Dérivation par rapport au domaine pour un problème aux limites avec condition de Signorini

This work deals with a nonlinear inverse problem of reconstructing an unknown boundary γ, the boundary conditions prescribed on γ being of Signorini type, by using boundary measurements. The problem is turned into an optimal shape design one, by constructing a Kohn-Vogelius-like cost function, the only minimum of which is proved to be the unknown boundary. Furthermore, we prove that the derivative of this cost function with respect to a direction θ depends only on the state u⁰, and not on its Lagrangian derivative u¹ (θ).     

Optimal design of the cooling plunger cavity

An axisymmetric system of mould, glass piece, plunger and plunger cavity is considered. The state problem is given as a stationary head conduction process. The system includes the glass piece representing the heat source and is cooled inside the plunger cavity by flowing water and outside by the environment of the mould. The design variable is taken to be the shape of the inner surface of the plunger cavity. The cost functional is the second power of the norm in the weighted space L _r ² of difference of trace of temperature from given constant, which is evaluated on the outward boundary of the plunger. Existence and uniqueness of the state problem solution and existence of a solution of the optimization problem are proved.     

Coupling of FEM and BEM in Shape Optimization

In the present paper we consider the numerical solution of shape optimization problems which arise from shape functionals of integral type over a compact region of the unknown shape, especially L ²-tracking type functionals. The underlying state equation is assumed to satisfy a Poisson equation with Dirichlet boundary conditions. We proof that the shape Hessian is not strictly H ^1/2-coercive at the optimal domain which implies ill-posedness of the optimization problem under consideration. Since the adjoint state depends directly on the state, we propose a coupling of finite element methods (FEM) and boundary element methods (BEM) to realize an efficient first order shape optimization algorithm. FEM is applied in the compact region while the rest is treated by BEM. The coupling of FEM and BEM essentially retains all the structural and computational advantages of treating the free boundary by boundary integral equations.This research has been carried out when the second author stayed at the Department of Mathematics, Utrecht University, The Netherlands, supported by the EU-IHP project Nonlinear Approximation and Adaptivity: Breaking Complexity in Numerical Modelling and Data Representation     

On the problem of determining the structure of a layered medium and the shape of an impulse source

For the equation of wave propagation in the half-space ? ₊ ² + = {(x, y) ∈ ?² | y > 0} we consider the problem of determining the speed of wave propagation that depends only on the variable y and the shape of a point impulse source on the boundary of the half-space. We show that, under some assumptions on the shape of the source and the structure of the medium, both unknown functions of one variable are uniquely determined by the displacements of boundary points of the medium. We estimate stability of a solution to the problem.     

Optimal boundary control by an elastic force at one end and a displacement at the other end for an arbitrary sufficiently large time interval in the string vibration problem

We consider a boundary value problem for the wave equation with given initial conditions and with boundary conditions of the second kind at one end of the string and boundary conditions of the first kind at the other end of the string. We assume the boundary conditions to ensure that the solution of the problem (in the class of generalized functions) satisfying the initial conditions at the initial time t = 0 satisfies given terminal conditions at the terminal time t = T. We clarify the relationship between the functions µ(t) and ν(t) in the boundary conditions and the given functions specifying the initial and terminal states. We obtain closed-form analytic expressions for the functions µ(t) and ν(t) minimizing the boundary energy functional.     

Stationary rotating surfaces in Euclidean space

A stationary rotating surface is a compact surface in Euclidean space whose mean curvature H at each point x satisfies 2H(x) = a r(x)² + b, where r(x) denotes the distance from x to a fixed straight-line L, and a and b are constants. These surfaces are solutions of a variational problem that describes the shape of a drop of incompressible fluid in equilibrium by the action of surface tension when it rotates about L with constant angular velocity. The effect of gravity is neglected. In this paper we study the geometric configurations of such surfaces, focusing the relationship between the geometry of the surface and the one of its boundary. As special cases, we will consider two families of such surfaces: axisymmetric surfaces and embedded surfaces with planar boundary.     

