficients with jumps at interfaces in 0 t open subset of n n 2 we consider a parabolic operator p t x t x x where the scalar coefficient t x is piecewise smooth in space yet discontinuous across a smooth interface s we prove a global in time local in space carleman estimate for p in the neighborhood of any point of the interface the observation region can be chosen independently of the sign of the jump of the coefficient at the considered point the derivation of this estimate relies on the separation of the problem into three microlocal regions related to high and low tangential frequencies at the interface in the high frequency regime we use calderon projectors in the low frequency regime we follow a more classical approach because of the parabolic nature of the problem we need to introduce weyl hormander anisotropic metrics symbol classes and pseudo differential operators each frequency regime and the associated technique require a different calculus a global in time and space carleman estimate on 0 t m m a manifold is also derived from the local result

aubry mather theory and hamilton jacobi equations we review and extend some results of jauslin kreiss moser and weinan e on the smooth approximation of mather sets

on symplectic fillings of lens spaces le st be the contact structure naturally induced on the lens space l p q s 3 z pz by the standard contact structure st on the three sphere s 3 we obtain a complete classification of the symplectic fillings of l p q st up to orientation preserving diffeomorphisms in view of our results we formulate a conjecture on the diffeomorphism types of the smoothings of complex two dimensional cyclic quotient singularities

carleman estimates for stratified media abstract we consider anisotropic elliptic and parabolic operators in a bounded stratified media in r n characterized by discontinuities of the coefficients in one direction the surfaces of discontinuities cross the boundary of the domain we prove carleman estimates for these operators with an arbitrary observation region

selection problems for a discount degenerate viscous hamilton jacobi equation abstract we prove that the solution of the discounted approximation of a degenerate viscous hamilton jacobi equation with convex hamiltonians converges to that of the associated ergodic problem we characterize the limit in terms of stochastic mather measures by using the nonlinear adjoint method and deriving a commutation lemma this convergence result was first proven by davini fathi iturriaga and zavidovique for first order hamilton jacobi equations

the structure of rational and ruled symplectic 4 manifolds this paper investigates the structure of compact symplectic 4 manifolds v w which contain a symplectically embedded copy c of s2 with nonnegative self intersection number such a pair v c w is called minimal if in addition the open manifold v c contains no exceptional curves i e symplectically embedded 2 spheres with self intersection 1 we show that every such pair v c w covers a minimal pair v c c which may be obtained from v by blowing down a finite number of disjoint exceptional curves in v c further the family of manifold pairs v c w under consideration is closed under blowing up and down we next give a complete list of the possible minimal pairs we show that v is symplectomorphic either to p2 with its standard form or to an s2 bundle over a compact surface with a symplectic structure which is uniquely determined by its cohomology class moreover this symplectomorphism may be chosen so that it takes c either to 2 a complex line or quadric in cp2 or in the case when v is a bundle to a fiber or section of the bundle department of mathematics state university of new york at stony brook stony brook new york 11794 3651 e mail address dusa math sunysb edu this content downloaded from 207 46 13 145 on wed 27 apr 2016 05 10 07 utc all use subject to http about jstor org terms

a simply connected surface of general type with p g 0 and k 2 2 in this paper we construct a simply connected minimal complex surface of general type with pg 0 and k2 2 using a rational blow down surgery and a gorenstein smoothing theory

carleman estimates and controllability results for the one dimensional heat equation with bv coefficients abstract we derive global carleman estimates for one dimensional linear parabolic equations t x c x with a coefficient of bounded variations these estimates are obtained by approximating c by piecewise constant coefficients c e and passing to the limit in the carleman estimates associated to the operators defined with c e such estimates yields observability inequalities for the considered linear parabolic equation which in turn yield controllability results for classes of semilinear equations

a lefschetz fibration on minimal symplectic fillings of a quotient surface singularity in this article we construct a genus 0 or genus 1 positive allowable lefschetz fibration on any minimal symplectic filling of the link of non cyclic quotient surface singularities as a byproduct we also show that any minimal symplectic filling of the link of quotient surface singularities can be obtained from a sequence of rational blowdowns from its minimal resolution

adjoint and compensated compactness methods for hamilton jacobi pde we investigate the vanishing viscosity limit for hamilton jacobi pde with nonconvex hamiltonians and present a new method to augment the standard viscosity solution approach the main idea is to introduce a solution e of the adjoint of the formal linearization and then to integrate by parts with respect to the density e this procedure leads to a natural phase space kinetic formulation and also to a new compensated compactness technique