phase transition with the line tension effect we make the connection between the geometric model for capillarity with line tension and the cahn hilliard model of two phase fluids to this aim we consider the energies f eps u eps int o du 2 1 over eps int o w u lambda int bd o v u where u is a scalar density function and w and v are double well potentials we show that the behaviour of f e in the limit e 0 and depends on the limit of e log if this limit is finite and strictly positive then the singular limit of the energies f e leads to a coupled problem of bulk and surface phase transitions and under certain assumptions agrees with the relaxation of the capillary energy with line tension these results were announced in abs1 and abs2

how to recognize convexity of a set from its marginals abstract we investigate the regularity of the marginals onto hyperplanes for sets of finite perimeter we prove in particular that if a set of finite perimeter has log concave marginals onto a e hyperplane then the set is convex our proof relies on measuring the perimeter of a set through a hilbertian fractional sobolev norm a fact that we believe has its own interest

existence and almost everywhere regularity of isoperimetric clusters for fractional perimeters abstract the existence of minimizers in the fractional isoperimetric problem with multiple volume constraints is proved together with a partial regularity result

a rate of convergence for monotone finite difference approximations to fully nonlinear uniformly elliptic pdes we obtain an algebraic rate of convergence for monotone and consistent finite difference approximations to lipschitz continuous viscosity solutions of uniformly elliptic partial differential equations 2007 wiley periodicals inc

the spherical fourier transform of rapidly decreasing functions a simple proof of a characterization due to harish chandra helgason trombi and varadarajan abstract harish chandra s helgason p c trombi and v s varadarajan have described the image of the l p schwartz spaces 0 p 2 under the spherical fourier transform we give a short proof of their results based on the paley wiener theorem

on orbit damage assessment for large space structures the need for monitoring the dynamic characteristics of large structural systems for the purpose of assessing the potential degradation of structural properties has been established this paper develops a theory for assessing the occurrence location and extent of potential damage utilizing on orbit response measurements the feasibility of the method is demonstrated using a simple structural system as an example

regularity theory for fully nonlinear integro differential equations we consider nonlinear integro differential equations like the ones that arise from stochastic control problems with purely jump levy processes we obtain a nonlocal version of the abp estimate harnack inequality and interior c 1 regularity for general fully nonlinear integro differential equations our estimates remain uniform as the degree of the equation approaches two so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations

ramanujanʼs master theorem for riemannian symmetric spaces abstract ramanujan s master theorem states that under suitable conditions the mellin transform of a power series provides an interpolation formula for the coefficients of this series based on the duality of compact and noncompact reductive riemannian symmetric spaces inside a common complexification we prove an analogue of ramanujan s master theorem for the spherical fourier transform of a spherical fourier series this extends the results proven by bertram for riemannian symmetric spaces of rank one

surface tension in ising systems with kac potentials we consider an ising spin system with kac potentials in a torus of d d 2 and fix the temperature below its lebowitz penrose critical value we prove that when the kac scaling parameter vanishes the log of the probability of an interface becomes proportional to its area and the surface tension related to the proportionality constant converges to the van der waals surface tension the results are based on the analysis of the rate functionals for gibbsian large deviations and on the proof that they converge to the perimeter functional of geometric measure theory which extends the notion of area our considerations include nonsmooth interfaces proving that the gibbsian probability of an interface depends only on its area and not on its regularity

wiener tauberian type theorems for radial sections of homogenous vector bundles on certain rank one riemannian symmetric spaces of noncompact type we will show that an uniform treatment yields wiener tauberian type results for various banach algebras and modules consisting of radial sections of some homogenous vector bundles on rank one riemannian symmetric spaces g k of noncompact type one example of such a vector bundle is the spinor bundle the algebras and modules we consider are natural generalizations of the commutative banach algebra of integrable radial functions on g k the first set of them are beurling algebras with analytic weights while the second set arises due to kunze stein phenomenon for noncompact semisimple lie groups

some monotonicity results for minimizers in the calculus of variations abstract we obtain monotonicity properties for minima and stable solutions of general energy functionals of the type f u u x d x under the assumption that a certain integral grows at most quadratically at infinity as a consequence we obtain several rigidity results of global solutions in low dimensions

