Boundary stabilization for a non‐linear beam on elastic bearings

In this paper, we study the equation under non‐linear boundary conditions which model the vibrations of a beam clamped at x=0 and supported by a non‐linear bearing at x=L. By adding only one damping mechanism at x=L, we prove the existence of a global solution and exponential decay of the energy. Copyright © 2001 John Wiley & Sons, Ltd.     

Existence of solutions for a class of nonlinear control problems

We consider problems of control and problems of optimal control, monitored by an abstract equation of the formEx=N _u x in a finite interval [0,T]; here,x is the state variable with values in a reflexive Banach space;u is the control variable with values in a metric space;E is linear and monotone; andN _u is nonlinear of the Nemitsky type. Thus, by well-known devices, the results apply also to parabolic partial differential equations in a cylinder [0,T]×G,G ⁿ , with Cauchy data fort=0 and Dirichlet or Neumann conditions on the lateral surface of the cylinder. We prove existence theorems for solutions and existence theorems for optimal solutions, by reduction to a theorem of Kemochi for reflexive Banach spaces.     

The convolution method for fourier expansions. The case of generalized transmission conditions of crack (screen) type in piecewise inhomogeneous media

We consider boundary value problems for the equation ? _x (K ? _x ?) + KL[?] = 0 in the space R ⁿ with generalized transmission conditions of the type of a strongly permeable crack or a weakly permeable screen on the surface x = 0. (Here L is an arbitrary linear differential operator with respect to the variables y ₁, …, y _n?1.) The coefficients K(x) > 0 are monotone functions of certain classes in the regions separated by the screen x = 0. The desired solution has arbitrary given singular points and satisfies a sufficiently weak condition at infinity.We derive formulas expressing the solutions of the above-mentioned problems in the form of simple quadratures via the solutions of the considered equation with a constant coefficient K for given singular points in the absence of a crack or a screen. In particular, the obtained formulas permit one to solve boundary value problems with generalized transmission conditions for equations with functional piecewise continuous coefficients in the framework of the theory of harmonic functions.     

The existence of ‐bounded solution operators of the thermoelastic plate equation with Dirichlet boundary conditions

We consider the linearized thermoelastic plate equation with the Dirichlet boundary condition in a general domain Ω, given by with the initial condition u|_(t=0)=u₀, u_t|_(t=0)=u₁, and θ|_(t=0)=θ₀ in Ω and the boundary condition u=∂_νu=θ=0 on Γ, where u=u(x,t) denotes a vertical displacement at time t at the point x=(x₁,⋯,x_n)∈Ω, while θ=θ(x,t) describes the temperature. This work extends the result obtained by Naito and Shibata that studied the problem in the half‐space case. We prove the existence of ‐bounded solution operators of the corresponding resolvent problem. Then, the generation of C₀ analytic semigroup and the maximal L_p‐L_q‐regularity of time‐dependent problem are derived.     

A new transformation of Burger’s equation for an exact solution in a bounded region necessary for certain boundary conditions

In this work, the transient analytic solution is found for the initial-boundary-value Burgers equation in 0?x?L. The boundary conditions are a homogeneous Dirichlet condition at x=0 and a constant total flux at x=L. The technique used consists of applying the transformation that reduces Burgers equation to a linear diffusion-advection equation. Previous work on this equation in a bounded region has only applied the Cole-Hopf transformation , which transforms Burgers equation to the linear diffusion equation. The Cole-Hopf transformation can only solve Burgers equation with constant Dirichlet boundary conditions, or time-dependent Dirichlet boundary conditions of the form u(0,t)=F₁(t) and u(L,t)=F₂(t),0?x?L. In this work, it is shown that the Cole-Hopf transformation will not solve Burgers equation in a bounded region with the boundary conditions dealt with in this work.     

Partitions of finite vector spaces into subspaces

Let V_n(q) denote a vector space of dimension n over the field with q elements. A set of subspaces of V_n(q) is a partition of V_n(q) if every nonzero element of V_n(q) is contained in exactly one element of . Suppose there exists a partition of V_n(q) into x_i subspaces of dimension n_i, 1 ≤ i ≤ k. Then x₁, …, x_k satisfy the Diophantine equation . However, not every solution of the Diophantine equation corresponds to a partition of V_n(q). In this article, we show that there exists a partition of V_n(2) into x subspaces of dimension 3 and y subspaces of dimension 2 if and only if 7x + 3y = 2ⁿ ? 1 and y ≠ 1. In doing so, we introduce techniques useful in constructing further partitions. We also show that partitions of V_n(q) induce uniformly resolvable designs on qⁿ points. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 329–341, 2008     

