an inverse boundary value problem for schrodinger operators with vector potentials we consider the schrodinger operator for a magnetic potential a and an electric potential q which are supported in a bounded domain in r n with n 3 we prove that knowledge of the dirichlet to neumann map associated to the schrodinger operator determines the magnetic field rot a and the electric potential 4 simultaneously provided rot a is small in the l topology

optimal isoparametric finite elements and error estimates for domains involving curved boundaries we describe a practical procedure for the triangulation of arbitrary n dimensional regular bounded domains by isoparametric simplicial elements preserving the optimal order of accuracy along similar lines we prove general error estimates for the finite element solution of second order elliptic problems various kinds of nonhomogeneous boundary conditions can be dealt with such as dirichlet neumann and robin boundary conditions

stability and resolution analysis for a topological derivative based imaging functional the aim of this paper is to study a topological derivative based anomaly detection algorithm we compare its performance with other imaging approaches such as multiple signal classification backpropagation and kirchhoff migration we also investigate its stability with respect to medium and measurement noises as well as its resolution a simple postprocessing of the data set is introduced and shown to be essential in order to obtain an efficient topological based imaging functional both in terms of resolution and stability

topological derivative for multi scale linear elasticity models applied to the synthesis of microstructures this paper proposes an algorithm for the synthesis optimization of microstructures based on an exact formula for the topological derivative of the macroscopic elasticity tensor and a level set domain representation the macroscopic elasticity tensor is estimated by a standard multi scale constitutive theory where the strain and stress tensors are volume averages of their microscopic counterparts over a representative volume element the algorithm is of simple computational implementation in particular it does not require artificial algorithmic parameters or strategies this is in sharp contrast with existing microstructural optimization procedures and follows as a natural consequence of the use of the topological derivative concept this concept provides the correct mathematical framework to treat topology changes such as those characterizing microstuctural optimization problems the effectiveness of the proposed methodology is illustrated in a set of finite element based numerical examples copyright 2010 john wiley sons ltd

optimal finite element interpolation on curved domains this paper is devoted to a general theory of approximation of functions in finite element spaces in particular the case is considered where the functions to be interpolated are on the one hand not very smooth and on the other are defined on curved domains thus a number of well known optimal interpolation results are generalized

finite element analysis of varitional crimes for a quasilinear elliptic problem in 3d we examine a finite element approximation of a quasilinear boundary value elliptic problem in a three dimensional bounded convex domain with a smooth boundary the domain is approximated by a polyhedron and a numerical integration is taken into account we apply linear tetrahedral finite elements and prove the convergence of approximate solutions on polyhedral domains in the w 1 2 norm to the true solution without any additional regularity assumptions

discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions one of the most important problems in numerical simulations is the preservation of qualitative properties of solutions of the mathematical models by computed approximations for problems of elliptic type one of the basic properties is the continuous maximum principle in our work we present several variants of the maximum principles and their discrete counterparts for scalar second order nonlinear elliptic problems with mixed boundary conditions the problems considered are numerically solved by the continuous piecewise linear finite element approximations built on simplicial meshes sufficient conditions providing the validity of the corresponding discrete maximum principles are presented geometrically they mean that the employed meshes have to be of acute or nonobtuse type depending of the type of the problem finally some examples of real life problems where the preservation of maximum principles plays an important role are presented

the liouville equation with singular data a concentration compactness principle via a local representation formula abstract for a bounded domain r 2 we establish a concentration compactness result for the following class of singular liouville equations u e u 4 j 1 m j p j in where pj j 0 and pj denotes the dirac measure with pole at point pj j 1 m our result extends brezis merle s theorem comm partial differential equations16 1991 1223 1253 concerning solution sequences for the regular liouville equation where the dirac measures are replaced by lp data p 1 in some particular case we also derive a mass quantization principle in the same spirit of li shafrir indiana univ math j 43 1994 1255 1270 our analysis was motivated by the study of the bogomol nyi equations arising in several self dual gauge field theories of interest in theoretical physics such as the chern simons theory self dual chern simons theories lecture notes in physics vol 36 springer verlag berlin 1995 and the electroweak theory selected papers on gauge theory of weak and electromagnetic interactions world scientific singapore