existence of solutions for a bi nonlocal fractional p kirchhoff type problem in this paper we are concerned with the existence of nonnegative solutions for a p kirchhoff type problem driven by a non local integro differential operator with homogeneous dirichlet boundary data as a particular case we study the following problem m x u s p p p s u f x u u s p p in u 0 in r n u s p p r 2 n u x u y p x y n p s d x d y where p s is a fractional p laplace operator is an open bounded subset of r n with lipschitz boundary m r 0 r is a continuous function and f r r 0 r is a continuous function satisfying the ambrosetti rabinowitz type condition the existence of nonnegative solutions is obtained by using the mountain pass theorem and an iterative scheme the main feature of this paper lies in the fact that the kirchhoff function m depends on x and the nonlinearity f depends on the energy of solutions

on an isoperimetric problem with a competing nonlocal term ii the general case this paper is the continuation of a previous paper h knupfer and c b muratov comm pure appl math 66 2013 1129 1162 we investigate the classical isoperimetric problem modified by an addition of a nonlocal repulsive term generated by a kernel given by an inverse power of the distance in this work we treat the case of a general space dimension we obtain basic existence results for minimizers with sufficiently small masses for certain ranges of the exponent in the kernel we also obtain nonexistence results for sufficiently large masses as well as a characterization of minimizers as balls for sufficiently small masses and low spatial dimensionality the physically important special case of three space dimensions and coulombic repulsion is included in all the results mentioned above in particular our work yields a negative answer to the question if stable atomic nuclei at arbitrarily high atomic numbers can exist in the framework of the classical liquid drop model of nuclear matter in all cases the minimal energy scales linearly with mass for large masses even if the infimum of energy cannot be attained 2014 wiley periodicals inc

existence results for the fractional nirenberg problem abstract we consider the fractional nirenberg problem on the standard sphere s n with n 4 using the theory of critical points at infinity we establish an euler hopf type formula and obtain some existence results for curvature satisfying assumptions of bahri coron type

isoperimetry and stability properties of balls with respect to nonlocal energies we obtain a sharp quantitative isoperimetric inequality for nonlocal s perimeters uniform with respect to s bounded away from 0 this allows us to address local and global minimality properties of balls with respect to the volume constrained minimization of a free energy consisting of a nonlocal s perimeter plus a non local repulsive interaction term in the particular case s 1 the s perimeter coincides with the classical perimeter and our results improve the ones of knuepfer and muratov comm pure appl math 66 7 1129 1162 2013 comm pure appl math 2014 concerning minimality of balls of small volume in isoperimetric problems with a competition between perimeter and a nonlocal potential term more precisely their result is extended to its maximal range of validity concerning the type of nonlocal potentials considered and is also generalized to the case where local perimeters are replaced by their nonlocal counterparts

selection of noisy measurement locations for error reduction in static parameter identification an incomplete set of noisy static force and displacement measurements is used for parameter identification of structures at the element level measurement location and the level of accuracy in the measured data can drastically affect the accuracy of the identified parameters a heuristic method is presented to select a limited number of degrees of freedom dof to perform a successful parameter identification and to reduce the impact of measurement errors on the identified parameters this pretest simulation uses an error sensitivity analysis to determine the effect of measurement errors on the parameter estimates the selected dof can be used for nondestructive testing and health monitoring of structures two numerical examples one for a truss and one for a frame are presented to demonstrate that using the measurements at the selected subset of dof can limit the error in the parameter estimates

image of the schwartz space under spectral projection let x g k be a riemannian symmetric space of non compact type and rank one the spectral projection p f of a function f on x can be written p f f where is the elementary spherical function corresponding to the complex parameter we characterize the image of the schwartz space mathcal s p x under the spectral projection for 0 p 2

stiffness matrix adjustment using mode data a procedure is introduced that uses in addition to mode data structural connectivity information to optimally adjust deficient stiffness matrices the adjustments performed are such that the percentage change to each stiffness coefficient is minimized the physical configuration of the analytical model is preserved and the adjusted model will exactly reproduce the modes used in the identification the theoretical development is presented and the procedure is demonstrated by numerical simulation of a test problem

