Hadamard Formula Research Articles

On second-order tensor representation of derivatives in shape optimization.

In this article, we study general properties of distributed shape derivatives admitting a volumetric tensor representation of order two. We obtain a general result providing a range of expressions for the shape derivative, with the distributed shape derivative at one end of the range and the standard Hadamard formula at the other end. We further apply this result to a cost functional depending on the solution of a fourth-order elliptic equation, and obtain the distributed shape derivative in the case of open sets, and the Hadamard formula for sets of class [Formula: see text]. We also consider the case of polygons, for which a description of the weak singularities of the solution appearing in the neighbourhood of the vertices is required to obtain the Hadamard formula. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Lattice walks confined to an octant in dimension 3: (non-)rationality of the second critical exponent

In the field of enumeration of walks in cones, it is known how to compute asymptotically the number of excursions (finite paths in the cone with fixed length, starting and ending points, using jumps from a given step set). As it turns out, the associated critical exponent is related to the eigenvalues of a certain Dirichlet problem on a spherical domain. An important underlying question is to decide whether this asymptotic exponent is a (non-)rational number, as this has important consequences on the algebraic nature of the associated generating function. In this paper, we ask whether such an excursion sequence might admit an asymptotic expansion with a first rational exponent and a second non-rational exponent. While the current state of the art does not give any access to such many-term expansions, we look at the associated continuous problem, involving Brownian motion in cones. Our main result is to prove that in dimension three there exists a cone such that the heat kernel (the continuous analog of the excursion sequence) has the desired rational/non-rational asymptotic property. Our techniques come from spectral theory and perturbation theory. More specifically, our main tool is a new Hadamard formula, which has an independent interest and allows us to compute the derivative of eigenvalues of spherical triangles along infinitesimal variations of the angles.

Shape optimization for the Laplacian eigenvalue over triangles and its application to interpolation error analysis

A computer-assisted proof is proposed for the Laplacian eigenvalue minimization problems over triangular domains under diameter constraints. The paper provides an elementary analysis to derive the Hadamard shape derivative formula for complicated problem settings that allow non-homogeneous Neumann boundary conditions. Moreover, an algorithm to rigorously evaluate the Hadamard formula is proposed by utilizing the guaranteed computation for both eigenvalues and eigenfunctions of differential operators, which is recently developed by the second author. Besides the model homogeneous Dirichlet eigenvalue problem, the eigenvalue problem associated with a non-homogeneous Neumann boundary condition, which is related to the Crouzeix–Raviart interpolation error constant, is considered. The computer-assisted proof tells that among the triangles with the unit diameter, the equilateral triangle minimizes the first eigenvalue for each concerned eigenvalue problem.

The Hadamard formula for nonlocal eigenvalue problems

The Hadamard formula for nonlocal eigenvalue problems

A generalized fractional Pohozaev identity and applications

Abstract We prove a fractional Pohozaev-type identity in a generalized framework and discuss its applications. Specifically, we shall consider applications to the nonexistence of solutions in the case of supercritical semilinear Dirichlet problems and regarding a Hadamard formula for the derivative of Dirichlet eigenvalues of the fractional Laplacian with respect to domain deformations. We also derive the simplicity of radial eigenvalues in the case of radial bounded domains and apply the Hadamard formula to this case.

Hadamard\u2019s variational formula in terms of stress and strain tensors

Starting from a Lagrangian action functional for two scalar fields we construct, by variational methods, the Laplacian Green function for a bounded domain and an appropriate stress tensor. By a further variation, imposed by a given vector field, we arrive at an interior version of the Hadamard variational formula, previously considered by P. Garabedian. It gives the variation of the Green function in terms of a pairing between the stress tensor and a strain tensor in the interior of the domain, this contrasting the classical Hadamard formula which is expressed as a pure boundary variation.

A fractional Hadamard formula and applications

We derive a shape derivative formula for the family of principal Dirichlet eigenvalues lambda _s(Omega ) of the fractional Laplacian (-Delta )^s associated with bounded open sets Omega subset mathbb {R}^N of class C^{1,1}. This extends, with a help of a new approach, a result in Dalibard and Gérard-Varet (Calc. Var. 19(4):976–1013, 2013) which was restricted to the case s=frac{1}{2}. As an application, we consider the maximization problem for lambda _s(Omega ) among annular-shaped domains of fixed volume of the type Bsetminus overline{B}', where B is a fixed ball and B' is ball whose position is varied within B. We prove that lambda _s(Bsetminus overline{B}') is maximal when the two balls are concentric. Our approach also allows to derive similar results for the fractional torsional rigidity. More generally, we will characterize one-sided shape derivatives for best constants of a family of subcritical fractional Sobolev embeddings.

