Analysis of a semigroup approach in the inverse problem of identifying an unknown coefficient

This article presents a semigroup approach to the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u_x) in the quasi‐linear parabolic equation u_t(x, t)=(k(u_x)u_x(x, t))_x+F(x, t), with Dirichlet boundary conditions u(0, t)=ψ₀, u(1, t)=ψ₁ and source function F(x, t). The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[·]: ?? → C¹[0, T], Ψ[·]: ?? → C¹[0, T] via semigroup theory. Copyright © 2008 John Wiley & Sons, Ltd.     

Wave equations with time-dependent spatial operators of higher order

We study the initial-boundary value problem for ?_t²u(t,x)+A(t)u(t,x)+B(t)?_tu(t,x)=f(t,x) on [0,T]×Ω(Ω??ⁿ) with a homogeneous Dirichlet boundary condition; here A(t) denotes a family of uniformly strongly elliptic operators of order 2m, B(t) denotes a family of spatial differential operators of order less than or equal to m, and u is a scalar function. We prove the existence of a unique strong solution u. Furthermore, an energy estimate for u is given.     

Perturbation theorems for linear hyperbolic mixed problems and applications to the compressible euler equations

The main result of this paper (which is completely new, apart from our previous and less general result proved in reference [9]) states that the nonlinear system of equations (1.11) (or, equivalently, (1.10)) that describes the motion of an inviscid, compressible (barotropic) fluid in a bounded domain Ω, gives rise to a strongly well-posed problem (in the Hadamard classical sense) in spaces H^k(Ω), k ≧ 3; see Theorem 1.4 below. Roughly speaking, if (a_n, ?_n) → (a, ?) in H^k × H^k and if f_n → f in ??²(0, T;H^k), then (v_n, g_n) → (v, g) in ?? (0, T; H^k × H^k). The method followed here (see also [8]) also applies to the non-barotropic case p = p(ρ, s) (see [10]) and to other nonlinear problems. These results are based upon an improvement of the structural-stability theorem for linear hyperbolic equations. See Theorem 1.2 below. Added in proof: The reader is referred to [29], Part I, for a concise explanation of some fundamental points in the method followed here. © 1993 John Wiley & Sons, Inc.     

The Dirichlet energy of mappings from ${\bf B^3}$ into a manifold: density results and gap phenomenon

Weak limits of graphs of smooth maps with equibounded Dirichlet integral give rise to elements of the space . We assume that the 2-homology group of has no torsion and that the Hurewicz homomorphism is injective. Then, in dimension n = 3, we prove that every element T in , which has no singular vertical part, can be approximated weakly in the sense of currents by a sequence of smooth graphs {u _k } with Dirichlet energies converging to the energy of T. We also show that the injectivity hypothesis on the Hurewicz map cannot be removed. We finally show that a similar topological obstruction on the target manifold holds for the approximation problem of the area functional.Received: 9 May 2003, Accepted: 5 June 2003, Published online: 25 February 2004     

The Dirichlet problem for the minimal surface system in arbitrary dimensions and codimensions

Let Ω be a bounded C² domain in ?ⁿ and ? ?Ω → ?^m be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map f : Ω → ?^m with f|_?Ω = ? and with the graph of f a minimal submanifold in ?^n+m. For m = 1, the Dirichlet problem was solved more than 30 years ago by Jenkins and Serrin [12] for any mean convex domains and the solutions are all smooth. This paper considers the Dirichlet problem for convex domains in arbitrary codimension m. We prove that if ψ : ¯Ω → ?^m satisfies 8nδ sup_Ω |D²ψ| + √2 sup_?Ω |Dψ| < 1, then the Dirichlet problem for ψ|_?Ω is solvable in smooth maps. Here δ is the diameter of Ω. Such a condition is necessary in view of an example of Lawson and Osserman [13]. In order to prove this result, we study the associated parabolic system and solve the Cauchy‐Dirichlet problem with ψ as initial data. © 2003 Wiley Periodicals, Inc.     

