uniform concentration compactness for sobolev spaces on variable domains abstract we present a new method for proving existence results in shape optimization problems involving the eigenvalues of the dirichlet laplace operator this method brings together the convergence theory and the concentration compactness principle given a sequence of open sets a n n n in r n not necessarily bounded but of uniformly bounded measure we prove a concentration compactness result in l l 2 r n for the sequence of resolvent operators r a n n n where r a n l 2 r n h 1 0 a n r a n 1

optimization relaxation and young measures we review the use of young measures in analyzing relaxed and generalized formulations for typical problems of optimization including variational principles optimal control problems models in materials science optimal design problems and nonlocal optimization problems

convergence in distribution of products of d d random matrices abstract in this paper we present limit theorems on the convergence in distribution of products of i i d matrices based on easily verifiable conditions on the support of the distribution

a multiphase shape optimization problem for eigenvalues qualitative study and numerical results in thie paper we consider the following multiphase shape optimization problem min big sum i 1 h lambda 1 omega i alpha omega i omega i open omega i subset d omega i cap omega

quantitative stability for the first dirichlet eigenvalue in reifenberg flat domains in rn abstract in this paper we prove that if and are close enough for the complementary hausdorff distance and their boundaries satisfy some geometrical and topological conditions then 1 1 c n where 1 resp 1 is the first dirichlet eigenvalue of the laplacian in resp and is the lebesgue measure of the symmetric difference here the constant 1 could be taken arbitrary close to 1 but strictly less and c is a constant depending on a lot of parameters including dimension n and some geometric properties of the domains

some necessary and sufficient conditions for the convergence of sequences of unilateral convex sets we establish some necessary and sufficient conditions for the convergence in w01 p of sequences of convex sets determined by obstacles these conditions are expressed in terms of properties of the p capacities of the level sets of the obstacles

nuclearity of semigroup c algebras and the connection to amenability abstract we study c algebras associated with subsemigroups of groups for a large class of such semigroups including positive cones in quasi lattice ordered groups and left ore semigroups we describe the corresponding semigroup c algebras as c algebras of inverse semigroups groupoid c algebras and full corners in associated group crossed products these descriptions allow us to characterize nuclearity of semigroup c algebras in terms of faithfulness of left regular representations and amenability of group actions moreover we also determine when boundary quotients of semigroup c algebras are uct kirchberg algebras this leads to a unified approach to cuntz algebras and ring c algebras

an existence result for optimal obstacles we consider the optimization problem min f g g x wheref g is a variational energy associated to the obstaclegand the classx of admissible obstacles is given byx g r g on g dx c with w1 p0 andc r fixed generally this problem does not have a solution and it may happen that the optimal obstacle is of relaxed form under a monotonicity assumption onf we prove the existence of a non relaxed optimal obstacle in the familyx through a new method based on the notions of andw convergences

the brauer group of a locally compact groupoid we define the brauer group br g of a locally compact groupoid g to be the set of morita equivalence classes of pairs a consisting of an elementary c bundle a over g 0 satisfying fell s condition and an action of g on a by isomorphisms when g is the transformation groupoid x h then br g is the equivariant brauer group br h x in addition to proving that br g is a group we prove three isomorphism results first we show that if g and h are equivalent groupoids then br g and br h are isomorphic this generalizes the result that if g and h are groups acting freely and properly on a space x say g on the left and h on the right then br g x h and br h g x are isomorphic secondly we show that the subgroup br0 g of br g consisting of classes a with a having trivial dixmier douady invariant is isomorphic to a quotient e g of the collection tw g of twists over g finally we prove that br g is isomorphic to the inductive limit ext g inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of the groups e g x where x varies over all principal g spaces x and g x is the imprimitivity groupoid associated to x

toeplitz algebras associated to endomorphisms of ore semigroups in this paper we consider the toeplitz algebra associated to actions of ore semigroups on c algebras in particular we consider injective and surjective actions of such semigroups we use the theory of groupoid dynamical systems to represent the toeplitz algebra as a groupoid crossed product we also discuss the k theory of the toeplitz algebra in some examples for instance we show that for the semigroup of positive matrices the k theory of the associated toeplitz algebra vanishes

optimization problems for an energy functional with mass constraint revisited abstract in this paper we report on a variational problem under a constraint on the mass which is motivated by the torsional rigidity and torsional creep following a device by alt caffarelli and friedman we treat instead a problem without constraint but with a penalty term we will complete some of the results of c bandle a wagner optimization problems for weighted sobolev constants calc var partial differential equations 29 2007 481 507 where the existence of a lipschitz continuous minimizer has been established in particular we prove qualitative properties of the optimal shape