wave equation on riemannian symmetric spaces we consider the wave equation on riemannian symmetric spaces of the noncompact type we prove that the solutions of the homogeneous equation have dispersion properties and we deduce strichartz type estimates for these solutions we use these estimates to treat nonlinear equations with power type as a byproduct we prove the global well posedness for small energy

asymptotics for the fractional allen cahn equation and stationary nonlocal minimal surfaces this article is mainly devoted to the asymptotic analysis of a fractional version of the elliptic allen cahn equation in a bounded domain omega subseteq mathbb r n with or without a source term in the right hand side of the equation commonly called chemical potential in contrast to the usual allen cahn equation the laplace operator is here replaced by the fractional laplacian delta s with s in 0 1 2 as defined in fourier space in the singular limit varepsilon to 0 we show that arbitrary solutions with uniformly bounded energy converge both in the energetic and geometric sense to surfaces of prescribed nonlocal mean curvature in whenever the chemical potential remains bounded in suitable sobolev spaces with no chemical potential the notion of surface of prescribed nonlocal mean curvature reduces to the stationary version of the nonlocal minimal surfaces introduced by caffarelli et al commun pure appl math 63 1111 1144 2010 under the same sobolev regularity assumption on the chemical potential we also prove that surfaces of prescribed nonlocal mean curvature have a minkowski codimension equal to one and that the associated sets have a locally finite fractional 2s prime perimeter in for every s prime in 0 1 2

characterization of eigenfunctions of the laplace beltrami operator using fourier multipliers abstract let x be a rank one riemannian symmetric space of noncompact type and be the laplace beltrami operator of x the space x can be identified with the quotient space g k where g is a connected noncompact semisimple lie group of real rank one with finite center and k is a maximal compact subgroup of g thus g acts naturally on x by left translations through this identification a function or measure on x is radial i e depends only on the distance from ek when it is invariant under the left action of k we consider right convolution operators on functions f on x defined by f f where is a radial possibly complex measure on x these operators will be called multipliers in particular is a radial average when is a radial probability measure notable examples of radial averages are ball sphere and annular averages and the heat operator in this paper we address problems of the following type fix a multiplier in particular an averaging operator suppose that f k k z is a bi infinite sequence of functions on x such that for all k z f k a f k 1 and f k m for some constants a c m 0 and a suitable norm from this hypothesis we try to infer that f 0 hence every f k is an eigenfunction of

fractional sobolev norms and bv functions on manifolds abstract the bounded variation seminorm and the sobolev seminorm on compact manifolds are represented as a limit of fractional sobolev seminorms this establishes a characterization of functions of bounded variation and of sobolev functions on compact manifolds as an application the special case of sets of finite perimeter is considered

sensor placement methodologies for dynamic testing two methods are presented for structural sensor placement the first scheme selects the most linearly independent impulse responses at all candidate sensor locations from a gram schmidt orthogonalization procedure the second scheme is based on a principal component analysis and iteratively removes sensors that do not contribute significant information to the fisher information matrix furthermore use of a model reduction criterion is proposed to address the optimality issue several sensor placement methods were implemented to compare results and were applied to an euler bernoulli beam and a cantilevered frame structure it is shown that the proposed frequency criterion appears to be a selective criterion for choosing optimum sensor locations finally the optimum measurement locations from several of the methods studied yield acceptable results based on data from an experimental study on the frame structure

asymptotics of harish chandra expansions bounded hypergeometric functions associated with root systems and applications a series expansion for heckman opdam hypergeometric functions phi lambda is obtained for all lambda is an element of alpha c as a consequence estimates for phi lambda away from the walls of a weyl chamber are established we also characterize the bounded hypergeometric functions and thus prove an analogue of the celebrated theorem of helgason and johnson on the bounded spherical functions on a riemannian symmetric space of the noncompact type the l p theory for the hypergeometric fourier transform is developed for 0 p 2 in particular an inversion formula is proved when 1 p 2 c 2013 elsevier inc all rights reserved

a quantitative isoperimetric inequality for fractional perimeters abstract recently frank and seiringer have shown an isoperimetric inequality for nonlocal perimeter functionals arising from sobolev seminorms of fractional order this isoperimetric inequality is improved here in a quantitative form