determining conductivity by boundary measurements a p calderon pose la question est il possible de determiner la conductivite thermique d un objet a partir de mesures statiques de la temperature et du flux de chaleur a la limite on demontre que ces mesures determinent de facon unique la conductivite thermique et toutes ses derivees a la limite

a compactness result for periodic multivortices in the electroweak theory we derive a priori uniform bounds for solutions of an elliptic system of liouville type equations first analyzed by j spruck and y yang comm math phys 144 1992 1 yielding periodic multivortices in the classical electroweak theory of glashow salam weinberg our proof is based on a concentration quantization result in the same spirit of brezis merle comm partial differential equations 16 8 9 1991 1223 and li shafrir indiana univ math j 43 4 1994 1255 for mean field equations on riemannian compact 2 manifolds

energy quantization for a singular super liouville boundary value problem in this paper we develop the blow up analysis and establish the energy quantization for solutions to super liouville type equations on riemann surfaces with conical singularities at the boundary in other problems in geometric analysis the blow up analysis usually strongly utilizes conformal invariance which yields a noether current from which strong estimates can be derived here however the conical singularities destroy conformal invariance therefore we develop another more general method that uses the vanishing of the pohozaev constant for such solutions to deduce the removability of boundary singularities

the finite element method for nonlinear elliptic equations with discontinuous coefficients the study of the finite element approximation to nonlinear second order elliptic boundary value problems with discontinuous coefficients is presented in the case of mixed dirichlet neumann boundary conditions the change in domain and numerical integration are taken into account with the assumptions which guarantee that the corresponding operator is strongly monotone and lipschitz continuous the following convergence results are proved 1 the rate of convergenceo h if the exact solutionu h 1 is piecewise of classh 1 0 1 2 the convergence without any rate of convergence ifu h 1 only

the super toda system and bubbling of spinors abstract we introduce the super toda system on riemann surfaces and study the blow up analysis for a sequence of solutions to the super toda system on a closed riemann surface with uniformly bounded energy in particular we show the energy identities for the spinor parts of a blow up sequence of solutions for which there are possibly four types of bubbling solutions namely finite energy solutions of the super liouville equation or the super toda system defined on r 2 or on r 2 0 this is achieved by showing some new energy gap results for the spinor parts of these four types of bubbling solutions

on the construction of a potential from cauchy data abstract we investigate the problem of recovering a potential q y in the differential equation u q y u 0 x y 0 1 0 1 u 0 y u 1 y u x 0 0 u x 1 f x u y x 1 g x the method of separation of variables reduces the recovery of q y to a nonstandard inverse sturm liouville problem an asymptotic formula is developed that suggests that under appropriate conditions on the cauchy pair f g q y is uniquely determined up to the mean moreover the recovery of q y is comparable to finding a function from its polynomial moments a reconstruction scheme is suggested and numerical examples are considered

finite element solution of nonlinear elliptic problems the study of the finite element approximation to nonlinear second order elliptic boundary value problems with mixed dirichlet neumann boundary conditions is presented in the discretization variational crimes are commited approximation of the given domain by a polygonal one numerical integration with the assumption that the corresponding operator is strongly monotone and lipschitz continuous and that the exact solutionu h1 the convergence of the method is proved under the additional assumptionu h2 the rate of convergenceo h is derived without the use of green s theorem

a priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems the effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied under similar assumption on the quadrature formula as for linear problems optimal error estimates in the l 2 and the h 1 norms are proved the numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh the analysis is valid for both simplicial and rectangular finite elements of arbitrary order numerical experiments corroborate the theoretical convergence rates

stability estimates for the radon transform with restricted data and applications in this article we prove a stability estimate going from the radon transform of a function with limited angle distance data to the l p norm of the function itself under some conditions on the support of the function we apply this theorem to obtain stability estimates for an inverse boundary value problem with partial data

