On the geometric form of solutions of a free boundary problem involving galvanization

In [6], T. I. Vogel studied a free boundary problem originating in the galvanization process. He showed that if the given boundary Γ* is starlike or convex, then so is the free boundary solution Γ. Our purpose is to generalize Vogel's second result by showing (under certain assumptions) that Γ cannot have more (local) maxima or minima (relative to a given direction) than Γ*; also that Γ cannot have more inflection points or greater total curvature than Γ*. The author has already proven analogous results for the Bernoulli free boundary problem in [1], [2] and [3].     

The elastic echo problem

We consider a homogeneous isotropic unbounded linear elastic medium Ω??³, having a free boundary Γ. A forcing f (t, x ) creates an incident displacement field u ₀(t, x ). This primary field is scattered by Γ giving rise to a secondary field or echo, for which we determine the asymptotic behaviour in time. These results are obtained via the use of an tension of the time‐dependent scattering theory of C. Wilcox. Copyright © 2003 John Wiley & Sons, Ltd.     

A conformal mapping algorithm for the Bernoulli free boundary value problem

We propose a new numerical method for the solution of the Bernoulli free boundary value problem for harmonic functions in a doubly connected domain D in where an unknown free boundary Γ₀ is determined by prescribed Cauchy data on Γ₀ in addition to a Dirichlet condition on the known boundary Γ₁. Our main idea is to involve the conformal mapping method as proposed and analyzed by Akduman, Haddar, and Kress for the solution of a related inverse boundary value problem. For this, we interpret the free boundary Γ₀ as the unknown boundary in the inverse problem to construct Γ₀ from the Dirichlet condition on Γ₀ and Cauchy data on the known boundary Γ₁. Our method for the Bernoulli problem iterates on the missing normal derivative on Γ₁ by alternating between the application of the conformal mapping method for the inverse problem and solving a mixed Dirichlet–Neumann boundary value problem in D. We present the mathematical foundations of our algorithm and prove a convergence result. Some numerical examples will serve as proof of concept of our approach. Copyright © 2015 John Wiley & Sons, Ltd.     

Probing for electrical inclusions with complex spherical waves

Let a physical body Ω in ?² or ?³ be given. Assume that the electric conductivity distribution inside Ω consists of conductive inclusions in a known smooth background. Further, assume that a subset Γ ? ?Ω is available for boundary measurements. It is proved using hyperbolic geometry that certain information about the location of the inclusions can be exactly recovered from static electric measurements on Γ. More precisely: given a ball B with center outside the convex hull of Ω and satisfying (B? ∩ ?Ω) ? Γ, boundary measurements on Γ with explicitly given Dirichlet data are enough to determine whether B intersects the inclusion. An approximate detection algorithm is introduced based on the theory. Numerical experiments in dimension two with simulated noisy data suggest that the algorithm finds the inclusion‐free domain near Γ and is robust against measurement noise. © 2007 Wiley Periodicals, Inc.     

Martin boundaries of random walks on Fuchsian groups

Let Γ be a finitely generated non-elementary Fuchsian group, and let μ be a probability measure with finite support on Γ such that supp μ generates Γ as a semigroup. If Γ contains no parabolic elements we show that for all but a small number of co-compact Γ, the Martin boundaryM of the random walk on Γ with distribution μ can be identified with the limit set Λ of Γ. If Γ has cusps, we prove that Γ can be deformed into a group Γ', abstractly isomorphic to Γ, such thatM can be identified with Λ', the limit set of Γ'. Our method uses the identification of Λ with a certain set of infinite reduced words in the generators of Γ described in [15]. The harmonic measure ν (ν is the hitting distribution of random paths in Γ on Λ) is a Gibbs measure on this space of infinite words, and the Poisson boundary of Γ, μ can be identified with Λ, ν.     

The Hele-Shaw problem with surface tension in a half-plane

In this paper, we consider the Hele-Shaw problem in a 2-dimensional fluid domain Ω(t) which is constrained to a half-plane. The boundary of Ω(t) consist of two components: Γ₀(t) which lies on the boundary of the half-plane, and Γ(t) which lies inside the half-plane. On Γ(t) we impose the classical boundary conditions with surface tension, and on Γ₀(t) we prescribe the normal derivative of the fluid pressure. At the point where Γ₀(t) and Γ(t) meet, there is an abrupt change in the boundary condition giving rise to a singularity in the fluid pressure. We prove that the problem has a unique solution with smooth free boundary Γ(t) for some small time interval.     