on a fractional ginzburg landau equation and 1 2 harmonic maps into spheres this paper is devoted to the asymptotic analysis of a fractional version of the ginzburg landau equation in bounded domains where the laplacian is replaced by an integro differential operator related to the square root laplacian as defined in fourier space in the singular ginzburg landau limit we show that solutions with uniformly bounded energy converge weakly to sphere valued 1 2 harmonic maps i e the fractional analogues of the usual harmonic maps in addition the convergence holds in smooth functions spaces away from a n 1 rectifiable closed set of finite n 1 hausdorff measure the proof relies on the representation of the square root laplacian as a dirichlet to neumann operator in one more dimension and on the analysis of a boundary version of the ginzburg landau equation besides the analysis of the fractional ginzburg landau equation we also give a general partial regularity result for stationary 1 2 harmonic maps in arbitrary dimension

chaotic dynamics of the heat semigroup on riemannian symmetric spaces abstract we show that the heat semigroup generated by certain perturbations of the laplace beltrami operator on the riemannian symmetric spaces of noncompact type is chaotic on their l p spaces when 2 p both the range of p and the range of chaos inducing perturbation are sharp this extends a result of ji and weber 17 where it was shown that under identical conditions the heat operator is subspace chaotic on these spaces

fattening and nonfattening phenomena for planar nonlocal curvature flows we discuss fattening phenomenon for the evolution of sets according to their nonlocal curvature more precisely we consider a class of generalized curvatures which correspond to the first variation of suitable nonlocal perimeter functionals defined in terms of an interaction kernel k which is symmetric nonnegative possibly singular at the origin and satisfies appropriate integrability conditions we prove a general result about uniqueness of the geometric evolutions starting from regular sets with positive k curvature in mathbb r n and we discuss the fattening phenomenon in mathbb r 2 for the evolution starting from the cross showing that this phenomenon is very sensitive to the strength of the interactions as a matter of fact we show that the fattening of the cross occurs for kernels with sufficiently large mass near the origin while for kernels that are sufficiently weak near the origin such a fattening phenomenon does not occur we also provide some further results in the case of the fractional mean curvature flow showing that strictly starshaped sets in mathbb r n have a unique geometric evolution moreover we exhibit two illustrative examples in mathbb r 2 of closed nonregular curves the first with a lipschitz type singularity and the second with a cusp type singularity given by two tangent circles of equal radius whose evolution develops fattening in the first case and is uniquely defined in the second thus remarking the high sensitivity of the fattening phenomenon in terms of the regularity of the initial datum the latter example is in striking contrast to the classical case of the local curvature flow where two tangent circles always develop fattening as a byproduct of our analysis we provide also a simple proof of the fact that the cross in mathbb r 2 is not a k minimal set for the nonlocal perimeter functional associated to k

nonlocal operators of order near zero abstract we study dirichlet forms defined by nonintegrable levy kernels whose singularity at the origin can be weaker than that of any fractional laplacian we show some properties of the associated sobolev type spaces in a bounded domain such as symmetrization estimates hardy inequalities compact inclusion in l 2 or the inclusion in some lorentz space we then apply those properties to study the associated nonlocal operator l and the dirichlet and neumann problems related to the equations l u f x and l u f u in

nonlocal curvature flows this paper aims at building a unified framework to deal with a wide class of local and nonlocal translation invariant geometric flows we introduce a class of nonlocal generalized mean curvatures and prove the existence and uniqueness for the level set formulation of the corresponding geometric flows we then introduce a class of generalized perimeters whose first variation is an admissible generalized curvature within this class we implement a minimizing movements scheme and we prove that it approximates the viscosity solution of the corresponding level set pde we also describe several examples and applications besides recovering and presenting in a unified way existence uniqueness and approximation results for several geometric motions already studied and scattered in the literature the theory developed in this paper also allows us to establish new results