Tracking and invariance problems for some classes of discrete delay systems

Consider the system with perturbation g _k ∈ ℝ ⁿ and output z _k = Cx _k . Here, A _k ,A _k ^(s) ∈ ℝ ^{n × n} , B _k ⁽¹⁾ ∈ ℝ ^{n × p} , B _k ⁽²⁾ ∈ ℝ ^{n × m} , C ∈ ℝ ^{p × n} . We construct a special Lyapunov-Krasovskii functional in order to synthesize controls u _k ⁽¹⁾ and u _k ⁽²⁾ for which the following properties are satisfied:

Successive projection method for solving the unbalanced Procrustes problem

We present a successive projection method for solving the unbalanced Procrustes problem: given matrix A∈Rn×n and B∈Rn×k, n>k, minimize the residual‖AQ-B‖F with the orthonormal constraint QTQ = Ik on the variant Q∈Rn×k. The presented algorithm consists of solving k least squares problems with quadratic constraints and an expanded balance problem at each sweep. We give a detailed convergence analysis. Numerical experiments reported in this paper show that our new algorithm is superior to other existing methods.     

Multidimensional Oscillations for Non-linear Hyperbolic Mixed Problem: The Justified Non-linear Geometric Optics Method

The mixed-Neumann problem for the non-linear wave equation □u−a(u)(∣∂_tu)∣²−∣∇u∣² = f_ε(z) is studied. The function f_ε(z) = ∑_k∈Kf_k(z,ε⁻¹ϕ_k(z),ε), ε∈[0,1], K is finite, f_k(z,θ_k,ε) are 2π-periodic with respect to θ_k. The existence of solution u_ε on a domain z = (t,x,y)∈[0,T]×ℝ₊×ℝ^d, d = 1 or 2, is proved when ε is sufficiently small; T does not depend on ε. By the non-linear geometric optics method the asymptotic (with respect to ε→0) solution ũ _ε is constructed. The estimation for the rest ε²r_ε = u_ε−ũ_ε is derived and the limit r_ε, ε→0, is studied.     

Convexity and all‐time C∞‐regularity of the interface in flame propagation

We consider the following one‐phase free boundary problem: Find (u, Ω) such that Ω = {u > 0} and with Q_T = ?ⁿ × (0, T). Under the condition that Ω_o is convex and log u_o is concave, we show that the convexity of Ω(t) and the concavity of log u(·, t) are preserved under the flow for 0 ≤ t ≤ T as long as ?Ω(t) and u on Ω(t) are smooth. As a consequence, we show the existence of a smooth‐up‐to‐the‐interface solution, on 0 < t < T_c, with T_c denoting the extinction time of Ω(t). We also provide a new proof of a short‐time existence with C^2,α initial data on the general domain. © 2002 John Wiley & Sons, Inc.     

Best Constants in the Miranda-Agmon Inequalities for Solutions of Elliptic Systems and the Classical Maximum Modulus Principle for Fluid and Elastic Half-spaces

The Dirichlet problem for elliptic systems of the second order with constant real and complex coefficients in the half-space  ^k ₊ = {x = (x ₁,…,x_k ): x_k > 0} is considered. It is assumed that the boundary values of a solution u = (u ₁,…,u _m) have the form ψ ₁ ξ ₁ + · · · + ψ _n ξ _n, 1 ≤ n ≤ m, where ξ ₁,· · ·,ξ _n is an orthogonal system of m-component normed vectors and ψ ₁,· · ·,ψ _n are continuous and bounded functions on ? ^k ₊. We study the mappings [C(? ^k ₊)] ⁿ ? (ψ ₁,…,ψ _n) → u(x) ?  ^m and [C(? ^k ₊)] ⁿ ? (ψ ₁,…,ψ _n) → u(x) ?  ^m generated by real and complex vector valued double layer potentials. We obtain representations for the sharp constants in inequalities between |u(x)| or |(z, u(x))| and ∥u| _xk =₀∥, where z is a fixed unit m-component vector, | · | is the length of a vector in a finite-dimensional unitary space or in Euclidean space, and (·,·) is the inner product in the same space. Explicit representations of these sharp constants for the Stokes and Lamé systems are given. We show, in particular, that if the velocity vector (the elastic displacement vector) is parallel to a constant vector at the boundary of a half-space and if the modulus of the boundary data does not exceed 1, then the velocity vector (the elastic displacement vector) is majorised by 1 at an arbitrary point of the half-space. An analogous classical maximum modulus principle is obtained for two components of the stress tensor of the planar deformed state as well as for the gradient of a biharmonic function in a half-plane.     