La monotonia di rappresentazioni del gruppo libero

Let Γ be a free nonabelian group on finitely many generators. Let Ω be the boundary of Γ, letC(Ω) be theC ^*-algebra of continuous functions on Ω, and let λ be the natural action of Γ onC(Ω). Aboundary representation is a representation of the crossed productC ^*-algebra Γ×_λ C(Ω). Given a unitary representation π of Γ onH, aboundary realization of π is an isometric Γ-inclusion ofH into the space of a boundary representation whose image is cyclic for that boundary representation. If the Γ-inclusion is bijective, we call, the realizationperfect. We prove below that if π admits an imperfect boundary realization, then there exists a nonzero vectorv ₀∈H satisfying $$\sum\limits_{|x| = n} {|\left\langle {v,\pi (x)v_0 } \right\rangle |^2 \leqslant |v|^2 } for each v \in {\mathcal{H}} (GVB)$$ If π is irreducible and weakly contained in the regular representation, and if no suchv ₀ exists, it follows that π satisfiesmonotony: up to equivalence, there exists exactly one realization of π, and that realization is perfect.     

Hölder norm estimate for the Hilbert transform in Clifford analysis

Let Ω ? ? ⁿ be a Jordan domain with d-summable boundary Γ. The main gol of this paper is to estimate the Hölder norm of a fractal version of the Hilbert transform in the Clifford analysis context acting from Hölder spaces of Clifford algebra valued functions defined on Γ. The explicit expression for the upper bound of the norm provided here is given in terms of the Hölder exponents, the diameter of Γ and certain d-sum (d > d) of the Whitney decomposition of Ω. The result obtained is applied to standard Hilbert transform for domains with left Ahlfors-David regular surface.     

Classical solutions of two-dimensional stokes problems on non-smooth domains. Part 1: The radon integral operators

The applicability of the Neumann indirect method of potentials to the Dirichlet and Neumann problems for the two-dimensional Stokes operator on a non-smooth boundary Γ is subject to two kinds of sufficient and/or necessary conditions on Γ. The first one, occurring in electrostatic, is equivalent to the boundedness on C(Γ) of the velocity double-layer potential W as well as to the existence of jump relations of potentials. The second condition, which forces Γ to be a simple rectifiable curve and which, compared to the Laplacian, is a stronger restriction on the corners of Γ, states that the Fredholm radius of W is greater than 2. Under these conditions, the Radon boundary integral equations defined by the above-mentioned jump relations are solvable by the Fredholm theory; the double- (for Dirichlet) and the single- (for Neumann) layer potentials corresponding to their solutions are classical solutions of the Stokes problems.     

On the solvability of the jump problem in clifford analysis

Let Ω be a bounded open and oriented connected subset of ? ⁿ which has a compact topological boundary Γ, let C be the Dirac operator in ? ⁿ , and let ?_0,n be the Clifford algebra constructed over the quadratic space ? ⁿ . An ?_0,n -valued smooth function f : Ω → ?_0,n in Ω is called monogenic in Ω if Df = 0 in Ω. The aim of this paper is to present the most general condition on Γ obtained so far for which a Hölder continuous function f can be decomposed as F ⁺ ? F ^? = f on Γ, where the components F ^± are extendable to monogenic functions in Ω^± with Ω⁺ := Ω, and Ω^? := ? ⁿ \ (Ω ? Γ), respectively.     

Polar boundary sets for superdiffusions and removable lateral singularities for nonlinear parabolic PDEs

Suppose L is a second-order elliptic differential operator in ℝ^d and D is a bounded, smooth domain in ℝ^d. Let 1 < α ≤ 2 and let Γ be a closed subset of ∂D. It is known [13] that the following three properties are equivalent: (α) Γ is ∂-polar; that is, Γ is not hit by the range of the corresponding (L, α)-superdiffusion in D; (β) the Poisson capacity of Γ is equal to 0; that is, the integral is equal to 0 or ∞ for every measure ν, where ρ(x) is the distance to the boundary and k(x, y) is the corresponding Poisson kernel; and (γ) Γ is a removable boundary singularity for the equation Lu = u^α in D; that is, if u ≥ 0 and Lu = u^α in D and if u = 0 on ∂D \ Γ, then u = 0. We investigate a similar problem for a parabolic operator in a smooth cylinder 𝒬 = ℝ₊ × D. Let Γ be a compact set on the lateral boundary of 𝒬. We show that the following three properties are equivalent: (a) Γ is 𝒢-polar; that is, Γ is not hit by the graph of the corresponding (L, α)-superdiffusion in 𝒬; (b) the Poisson capacity of Γ is equal to 0; that is, the integral is equal to 0 or ∞ for every measure ν, where k(r, x; t, y) is the corresponding (parabolic) Poisson kernel; and (c) Γ is a removable lateral singularity for the equation u˙ + Lu = u^α in 𝒬; that is, if u ≥ 0 and u˙ + Lu = u^α in 𝒬 and if u = 0 on ∂𝒬 \ Γ and on {∞} × D, then u = 0. © 1998 John Wiley & Sons, Inc.     