a new reconstruction method for the inverse potential problem abstract the inverse potential problem consists in reconstructing an unknown measure with support in a geometrical domain from a single boundary measurement in order to deal with this severely ill posed inverse problem we rewrite it as an optimization problem where a kohn vogelius type functional measuring the misfit between the solutions of two auxiliary problems is minimized one auxiliary problem contains information on the boundary measurement while the other one corresponds to the boundary excitation the solutions of the auxiliary problems coincide once the inverse problem is solved in order to minimize the kohn vogelius criterion its total variation with respect to a set of ball shaped perturbations on the measure is explicitly evaluated then a new method for solving the inverse potential problem based on the expression obtained is devised finally some numerical results are presented in order to show the effectiveness of the proposed reconstruction algorithm

finite element approximation of a problem with a nonlinear newton boundary condition the paper is concerned with the study of an elliptic boundary value problem with a nonlinear newton boundary condition the existence and uniqueness of the solution of the continuous problem is established with the aid of the monotone operator theory the main attention is paid to the investigation of the finite element approximation using numerical integration for the computation of nonlinear boundary integrals the solvability of the discrete finite element problem is proved and the convergence of the approximate solutions to the exact one is analysed

topological derivative based topology optimization of structures subject to drucker prager stress constraints abstract an algorithm for topology optimization of elastic structures under plane stress subject to the drucker prager stress constraint is presented the algorithm is based on the use of the topological derivative of the associated objective functional in conjunction with a level set representation of the structure domain in this context a penalty functional is proposed to enforce the point wise stress constraint and a closed formula for its topological derivative is derived the resulting algorithm is of remarkably simple computational implementation it does not require post processing procedures of any kind and features only a minimal number of user defined algorithmic parameters this is in sharp contrast with current procedures for topological structural optimization with local stress constraints the effectiveness and efficiency of the algorithm presented here are demonstrated by means of numerical examples the examples show in particular that it can easily handle structural optimization problems with underlying materials featuring strong asymmetry in their tensile and compressive yield strengths

finite element solution of flow problems with trailing conditions abstract the paper deals with the finite element solution of stream function problems describing nonviscous subsonic irrotational flows past profiles the main emphasis is laid on the treatment of the nonstandard trailing stagnation conditions which lead to physically admissible solutions the paper presents a general conception of stream function finite element modelling of complicated flow problems and a complete theory of the finite element approximations including the investigation of the existence and uniqueness of the solution of the nonsymmetric discrete problem and the convergence of approximate solutions to the exact solution

a quantization property for blow up solutions of singular liouville type equations abstract motivated by the study of self dual vortices in gauge field theory cf vortices and monopoles birkhauser boston 1980 solitons in field theory and nonlinear analysis monographs in mathematics springer new york 2001 we consider a concentrating solution sequence u k satisfying u k z 2 k v k z e u k in b 1 z 2 k v k z e u k z 0 weakly in the sense of measures in b 1 where i k r k ii v k c 0 1 b 1 0 a v k b and v k a in b 1 we prove that necessarily 8 n 8 1 8 z the result is sharp as shown by explicit examples and should be compared with that obtained by li shafrir ind univ math j 43 4 1994 1255 concerning the situation where k 0 k n

an inverse polynomial method for the identification of the leading coefficient in the sturm liouville operator from boundary measurements an inverse polynomial method of determining the unknown leading coefficient k k x of the linear sturm liouville operator au k x u x q x u x x 0 1 is presented as an additional condition only two measured data at the boundary x 0 x 1 are used in absence of a singular point u x 0 u x 0 x 0 1 the inverse problem is classified as a well conditioned if there exists at least one singular point then the inverse problem is classified as moderately ill conditioned u x 0 0 x 0 0 1 u x 0 x x 0 u x 0 x 0 1 and severely ill conditioned u x 0 u x 0 0 x 0 0 1 u x 0 u x 0 x x 0 for each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a cauchy problem for to the first order differential equation with respect the unknown function k k x an approximate analytical solution of the each cauchy problems are derived in explicit form numerical simulations all the above cases are given for noise free and noisy data an accuracy of the presented approach is demonstrated on numerical test solutions