On the instability of resonances in parallel-plane waveguides under local perturbations of the boundary

We study the large-time asymptotics for solutions u( x , t) of the wave equation with Dirichlet boundary data, generated by a time-harmonic force distribution of frequency ω, in a class of domains with non-compact boundaries and show that the results obtained in [11] for a special class of local perturbations of Ω₀ ? ?² × (0,1) can be extended to arbitrary smooth local perturbations Ω of Ω₀. In particular, we prove that u is bounded as t → ∞ if Ω does not allow admissible standing waves of frequency ω in the sense of [8]. This implies in connection with [8]. Theorem 3.1 that the logarithmic resonances of the unperturbed domain Ω₀ at the frequencies ω = πk (k = 1, 2,…) observed in [14] can be simultaneously removed by small perturbations of the boundary. As a main step of our analysis, the determination of admissible solutions of the boundary value problem ΔU + κ²U = ? f in Ω, U = 0 on ?Ω is reduced to a compact operator equation.     

Semigroup approach for identification of the unknown diffusion coefficient in a quasi‐linear parabolic equation

This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u(x,t)) in the quasi‐linear parabolic equation u_t(x,t)=(k(u(x,t))u_x(x,t))_x, with Dirichlet boundary conditions u(0,t)=ψ₀, u(1,t)=ψ₁. The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[?]:?? →C¹[0,T], Ψ[?]:??→C¹[0,T] via semigroup theory. In this paper, it is shown that if the null space of the semigroup T(t) consists of only zero function, then the input–output mappings Φ[?] and Ψ[?] have the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of these mappings. Moreover, under the light of measured output data (boundary observations) f(t):=k(u(0,t))u_x(0,t) or/and h(t):=k(u(1,t))u_x(1,t), the values k(ψ₀) and k(ψ₁) of the unknown diffusion coefficient k(u(x,t)) at (x,t)=(0,0) and (x,t)=(1,0), respectively, can be determined explicitly. In addition to these, the values k_u(ψ₀) and k_u(ψ₁) of the unknown coefficient k(u(x,t)) at (x,t)=(0,0) and (x,t)=(1,0), respectively, are also determined via the input data. Furthermore, it is shown that measured output data f(t) and h(t) can be determined analytically by an integral representation. Hence the input–output mappings Φ[?]:??→ C¹[0,T], Ψ[?]:??→C¹[0,T] are given explicitly in terms of the semigroup. Copyright © 2007 John Wiley & Sons, Ltd.     

Analysis for the identification of an unknown diffusion coefficient via semigroup approach

This paper presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u_x) in the inhomogenenous quasi‐linear parabolic equation u_t(x, t)=(k(u_x)u_x(x, t))_x +F(u), with the Dirichlet boundary conditions u(0, t)=ψ₀, u(1, t)=ψ₁ and source function F(u). The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[·]:??→C¹[0, T], Ψ[·]:??→C¹[0, T] via semigroup theory. Copyright © 2009 John Wiley & Sons, Ltd.     

EMBEDDINGS OF ONE KIND OF GRAPHS

The purpose of this paper is to display a new kind of simple graphs which belong to B. inwhich any graph has its orientable genus n,n≥3. Furthermore, for any integer k,1≤k≤n,there exists a graph B^kn of B. such that the non-orientable genus of B^kn is k.     