The heat equation under conditions on the moments in higher dimensions

We consider the heat equation on the N‐dimensional cube (0, 1)^N and impose different classes of integral conditions, instead of usual boundary ones. Well‐posedness results for the heat equation under the condition that the moments of order 0 and 1 are conserved had been known so far only in the case of , for which such conditions can be easily interpreted as conservation of mass and barycenter. In this paper we show that in the case of general N the heat equation with such integral conditions is still well‐posed, upon suitably relaxing the notion of solution. Existence of solutions with general initial data in a suitable space of distributions over (0, 1)^N are proved by introducing two appropriate realizations of the Laplacian and checking by form methods that they generate analytic semigroups. The solution thus obtained turns out to solve the heat equation only in a certain distributional sense. However, one of these realizations is tightly related to a well‐known object of operator theory, the Krein–von Neumann extension of the Laplacian. This connection also establishes well‐posedness in a classical sense, as long as the initial data are L²‐functions.     

Uniform convergence of the horizontal line method for solutions and free boundaries in Stefan evolution inequalities

The change of variable for the temperature Θ in the one-phase Stefan problem leads to the evolution inequality, (u_t – Δu – f)(v – u) ? 0 for all regular v ? 0, where u ? 0 is required. This inequality is to hold over a space-time domain D = Ω × (0, T) with a Dirichlet boundary condition imposed on ? Ω × (0, T) and a zero initial condition. The free boundary phase interface is given in one space dimension by The fully implicit divided difference scheme leads to a sequence of elliptic variational inequalities for {u_m}. The sequence {u_m} may be interpolated linearly in t to obtain an approximation U_Δt of u. The following results are obtained in this paper: (i) a two-sided weak maximum principle for u_m – u_m-1 in N space dimensions, hence the free boundary approximation for N = 1, is a monotone increasing step function; (ii) the uniform convergence of U_Δt and ?U_Δt, to u and ?u, respectively, on D ; (iii) the uniform convergence to the Hölder continuous, monotone increasing free boundary x on [0, T] of the piecewise linear sequence x_Δt, where x_Δt interpolates x _Δt, in one space dimension; (iv) a constructive existence proof for u and x in prescribed regularity classes.     

Non trivial L q solutions to the Ginzburg-Landau equation

It is shown that the contour problem for the stationary Ginzburg-Landau equation where =x/r with r=|x|, is well posed in L⁴(ⁿ) for a class of small data fL²(S^n–1).Mathematics Subject Classification (2000):35J05, 43A32Supported by a grant from Spanish Ministry of Education and CultureReceived: 16 December 2001     

Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity

We study the branch of semistable and unstable solutions (i.e., those whose Morse index is at most 1) of the Dirichlet boundary value problem ? Δu = λf(x)/(1 ? u)² on a bounded domain Ω ? ?^N, which models—among other things—a simple electrostatic microelectromechanical system (MEMS) device. We extend the results of 11 relating to the minimal branch, by obtaining compactness along unstable branches for 1 ≤ N ≤ 7 on any domain Ω and for a large class of “permittivity profiles” f. We also show the remarkable fact that powerlike profiles f(x) ? |x|^α can push back the critical dimension N = 7 of this problem by establishing compactness for the semistable branch on the unit ball, also for N ≥ 8 and as long as As a byproduct, we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for λ in a natural range. In all these results, the conditions on the space dimension and on the power of the profile are essentially sharp. © 2007 Wiley Periodicals, Inc.     

PiecewiseL-splines

Leta=x ₀<x ₁<...<x _N =b be a partition of the interval [a, b] and letL be a normalm-th order linear differential operator. The purpose of this note is to point out that spline functions in one variable need not be excluded to piecewise fits of functions belonging to the null spaceN(L ^* L) on each closed subinterval [x _i,x _i+1], 0in-1 but may be extended to piecewise fits of functions belonging toN(L _i ^* L _i) on each subinterval [x _i,x _i+1] provided theL _i's are selected from a uniformly bounded family of normal linear differential operators. Furthermore when theL _i's are so selected one obtains the usual integral relations and error estimates obtained for splines [2, 8 and 9] including the extended error estimates obtained by Swartz and Varga [10].     

Uniqueness of Solutions of Some Nonlocal Boundary-Value Problems for Operator-Differential Equations on a Finite Segment

For the equation L ₀ x(t) + L ₁ x ⁽¹⁾(t) + ... + L _n x ⁽ⁿ⁾(t) = 0, where L _k, k = 0, 1, ... , n, are operators acting in a Banach space, we formulate conditions under which a solution x(t) that satisfies some nonlocal homogeneous boundary conditions is equal to zero.     