On exact solutions of Stokes second problem for a Burgers�� fluid, II. The cases �� =?��2/4 and �� > ��2/4

In this study, the exact solutions of the Stokes second problem for a Burgers?? fluid are presented when the relaxation time satisfies the conditions ?? =???²/4 and ?? >???²/4. The velocity field and the associated tangential stress, when only one initial condition is necessary for velocity, are determined by means of the Laplace transform. The physical interpretation for the emerging parameters is discussed with the help of graphical illustrations. The similar solutions for the Stokes?? first problem are obtained as the limiting cases of our solutions.     

Elliptic boundary value problems with λ-dependent boundary conditions

In this paper second order elliptic boundary value problems on bounded domains Ω⊂Rn with boundary conditions on ∂Ω depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework. For a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure. In the special case of rational Nevanlinna or Riesz-Herglotz functions on the boundary the solution operator is obtained in an explicit form in the product Hilbert space L²(Ω)⊕(L²m(∂Ω)), which is a natural generalization of known results on λ-linear elliptic boundary value problems and λ-rational boundary value problems for ordinary second order differential equations.     

Estimates of solutions of a general initial-boundary-value problem for the linear nonstationary system of Navier-Stokes equations in a half-space

In the half-spaceX ₃>0 one considers the initial-boundary-value problem for the Stokes system in which the boundary conditions are given by means of an arbitrary matricial differential operator of size 3×4. One proves that, under certain restrictions on this operator, the solution satisfies coercive estimates in the norms of W_ρ ^2?,?.     

Dirichlet problem for pluriholomorphic functions of two complex variables

In this paper the Dirichlet problem for pluriholomorphic functions of two complex variables is investigated. The key point is the relation between pluriholomorphic functions and pluriharmonic functions. The link is constituted by the Fueter-regular functions of one quaternionic variable. Previous results about the boundary values of pluriharmonic functions and new results on L² traces of regular functions are applied to obtain a characterization of the traces of pluriholomorphic functions.     

A spectral method for parabolic differential equations

We present a spectral method for parabolic partial differential equations with zero Dirichlet boundary conditions. The region Ω for the problem is assumed to be simply-connected and bounded, and its boundary is assumed to be a smooth surface. An error analysis is given, showing that spectral convergence is obtained for sufficiently smooth solution functions. Numerical examples are given in both ?² and ?³.     

Estimate of solution stability in a two-dimensional inverse problem for elasticity equations

The problem of determining the density of the medium and one of its elasticity moduli is considered. Properties of the elastic medium and external forces are assumed to be independent of the coordinate x ₃. In this case, the third component of the displacement vector satisfies a scalar equation of the second order, which contains the density ρ of the medium and elasticity modulus μ as coefficients. The parameters ρ and μ are known to be positive and constant everywhere outside some compact domain D ? ?², but they are unknown inside D. The problem of determining these coefficients in D via information, given on the boundary of the domain D for some finite time interval, about a solution of two direct problems is considered. An estimate of the conditional stability of a solution of the inverse problem under consideration is established.     

A highly accurate homogeneous scheme for solving the laplace equation on a rectangular parallelepiped with boundary values in C k, 1

In this paper, a homogeneous scheme with 26-point averaging operator for the solution of Dirichlet problem for Laplace??s equation on rectangular parallelepiped is analyzed. It is proved that the order of convergence is O(h ⁴), where h is the mesh step, when the boundary functions are from C ^{3, 1}, and the compatibility condition, which results from the Laplace equation, for the second order derivatives on the adjacent faces is satisfied on the edges. Futhermore, it is proved that the order of convergence is O(h ⁶(|lnh| + 1)), when the boundary functions are from C ^{5, 1}, and the compatibility condition for the fourth order derivatives is satisfied. These estimations can be used to justify different versions of domain decomposition methods.     

Compact gradient tracking in shape optimization

In the present paper we consider the minimization of gradient tracking functionals defined on a compact and fixed subdomain of the domain of interest. The underlying state is assumed to satisfy a Poisson equation with Dirichlet boundary conditions. We proof that, in contrast to the situation of gradient tracking on the whole domain, the shape Hessian is not strictly H ^1/2-coercive at the optimal domain which implies ill-posedness of the shape problem under consideration. Shape functional and gradient require only knowledge of the Cauchy data of the state and its adjoint on the boundaries of the domain and the subdomain. These data can be computed by means of boundary integral equations when reformulating the underlying differential equations as transmission problems. Thanks to fast boundary element techniques, we derive an efficient algorithm to solve the problem under consideration.     