regularity for shape optimizers the nondegenerate case we consider minimizers of f 1 n 1 where f is a function strictly increasing in each parameter and k is the kth dirichlet eigenvalue of our main result is that the reduced boundary of the minimizer is composed of c1 graphs and exhausts the topological boundary except for a set of hausdorff dimension at most n 3 we also obtain a new regularity result for vector valued bernoulli type free boundary problems 2018 wiley periodicals inc

an existence result for a class of shape optimization problems given a bounded open subset of rn we prove the existence of a minimum point for a functional f defined on the family a of all quasiopen subsets of under the assumption that f is decreasing with respect to set inclusion and that f is lower semicontinuous on a with respect to a suitable topology related to the resolvents of the laplace operator with dirichlet boundary condition applications are given to the existence of sets of prescribed volume with minimal kth eigenvalue or with minimal capacity with respect to a given elliptic operator

shape optimisation problems governed by nonlinear state equations the purpose of this paper is to give a compactness continuity result for the solution of a nonlinear dirichlet problem in terms of its domain variation the topology in the family of domains is given by the hausdorff metric and continuity is obtained under capacity conditions a generalisation of sverak s result in iv dimensions is deduced as a particular case

lipschitz regularity of the eigenfunctions on optimal domains we study the optimal sets omega ast subseteq mathbb r d for spectral functionals of the form f big lambda 1 omega ldots lambda p omega big which are bi lipschitz with respect to each of the eigenvalues lambda 1 omega lambda 2 omega ldots lambda p omega of the dirichlet laplacian on omega a prototype being the problem min big lambda 1 omega cdots lambda p omega omega subseteq mathbb r d omega 1 big we prove the lipschitz regularity of the eigenfunctions u 1 ldots u p of the dirichlet laplacian on the optimal set omega ast and as a corollary we deduce that omega ast is open for functionals depending only on a generic subset of the spectrum as for example lambda k omega our result proves only the existence of a lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved

shape optimization problems on metric measure spaces abstract we consider shape optimization problems of the form min j x m c where x is a metric measure space and j is a suitable shape functional we adapt the notions of convergence and weak convergence to this new general abstract setting to prove the existence of an optimal domain several examples are pointed out and discussed

worst case shape optimization for the dirichlet energy abstract we consider the optimization problem for a shape cost functional f f which depends on a domain varying in a suitable admissible class and on a right hand side f more precisely the cost functional f is given by an integral which involves the solution u of an elliptic pde in with right hand side f the boundary conditions considered are of the dirichlet type when the function f is only known up to some degree of uncertainty our goal is to obtain the existence of an optimal shape in the worst possible situation some numerical simulations are provided showing the difference in the optimal shape between the case when f is perfectly known and the case when only the worst situation is optimized

minimization of the k th eigenvalue of the robin laplacian abstract the paper is concerned with the minimization of the k th eigenvalue of the laplace operator with robin boundary conditions among all open sets of r n satisfying a volume constraint we prove the existence of a solution in a relaxed framework and find some qualitative properties of the optimal sets the main idea is to see these spectral shape optimization questions as free discontinuity problems in the framework of special functions of bounded variation one of the key difficulties for k 3 comes from the fact that the eigenvalues are critical points

characterization for the kuratowski limits of a sequence of sobolev spaces the purpose of this paper is to study the strong lower and the weak upper limits in the sense of kuratowski of a sequence of sobolev spaces w1 p0 n n nand compare them to a fixed spacew1 p0 the results are expressed in terms of the behavior of the local capacity in balls of the complementary sets if these two limits coeincide we obtain necessary and sufficient conditions for the convergence in the sense of mosco of the sobolev spaces hence for the continuity of the solution of a dirichlet problem associated to thep laplacian in terms of the geometric domain variation

wiener hopf operators on ordered homogeneous spaces i abstract the c algebra w p of wiener hopf operators related to an ordered homogeneous space g h is shown to be the homomorphic image of a groupoid c algebra to obtain this groupoid c algebra we construct an ordered compactification of g h the regularity of this compactification permits to show that w p contains the ideal of compact operators

minimization of the k th eigenvalue of the dirichlet laplacian for every k in mathbb n we prove the existence of a quasi open set minimizing the k th eigenvalue of the dirichlet laplacian among all sets of prescribed lebesgue measure moreover we prove that every minimizer is bounded and has a finite perimeter the key point is the observation that such quasi open sets are shape subsolutions for an energy minimizing free boundary problem