optimum sensitivity based statistical parameters estimation from modal response based on the concept of optimum sensitivity we present a method for estimating the mean and covariance of parameters of a mechanical system from the statistics of its measured modal response the optimum sensitivity defined as the sensitivity of system parameters with respect to observed output is obtained by direct differentiation of the kuhn tucker optimality criteria for a nonlinear least squares output error estimator with the optimum sensitivity derivatives up to the second order we can estimate the second order approximation of both the mean and covariance of the system parameters by applying methods developed originally for evaluating the output of uncertain systems based on the more conventional notion of sensitivity the sensitivity of system response with respect to system parameters the present approach allows us to assess the bias due to nonlinearities in the least squares estimator whereas conventional sensitivity based methods do not furthermore the present method is generally much more efficient than monte carlo simulation because nonlinear optimization is performed only once we demonstrate through example problems that compared to the conventional sensitivity based methods the present method provides statistical indices that are more consistent with those obtained by monte carlo simulation

boundary behavior of nonlocal minimal surfaces abstract we consider the behavior of the nonlocal minimal surfaces in the vicinity of the boundary by a series of detailed examples we show that nonlocal minimal surfaces may stick at the boundary of the domain even when the domain is smooth and convex this is a purely nonlocal phenomenon and it is in sharp contrast with the boundary properties of the classical minimal surfaces in particular we show stickiness phenomena to half balls when the datum outside the ball is a small half ring and to the side of a two dimensional box when the oscillation between the datum on the right and on the left is large enough when the fractional parameter is small the sticking effects may become more and more evident moreover we show that lines in the plane are unstable at the boundary namely small compactly supported perturbations of lines cause the minimizers in a slab to stick at the boundary by a quantity that is proportional to a power of the perturbation in all the examples we present concrete estimates on the stickiness phenomena also we construct a family of compactly supported barriers which can have independent interest

regularity properties of nonlocal minimal surfaces via limiting arguments abstract we prove an improvement of flatness result for nonlocal minimal surfaces which is independent of the fractional parameter s when s 1 as a consequence we obtain that all the nonlocal minimal cones are flat and that all the nonlocal minimal surfaces are smooth when the dimension of the ambient space is less or equal than 7 and s is close to 1

finite element discretizations of nonlocal minimal graphs convergence abstract in this paper we propose and analyze a finite element discretization for the computation of fractional minimal graphs of order s 0 1 2 on a bounded domain such a plateau problem of order s can be reinterpreted as a dirichlet problem for a nonlocal nonlinear degenerate operator of order s 1 2 we prove that our numerical scheme converges in w 1 2 r for all r s where w 1 2 s is closely related to the natural energy space moreover we introduce a geometric notion of error that for any pair of h 1 functions in the limit s 1 2 recovers a weighted l 2 discrepancy between the normal vectors to their graphs we derive error bounds with respect to this novel geometric quantity as well in spite of performing approximations with continuous piecewise linear lagrangian finite elements the so called stickiness phenomenon becomes apparent in the numerical experiments we present

sensitivity analysis of discrete structural systems methods for calculating sensitivity derivatives for discrete structural systems are surveyed primarily covering literature published during the past two decades methods are described for calculating derivatives of static displacements and stresses eigenvalues and eigenvectors transient structural response and derivatives of optimum structural designs with respect to problem parameters the survey is focused on publications developed in nonstructural fields such as electronics controls and physical chemistry which are directly applicable to structural problems most notable among the nonstructural based methods are the adjoint variable technique from control theory and the green s function and fast methods from physical chemistry

nonlocal minimal clusters in the plane abstract we prove existence of partitions of an open set with a given number of phases which minimize the sum of the fractional perimeters of all the phases with dirichlet boundary conditions in two dimensions we show that if the fractional parameter s is sufficiently close to 1 the only singular minimal cone that is the only minimal partition invariant by dilations and with a singular point is given by three half lines meeting at 120 degrees in the case of a weighted sum of fractional perimeters we show that there exists a unique minimal cone with three phases

nonlocal minimal graphs in the plane are generically sticky we prove that nonlocal minimal graphs in the plane exhibit generically stickiness effects and boundary discontinuities more precisely we show that if a nonlocal minimal graph in a slab is continuous up to the boundary then arbitrarily small perturbations of the far away data produce boundary discontinuities hence either a nonlocal minimal graph is discontinuous at the boundary or a small perturbation of the prescribed conditions produces boundary discontinuities the proof relies on a sliding method combined with a fine boundary regularity analysis based on a discontinuity smoothness alternative namely we establish that nonlocal minimal graphs are either discontinuous at the boundary or their derivative is holder continuous up to the boundary in this spirit we prove that the boundary regularity of nonlocal minimal graphs in the plane jumps from discontinuous to c 1 gamma with no intermediate possibilities allowed in particular we deduce that the nonlocal curvature equation is always satisfied up to the boundary as an interesting byproduct of our analysis one obtains a detailed understanding of the switch between the regime of continuous and hence differentiable nonlocal minimal graphs to that of discontinuous and hence with differentiable inverse ones