Identification of the unknown diffusion coefficient in a quasi‐linear parabolic equation by semigroup approach with mixed boundary conditions

In this article, a semigroup approach is presented for the mathematical analysis of the inverse coefficient problems of identifying the unknown diffusion coefficient k(u(x, t)) in the quasi‐linear parabolic equation u_t(x, t)=(k(u(x, t))u_x(x, t))_x, with Dirichlet boundary conditions u_x(0, t)=ψ₀, u(1, t)=ψ₁. The main purpose of this work is to analyze the distinguishability of the input–output mappings Φ[·] : ??→C¹[0, T], Ψ[·] : ??→C¹[0, T] using semigroup theory. In this article, it is shown that if the null space of semigroups T(t) and S(t) consists of only a zero function, then the input–output mappings Φ[·] and Ψ[·] have the distinguishability property. Copyright © 2008 John Wiley & Sons, Ltd.     

Improving the Rate of Convergence of High-Order Finite Elements on Polyhedra I: A Priori Estimates

ABSTRACT

Let 𝒯 _k be a sequence of triangulations of a polyhedron Ω ? ? ⁿ and let S _k be the associated finite element space of continuous, piecewise polynomials of degree m. Let u _k  ∈ S _k be the finite element approximation of the solution u of a second-order, strongly elliptic system Pu = f with zero Dirichlet boundary conditions. We show that a weak approximation property of the sequence S _k ensures optimal rates of convergence for the sequence u _k . The method relies on certain a priori estimates in weighted Sobolev spaces for the system Pu = 0 that we establish. The weight is the distance to the set of singular boundary points. We obtain similar results for the Poisson problem with mixed Dirichlet–Neumann boundary conditions on a polygon.     

A note on the Picard bundle over a moduli space of vector bundles

Let ??(n , d ) be a coprime moduli space of stable vector bundles of rank n ≥ 2 and degree d over a complex irreducible smooth projective curve X of genus g ≥ 2 and ??_ξ ? ??(n , d ) a fixed determinant moduli space. Assuming that the degree d is sufficiently large, denote by ?? the vector bundle over X ×??(n , d ) defined by the kernel of the evaluation map H ⁰(X , E ) → E_x , where E ∈??(n , d ) and x ∈ X . We prove that ?? and its restriction ??_ξ to X × ??ξ are stable. The space of all infinitesimal deformations of ?? over X ×??(n , d ) is proved to be of dimension 3g and that of ??ξ over X × ??ξ of dimension 2g , assuming that g ≥ 3 and if g = 3 then n ≥ 4 and if g = 4 then n ≥ 3. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Edge intersection graphs of systems of paths on a grid with a bounded number of bends

We answer some of the questions raised by Golumbic, Lipshteyn and Stern and obtain some other results about edge intersection graphs of paths on a grid (EPG graphs). We show that for any d≥4, in order to represent every n vertex graph with maximum degree d as an edge intersection graph of n paths on a grid, a grid of area Θ(n²) is needed. We also show several results related to the classes B_k-EPG, where B_k-EPG denotes the class of graphs that have an EPG representation such that each path has at most k bends. In particular, we prove: For a fixed k and a sufficiently large n, the complete bipartite graph K_m,n does not belong to B_2m−3-EPG (it is known that this graph belongs to B_2m−2-EPG); for any odd integer k we have B_k-EPG B_k+1-EPG; there is no number k such that all graphs belong to B_k-EPG; only 2^O(knlog(kn)) out of all the labeled graphs with n vertices are in B_k-EPG.     

Weak convergence of currents

We give certain conditions to guarantee weak convergence u _k T _k → uT, where u _k , u are plurisubharmonic functions and T _k , T are positive closed currents. As applications we obtain that convergence in capacity of plurisubharmonic functions u _k implies weak convergence of the complex Monge-Ampère measures (dd ^c u _k ) ⁿ if all of the plurisubharmonic functions u _k are bounded below by one of some sorts of plurisubharmonic functions.     

Nonstationary weak limit of a stationary harmonic map sequence

Let M and N be two compact Riemannian manifolds. Let u_k be a sequence of stationary harmonic maps from M to N with bounded energies. We may assume that it converges weakly to a weakly harmonic map u in H^1,2 (M, N) as k → ∞. In this paper, we construct an example to show that the limit map u may not be stationary. © 2002 Wiley Periodicals, Inc.