manifold derivative in the laplace beltrami equation abstract this paper is concerned with the derivative of the solution with respect to the manifold more precisely with theshape tangentialsensitivity analysis of the solution to the laplace beltrami boundary value problem with homogeneous dirichlet boundary conditions the domain is an open subset of a smooth compact manifold of r n the flow of a vector fieldv t changes in t and in t the relative boundary tof tin tis smooth enough andy t is the solution in tof the laplace dirichlet problem with zero boundary value on t the shape tangential derivative is characterized as being the solution of a similar non homogeneous boundary value problem that elementy v can be simply defined by the restriction to ofy y vwhereyis the material derivative ofyand yis the tangential gradient ofy the study splits in two parts whether the relative boundary of is empty or not in both cases the shape derivative depends on the deviatoric part of the second fundamental form of the surface on the fieldv 0 through its normal component on and on the tangential fieldv 0 through its normal component on the relative boundary we extend the structure results for theshape tangentialderivative making use ofintrinsic geometryapproach and intensive use of extension operators

relaxation of multidimensional variational problems with constraints of general form this paper is devoted to further developement of an idea of a well known theorem of bogolubov 2 here we construct a relaxation of multidimensional variational problems with constraints of rather general form on gradients of admissible functions it is assumed that the gradient of an admissible function belongs to an arbitrary bounded set this relaxation involves as a class of admissible functions the closure of the class of admissible functions of the original problem in the topology of uniform convergence and uses a theorem characterizing this closure which is proved in 15 the case when the gradient of an admissible function is constrained within a bounded closed convex body is studied in the works 13 15 19 the study of multidimensional variational problems was started in 1970s by ekeland and temam 13 the existing literature on relaxation of variational problems including two monographs by buttazzo 3 and dacorogna 9 and the review paper by marcellini 18 containing a considerable list of references is quite rich however the author failed to and a setting similar to that of the paper for the most recent results on relaxation and related topics see 1 4 8 11 14 this paper deals with the case where an integrand depends on a scalar function of several variables at the end of the paper we will make a conjecture on generalization of the main relaxation result of the paper to the case of an integrand depending on a vector function of several variables we also make a conjecture on generalization of

existence of minimizers for eigenvalues of the dirichlet laplacian with a drift abstract this paper deals with the eigenvalue problem for the operator l x with dirichlet boundary conditions we are interested in proving the existence of a set minimizing any eigenvalue k of l under a suitable measure constraint suggested by the structure of the operator more precisely we prove that for any c 0 and k n the following minimization problem min k quasi open set e x 2 2 d x c has a solution

minimization of the third eigenvalue of the dirichlet laplacian for every k in mathbb n we prove the existence of a quasi open set minimizing the k th eigenvalue of the dirichlet laplacian among all sets of prescribed lebesgue measure moreover we prove that every minimizer is bounded and has a finite perimeter the key point is the observation that such quasi open sets are shape subsolutions for an energy minimizing free boundary problem

multiphase shape optimization problems this paper is devoted to the analysis of multiphase shape optimization problems which can formally be written as min g f 1 omega 1 dots f h omega h m bigcup i 1 h omega i omega i subset d omega i cap omega j emptyset where d subseteq mathbb r d is a given bounded open set omega i is the lebesgue measure of omega i and m is a positive constant for a large class of such functionals we analyze qualitative properties of the cells and the interaction between them each cell is itself a subsolution for a single phase shape optimization problem from which we deduce properties like finite perimeter inner density separation by open sets absence of triple junction points etc as main examples we consider functionals involving the eigenvalues of the dirichlet laplacian of each cell i e f i lambda k i

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

minimizing the second eigenvalue of the laplace operator with dirichlet boundary conditions in this paper we are interested in the minimization of the second eigenvalue of the laplacian with dirichlet boundary conditions amongst convex plane domains with given area the natural candidate to be the optimum was the stadium a convex hull of two identical tangent disks we refute this conjecture nevertheless we prove the existence of a minimizer we also study some qualitative properties of the minimizer regularity geometric properties

semigroup c algebras and amenability of semigroups we construct reduced and full semigroup c algebras for left cancellative semigroups our new construction covers particular cases already considered by a nica and also toeplitz algebras attached to rings of integers in number fields due to j cuntz moreover we show how left amenability of semigroups can be expressed in terms of these semigroup c algebras in analogy to the group case