A note on a third‐order multi‐point boundary value problem at resonance

Based on the coincidence degree theory of Mawhin, we prove some existence results for the following third‐order multi‐point boundary value problem at resonance where f: [0, 1] × R³ → R is a continuous function, 0 < ξ₁ < ??? < ξ_m < 1, α_i ∈ R, i = 1, …, m, m ≥ 1 and 0 < η₁ < η₂ < ??? < η_n < 1, β_j ∈ R, j = 1, 2, …, n, n ≥ 2. In this paper, the dimension of the linear space Ker L (linear operator L is defined by Lx = x′^′′) is equal to 2. Since all the existence results for third‐order differential equations obtained in previous papers are for the case dim Ker L = 1, our work is new.     

Tame Supercuspidal Representations of GL(n) Distinguished by a Unitary Group

This paper analyzes the space Hom_H(, 1), where is an irreducible, tame supercuspidal representation of GL(n) over a p-adic field and H is a unitary group in n variables contained in GL(n). It is shown that this space of linear forms has dimension at most one. The representations which admit nonzero H-invariant linear forms are characterized in several ways, for example, as the irreducible, tame supercuspidal representations which are quadratic base change lifts.Research supported in part by NSA grant #MDA904-99-1-0065.Research supported in part by NSERC     

On the global well‐posedness of discrete Boltzmann systems with chemical reaction

We study an initial‐boundary value problem in one‐space dimension for the discrete Boltzmann equation extended to a diatomic gas undergoing both elastic multiple collisions and chemical reactions. By integration of conservation equations, we prove a global existence result in the half‐space for small initial data N₀∈??∩L¹. Copyright © 2005 John Wiley & Sons, Ltd.     

Conditions for Correct Solvability of a Simplest Singular Boundary Value Problem

We consider a boundary value problem where f(x) ∈ L_p(R), p ∈ [1,∞] (L_∞(R) ≔ C(R) and 0 ≤ q(x) ∈ L^loc₁( R). Boundary value problem (0.1) is called correctly solvable in the given space L_p(R) if for any f(x) ∈ L_p(R) there is a unique solution y(x) ∞ L_p(R) and the following inequality holds with absolute constant c(p) ∈ (0,∞). We find criteria for correct solvability of the problem (0.1) in L_p(R).     

On boundary controllability of one-dimensional vibrating systems by W,p-controls for p ∈ [2, ∞]

This paper is concerned with boundary control of one-dimensional vibrating media whose motion is governed by a wave equation with a 2n-order spatial self-adjoint and positive-definite linear differential operator with respect to 2n boundary conditions. Control is applied to one of the boundary conditions and the control function is allowed to vary in the Sobolev space W, ^p for p∈[2, ∞] With the aid of Banach space theory of trigonometric moment problems, necessary and sufficient conditions for null-controllability are derived and applied to vibrating strings and Euler beams. For vibrating strings also, null-controllability by L^p-controls on the boundary is shown by a direct method which makes use of d'Alembert's solution formula for the wave equation.     

Maximum estimates for oblique derivative problems with right hand side in L p , p<n

If u is a solution of the second order elliptic differential equation Lu=f in some Lipschitz domain in ⁿ , we estimate the maximum of u in terms of some oblique derivative prescribed on the boundary of the domain and in terms of the L ^p norm of f with p<n.     

On the strong controllability of the diffusion equation

Let (x,t)y (x,t),x[0, 1],t[0,T], be the solution of the diffusion equation in one spatial variable corresponding to zero initial conditions and boundary controluL ₂(0,T). GivenfL ₂(0, 1), it is not possible, in general, to find a controlu such thaty(·,T)=f. We extend the space of controls in such a manner thatL ₂(0,T) can be considered to be a subset of a new spaceS of control elements; this space contains elements which do provide a solution to the problem of moments associated with the problem of makingy(·,T)=f inL ₂(0, 1). We show then that the action of the elements ofS can be approximated by that of control functions inL ₂(0,T) in a suitable manner.     

Spectral convergence for vibrating systems containing a part with negligible mass

We consider a set of Neumann (mixed, respectively) eigenvalue problems for the Laplace operator. Each problem is posed in a bounded domain Ω_R of ?ⁿ, with n=2,3, which contains a fixed bounded domain B where the density takes the value 1 and 0 outside. Ω_R has a diameter depending on a parameter R, with R?1, diam(Ω_R) →∞ as R→∞ and the union of these sets is the whole space ?ⁿ (the half space {x∈?ⁿ/x_n<0}, respectively). Depending on the dimension of the space n, and on the boundary conditions, we describe the asymptotic behaviour of the eigenelements as R→∞. We apply these asymptotics in order to derive important spectral properties for vibrating systems with concentrated masses. Copyright © 2005 John Wiley & Sons, Ltd.