a new method for inverse electromagnetic casting problems based on the topological derivative the inverse electromagnetic casting problem consists in looking for a suitable set of electric wires such that the electromagnetic field induced by an alternating current passing through them makes a given mass of liquid metal acquire a predefined shape in this paper we propose a new method for the topology design of such inductors the inverse electromagnetic casting problem is formulated as an optimization problem and topological derivatives are considered in order to locate new wires in the right position several numerical examples are presented showing that the proposed technique is effective to design suitable inductors

higher order topological sensitivity for 2 d potential problems application to fast identification of inclusions abstract this article concerns an extension of the topological derivative concept for 2 d potential problems involving penetrable inclusions whereby a cost function j is expanded in powers of the characteristic size e of a small inclusion the o e4 approximation of j is established for a small inclusion of given location shape and conductivity embedded in a 2 d region of arbitrary shape and conductivity and then generalized to several such inclusions simpler and more explicit versions of this result are obtained for a centrally symmetric inclusion and a circular inclusion numerical tests are performed on a sample configuration for i the o e4 expansion of potential energy and ii the identification of a hidden inclusion for the latter problem a simple approximate global search procedure based on minimizing the o e4 approximation of j over a dense search grid is proposed and demonstrated

on galerkin approximations of a quasilinear nonpotential elliptic problem of a nonmonotone type abstract we prove the convergence of galerkin approximations for a second order quasilinear elliptic problem of a divergence form with nonlinear mixed boundary conditions the existence and uniqueness of a weak and approximate solution is studied as well

on the finite element approximation of a cascade flow problem the finite element analysis of a cascade flow problem with a given velocity circulation round profiles is presented the nonlinear problem for the stream function with nonstandard boundary conditions is discretized by conforming linear triangular elements we deal with the properties of the discrete problem and study the convergence of the method both for polygonal and nonpolygonal domains including the effect of numerical integration

prescribing curvature on compact surfaces with conical singularities we study the berger nirenberg problem on surfaces with conical singularities i e we discuss conditions under which a function on a riemann surface is the gaussian curvature of some conformal metric with a prescribed set of singularities of conical types

on the continuum limit of a discrete inverse spectral problem on optimal finite difference grids we consider finite difference approximations of solutions of inverse stur m liouville problems in bounded intervals using three point finite difference sc hemes we discretize the equations on so called optimal grids constructed as follow s for a staggered grid with 2k points we ask that the finite difference operator a k k jacobi matrix and the sturm liouville differential operator share the k lowest eigenvalues and the values of the orthonormal eigenfunctions at one end of the interval this requirement determines uniquely the entries in the jacobi matrix which are grid cell averages of the coefficients in the continuum problem if these coefficients are known we can find the grid which we call optimal becau se it gives by design a finite difference operator with a prescribed spectr al measure we focus attention on the inverse problem where neither the coefficients n or the grid are known a key question in inversion is how to parametrize the coefficients i e how to choose the grid it is clear that to be successful this grid must be close to the optimal one which is unknown fortunately as we show here the grid dependence on the unknown coefficients is weak so the inversion can be done on a pr ecomputed grid for an a priori guess of the unknown coefficients this obse rvation leads to a simple yet efficient inversion algorithm which gives coefficien ts that converge pointwise to the true solution as the number k of data points tends to infinity the cornerstone of our convergence proof is showing that op timal grids provide an implicit natural regularization of the inverse problem by givin g reconstructions with uniformly bounded total variation the analysis is based on a novel explicit perturbation analysis of lanczos recursions and on a discrete

vanishing pohozaev constant and removability of singularities conformal invariance of two dimensional variational problems is a condition known to enable a blow up analysis of solutions and to deduce the removability of singularities in this paper we identify another condition that is not only sufficient but also necessary for such a removability of singularities this is the validity of the pohozaev identity in situations where such an identity fails to hold we introduce a new quantity called the pohozaev constant which on one hand measures the extent to which the pohozaev identity fails and on the other hand provides a characterization of the singular behavior of a solution at an isolated singularity we apply this to the blow up analysis for super liouville type equations on riemann surfaces with conical singularities because in the presence of such singularities conformal invariance no longer holds and a local singularity is in general non removable unless the pohozaev constant is vanishing