On Shape Optimization Theory with Fractional Laplacian

The fractional Laplacian is a nonlocal operator that appears in biology, in physic, in fluids dynamic, in financial mathematics and probability. This paper deals with shape optimization problem associated to the fractional laplacian ∆^s, 0 < s < 1. We focus on functional of the form <I>J</I>(Ω) = <i>j</i>(Ω,<i>u</i><SUB>Ω</SUB>) where <i>u</i><SUB>Ω</SUB> is solution to the fractional laplacian. A brief review of results related to fractional laplacian and fractional Sobolev spaces are first given. By a variational approach, we show the existence of a weak solution uΩbelonging to the fractional Sobolev spaces <I>D</I>^{s, 2}(Ω) of the boundary value problem considered. Then, we study the existence of an optimal shape of the functional <I>J</I>(Ω) on the class of admissible sets <img width="10" height="10" src="http://article.sciencepublishinggroup.com/journal/147/1471488/image001.png" /> under constraints volume. Finally, shape derivative of the functional is established by using Hadamard formula’s and an optimality condition is also given.

Hadamard asymptotics for eigenvalues of the Dirichlet Laplacian

This paper is dedicated to the classical Hadamard formula for asymptotics of eigenvalues of the Dirichlet-Laplacian under perturbations of the boundary. We prove that the Hadamard formula still holds for C1-domains with C1-perturbations. We also derive an optimal estimate for the remainder term in the C1,α-case. Furthermore, if the boundary is merely Lipschitz, we show that the Hadamard formula is not valid.

A localized domain perturbation which splits the spectrum of the Laplacian

For any Lipschitz domain we construct an arbitrarily small, localized perturbation which splits the spectrum of the Dirichlet, Neumann or Robin Laplacian into simple eigenvalues. We showcase two different approaches. The first one consists in the excision of a hole inside the domain and the perturbation of its boundary, and is based on a Hadamard's formula and sharp spectral stability estimates. The second one consists in the deformation of the boundary of the domain itself, and requires more technical properties of the bilinear form of the variational problem.

EIGENVALUES OF THE FINSLER p ‐LAPLACIAN ON VARYING DOMAINS

We study the dependence of the first eigenvalue of the Finsler $p$-Laplacian and the corresponding eigenfunctions upon perturbation of the domain and we generalize a few results known for the standard $p$-Laplacian. In particular, we prove a Frechet differentiability result for the eigenvalues, we compute the corresponding Hadamard formulas and we prove a continuity result for the eigenfunctions. Finally, we briefly discuss a well-known overdetermined problem and we show how to deduce the Rellich-Pohozaev identity for the Finsler $p$-Laplacian from the Hadamard formula.

Nilpotent Gelfand pairs and Schwartz extensions of spherical transforms via quotient pairs

It has been shown [1,2,9,10] that for several nilpotent Gelfand pairs (N,K) (i.e., with N a nilpotent Lie group, K a compact group of automorphisms of N and the algebra L1(N)K commutative) the spherical transform establishes a 1-to-1 correspondence between the space S(N)K of K-invariant Schwartz functions on N and the space S(Σ) of functions on the Gelfand spectrum Σ of L1(N)K which extend to Schwartz functions on Rd, once Σ is suitably embedded in Rd. We call this property (S).We present here a general bootstrapping method which allows to establish property (S) to new nilpotent pairs (N,K), once the same property is known for a class of quotient pairs of (N,K) and a K-invariant form of Hadamard formula holds on N.We finally show how our method can be recursively applied to prove property (S) for a significant class of nilpotent Gelfand pairs.

Holomorphic automorphisms of noncommutative polyballs

In this paper, we study free holomorphic functions on regular polyballs and provide analogues of several classical results from complex analysis such as: Abel theorem, Hadamard formula, Cauchy inequality, Schwarz lemma, and maximum principle. These results are used together with a class of noncommutative Berezin transforms to obtain a complete description of the group Aut(B_n) of all free holomorphic automorphisms of the polyball. The abstract polyball B_n has a universal model S consisting of left creation operators acting on tensor products of full Fock spaces. We determine: the group of automorphisms of the Cuntz-Toeplitz algebra C*(S) which leaves invariant the noncommutative polyball algebra A_n; the group of unitarily implemented automorphisms of the polyball algebra A_n and the noncommutative Hardy algebra F_n^\infty, respectively. We prove that the free holomorphic automorphism group Aut(B_n) is a sigma-compact, locally compact topological group with respect to the topology induced by an appropriate metric. Finally, we obtain a concrete unitary projective representation of the topological group Aut(B_n)in terms of noncommutative Berezin kernels associated with regular polyballs.