strategy for finding a near optimal measurement set for parameter estimation from modal response algorithms that estimate structural parameters from modal response using least squares minimization of force or displacement residuals generally do not have unique solutions when the data are spatially sparse the number and character of the multiple solutions depend upon the physical features of the structure and the locations of the response measurements it has been observed that both the number of solutions and the sensitivity of the parameter estimates to measurement noise is greatly influenced by the choice of measurement locations in this paper we present a heuristic method to select a near optimal subset of measurement locations starting from a particular set of measurements by minimizing the sensitivity of the parameter estimates with respect to observed response the statistical properties of solution clusters generated from a monte carlo samples of noisy data are used to determine the best candidate measurement to be dropped from the current set the process is repeated until solution sensitivity cannot be significantly reduced we also show that the laborious monte carlo computations can be avoided in certain cases by using a direct computation of sensitivity estimates a numerical example is provided to illustrate the method and to examine the performance of the proposed algorithm

interpolations and fractional sobolev spaces in carnot groups abstract in this paper we present an interpolation approach to the fractional sobolev spaces in carnot groups using the k method this approach provides us with a different characterization of these sobolev spaces moreover it provides us with the limiting behavior of the fractional sobolev norms at the end points this allows us to deduce results similar to the bourgain brezis mironescu and maz ya shaposhnikova in the case p 1 and davila s result in the case p 1 also this allows us to deduce the limiting behavior of the fractional perimeter in carnot groups

approximation schemes for propagation of fronts with nonlocal velocities and neumann boundary conditions the convergence of schemes for propagation of fronts in a bounded domain moving with normal velocities is studied the velocities considered depend on the principal curvatures the normal direction the location as well as some nonlocal properties of the front most of the schemes considered are in essence threshold dynamics type approximation schemes modified for neumann boundary conditions and nonlocal terms the existence and uniqueness of appropriately defined viscosity solutions of the level set equations describing the nonlocal motions is also shown

existence of solutions for kirchhoff type problem involving the non local fractional p laplacian abstract the purpose of this paper is to investigate the existence of weak solutions for a kirchhoff type problem driven by a non local integro differential operator of elliptic type with homogeneous dirichlet boundary conditions as follows m r 2 n u x u y p k x y d x d y l k p u f x u in u 0 in r n where l k p is a non local operator with singular kernel k is an open bounded subset of r n with lipshcitz boundary m is a continuous function and f is a caratheodory function satisfying the ambrosetti rabinowitz type condition we discuss the above mentioned problem in two cases when f satisfies sublinear growth condition the existence of nontrivial weak solutions is obtained by applying the direct method in variational methods when f satisfies suplinear growth condition the existence of two nontrivial weak solutions is obtained by using the mountain pass theorem

on higher dimensional singularities for the fractional yamabe problem a nonlocal mazzeo pacard program we consider the problem of constructing solutions to the fractional yamabe problem which are singular at a given smooth submanifold for which we establish the classical gluing method of mazzeo and pacard j differential geom 1996 for the scalar curvature in the fractional setting this proof is based on the analysis of the model linearized operator which amounts to the study of a fractional order ordinary differential equation ode thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of nonlocal odes note however that no traditional phase plane analysis is available here instead we first provide a rigorous construction of radial fast decaying solutions by a blowup argument and a bifurcation method then second we use conformal geometry to rewrite this nonlocal ode giving a hint of what a nonlocal phase plane analysis should be third for the linear theory we use complex analysis and some non euclidean harmonic analysis to examine a fractional schrodinger equation with a hardy type critical potential we construct its green s function deduce fredholm properties and analyze its asymptotics at the singular points in the spirit of frobenius method surprisingly enough a fractional linear ode may still have a 2 dimensional kernel as in the second order case