Local existence and non-relativistic limits of shock solutions to a multidimensional piston problem for the relativistic Euler equations

The multidimensional piston problem is a special initial-boundary value problem. The boundary conditions are given in two conical surfaces: one is the boundary of the piston, and the other is the shock whose location is to be determined later. In this paper, we are concerned with spherically symmetric piston problem for the relativistic Euler equations. A local shock front solution with the state equation p = a ² ρ, a is a constant and has been established by the Newton iteration. To overcome the difficulty caused by the free boundary, we introduce a coordinate transformation to fix it and employ the linear iteration scheme to establish a sequence of approximate solutions to the auxiliary problems by iteration. In each step, the value of the solution of the previous problem is taken as the data to determine the solution of the next problem. We obtain the existence of the original problem by establishing the convergence of these sequences. Meanwhile, we establish the convergence of the local solution as c → ∞ to the corresponding solution of the classical non-relativistic Euler equations.     

Trigonometric variable shape parameter and exponent strategy for generalized multiquadric radial basis function approximation

The generalized multiquadric radial basis function (φ_j=[(x-x_j)²+c²]^β) has the exponent β and shape parameter c that play an important role in the accuracy of the approximation. In this study, we present a trigonometric variable shape parameter and exponent strategy and apply it to function interpolations and linear boundary value problems. Several numerical experiments with the uniformly spaced nodes show that the inverse multiquadric radial basis function (β = −0.5) with the trigonometric variable shape parameter c strategy results in the best accuracy for the one-dimensional interpolations; the trigonometric variable shape parameters and exponent strategy produces the best accuracy for the two-dimensional interpolations and linear boundary value problems. For the non-uniformly spaced nodes, the random variable shape parameter c and exponent β strategy produces the best accuracy for the two-dimensional boundary value problem.     

Characterization of solutions to the generalized Cauchy-Riemann system

For a generalized biaxially symmetric potential U on a semi-disk D⁺, a harmonic conjugate V is defined by the generalized Cauchy-Riemann system. There is an associated boundary value theory for the Dirichlet problem. The converse to the Dirichlet problem is considered by determining the boundary functions to which U and V converge. The unique limits are hyperfunctions on the ?D⁺. In fact, the space of hyperfunctions is isomorphic to the spaces of generalized biaxially symmetric potentials and their harmonic conjugates. A representation theorem is given for U and V terms of convolutions of certain Poisson kernels with continuous functions that satisfy a growth condition on the ?D⁺.     

Asymptotic expansions for interior penalty solutions of control constrained linear-quadratic problems

We consider a quadratic optimal control problem governed by a nonautonomous affine ordinary differential equation subject to nonnegativity control constraints. For a general class of interior penalty functions, we provide a first order expansion for the penalized states and adjoint states around the state and adjoint state of the original problem. Our main argument relies on the following fact: if the optimal control satisfies strict complementarity conditions for its Hamiltonian except for a set of times with null Lebesgue measure, the functional estimates for the penalized optimal control problem can be derived from the estimates of a related finite dimensional problem. Our results provide several types of efficiency measures of the penalization technique: error estimates of the control for L ^s norms (s in [1, +∞]), error estimates of the state and the adjoint state in Sobolev spaces W ^1,s (s in [1, +∞)) and also error estimates for the value function. For the L ¹ norm and the logarithmic penalty, the sharpest results are given, by establishing an error estimate for the penalized control of order ${O(\varepsilon|\log\epsilon|)}$ where ${\varepsilon >0 }$ is the (small) penalty parameter.     

Optimal control of the obstacle in a quasilinear elliptic variational inequality

In this paper we consider an obstacle control problem where the state satisfies a quasilinear elliptic variational inequality and the control function is the obstacle. The state is chosen to be close to the desire profile while the H² norms of the obstacle is not too large. Existence and necessary conditions for the optimal control are established.     

Boundary value problems for two types of degenerate elliptic systems in <Emphasis Type="Italic">R</Emphasis><Superscript>4</Superscript>

Firstly, the Riemann boundary value problem for a kind of degenerate elliptic system of the first order equations in R ⁴ is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system’s solution, the boundary value problem as stated above is transformed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R ⁴ are derived.