On differentiability of eigenvalues of second order elliptic operators on non-smooth domains

The regularity of eigenvalues of elliptic operators upon deformations of a given bounded domain is a classical problem in elliptic PDEs which has been focused by many authors. We establish a theorem on Cr dependence of algebraically simple eigenvalues and eigenfunctions with respect to perturbations of C1 class of non-smooth domains and of Cr class of coefficients of elliptic operators. Moreover, we also compute the first variation of these eigenvalues in relation to both parameters for non-smooth domains. As a byproduct, we extend Hadamard's formula to second order elliptic operators for domains of C2 class and other non-smooth ones.

A new method of convergence acceleration of series expansion for analytic functions in the complex domain

This paper proposes a new method of convergence acceleration of series expansion of complex functions which are analytic on and inside the unit circle in the complex plane. This class of complex functions may have some singularities outside the unit circle, which dominate convergence of series expansion. In the proposed method, the singular points are moved away from the origin using conformal mapping, and the function is expanded using a sequence of polynomials orthogonalized on the boundary of the mapped complex domain. The decay rate of coefficients of the orthogonal polynomial expansion can be related to the convergence region in a similar form to the Cauchy–Hadamard formula for power series. Using this relation, we quantitatively evaluate and maximize the convergence rate of the improved series. Numerical examples demonstrate that the proposed method is effective for slow convergent series, and may converge faster than Pade approximants.

On the Hadamard Formula for Second Order Systems in Non-Smooth Domains

Perturbations of eigenvalues of the Dirichlet problem for a second order elliptic system in a bounded domain Ω in ℝ n are studied under variations of the domain Ω. We investigate the case when the perturbed domain is located in a d-neighborhood of the reference Lipschitz domain. A new asymptotic formula is derived; it contains terms that are absent in the classical formula of Hadamard. The latter is valid only for smooth domains and smooth perturbations. We give conditions that guarantee the validity of the Hadamard formula. The general asymptotic formula is applied when Ω is perturbed by small curvilinear and circular cuts. Most of the results are new even for the Laplace operator.

Analog of the Hadamard formula for the first ellipse of meromorphy

Suppose that Pn are orthonormal polynomials on the closed interval [−1, 1] which are constructed from a weight function satisfying the Szego condition. In this paper, we obtain the first ellipse of meromorphy of the function F(z) = Σn=0∞FnPn(z), i.e., the maximal ellipse with foci at the points ±1 to which the function F can be extended as a meromorphic function having at most one pole.

Generalized Tomonaga-Schwinger equation from the Hadamard formula

A generalized Tomonaga-Schwinger equation, holding on the entire boundary of a finite spacetime region, has recently been considered as a tool for studying particle scattering amplitudes in background-independent quantum field theory. The equation has been derived using lattice techniques under assumptions on the existence of the continuum limit. Here I show that in the context of continuous Euclidean field theory the equation can be directly derived from the functional integral formalism, using a technique based on Hadamard's formula for the variation of the propagator.

Integrable Structure of the Dirichlet Boundary Problem in Two Dimensions

We study how the solution of the two-dimensional Dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem. We show that the Hadamard formula for the variation of the Dirichlet Green function under deformations of the domain reveals an integrable structure. The independent variables corresponding to the infinite set of commuting flows are identified with harmonic moments of the domain. The solution to the Dirichlet boundary problem is expressed through the tau-function of the dispersionless Toda hierarchy. We also discuss a degenerate case of the Dirichlet problem on the plane with a gap. In this case the tau-function is identical to the partition function of the planar large $N$ limit of the Hermitean one-matrix model.

Dual boundary element analysis using complex variables for potential problems with or without a degenerate boundary

The dual boundary element method in the real domain proposed by Hong and Chen in 1988 is extended to the complex variable dual boundary element method. This novel method can simplify the calculation for a hypersingular integral, and an exact integration for the influence coefficients is obtained. In addition, the Hadamard integral formula is obtained by taking the derivative of the Cauchy integral formula. The two equations (the Cauchy and Hadamard integral formula) constitute the basis for the complex variable dual boundary integral equations. After discretizing the two equations, the complex variable dual boundary element method is implemented. In determining the influence coefficients, the residue for a single-order pole in the Cauchy formula is extended to one of higher order in the Hadamard formula. In addition, the use of a simple solution and equilibrium condition is employed to check the influence matrices. To extract the finite part in the Hadamard formula, the extended residue theorem is employed. The role of the Hadamard integral formula is examined for the boundary value problems with a degenerate boundary. Finally, some numerical examples, including the potential flow with a sheet pile and the torsion problem for a cracked bar, are considered to verify the validity of the proposed formulation. The results are compared with those of real dual BEM and analytical solutions where available. A good agreement is obtained.