Asymptotic windings of horocycles

Let Γ be a torsion-free hyperbolic group. We study Γ-limit groups which, unlike the fundamental case in which Γ is free, may not be finitely presentable or geometrically tractable. We define model Γ-limit groups, which always have good geometric properties (in particular, they are always relatively hyperbolic). Given a strict resolution of an arbitrary Γ-limit group L, we canonically construct a strict resolution of a model Γ-limit group, which encodes all homomorphisms L → Γ that factor through the given resolution. We propose this as the correct framework in which to study Γ-limit groups algorithmically. We enumerate all Γ-limit groups in this framework.     

Numerical analysis of a mixed elliptic problem with flux and convective boundary conditions to obtain a discrete solution of nonconstant sign

We consider a material that occupies a convex polygonal bounded domain Ω ⊂ ℝⁿ, with regular boundary Γ = Γ₁ ∪ Γ₂ (with Γ ∩ Γ = ∅︁) with meas (Γ₁) = |Γ₁| > 0 and |Γ₂| > 0. We assume, without loss of generality, that the melting temperature is 0°C. We consider the following steady‐state heat conduction problem in Ω: with α, q, B = Const > 0, and q and α represent the heat flux on Γ₂ and the heat transfer coefficient on Γ₁, respectively. In a previous article (Tabacman‐ Tarzia, J Diff Eq 77 (1989), 16– 37) sufficient and/or necessary conditions on data α, q, B, Ω, Γ₁, Γ₂ to obtain a temperature u of nonconstant sign in Ω (that is, a multidimensional steady‐state, two‐phase, Stefan problem) were studied. In this article, we consider a regular triangulation by finite element method of the domain Ω with Lagrange triangles of the type 1, with h > 0 the parameter of the discretization. We study sufficient (and/or necessary) conditions on data α, q, B, Ω, Γ₁, and Γ₂ to obtain a change of phase (steady‐state, two‐phase, discretized Stefan problem) in corresponding discretized domain, that is, a discrete temperature of nonconstant sign in Ω. Moreover, error bounds as a function of the parameter h, are also obtained. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq. 15: 355–369, 1999     

Boundary problems for systems of partial differential operators of mixed order

On a compact n-dimensional C^∞ manifold $\text{Ω}$ with boundary Γ we consider a matrix A of linear partial differential operators that are not all of the same order. For such systems it is not evident what to regard as Cauchy data (on Γ). We introduce a definition of the so-called reduced Cauchy data, ranging in a vector bundle over Γ, which allows us (1) to set up a Cauchy problem (well posed in the noncharacteristic, analytic case) without the usual extra compatibility condition; (2) to construct boundary projectors in the space of reduced Cauchy data for Douglis-Nirenberg elliptic systems, with which the study of boundary problems is carried over to a (global) study of a pseudodifferential problems over Γ. Further applications (e.g. to spectral theory, outlined in [4]) will be given elsewhere.     

The existence of solutions of a general boundary value problem for the divergence

This paper is concerned with the question of necessary and sufficient conditions to find a vector field V∈ Γ(TM) solving the equation div V = Φ under inhomogeneous boundary conditions V|_?M = Z|_?M with Z∈ Γ (TM) An existence and regularity result is given for an arbitrary Riemannian manifold with boundary, M. The proof is based on the Hodge theory of differential forms.     

Regarding Some Problems of the Kutta — Joukovskii Condition in Lifting Surface Theory

We are interested in finding the velocity distribution at the wings of an aeroplane. Within the scope of a three — dimensional linear theory we analyse a model which is formulated as a mixed screen boundary value problem for the Helmholtz equation (Δ +k²)Φ = 0 in ?³\s where Φ denotes the perturbation velocity potential, induced by the presence of the wings and s :=L UW with the projection L of the wings onto the (x,y)- plane and the wake W. Not all Cauchy data are given explicitly on L, respectively W. These missing Cauchy data depend on the wing circulation Γ· Γ has to be fixed by the Kutta–Joukovskii condition: Λ Φ should be finite near the trailing edge x_t of L. To fulfil this condition in a way that all appearing terms can be defined mathematically exactly and belong to spaces which are physically meaningful, we propose to fix Γ by the condition of vanishing stress intensity factors of Φ near x_t up to a certain order such that ΛΦ|_xt ?W₂^?(x_t)? L₂(x_t),?>0. In the two–dimensional case, and if L is the left half–plane in ?², we have an explicit formula to calculate Γ and we can control the regularity of Γ and Φ.     

Finite element solutions for three‐dimensional elliptic boundary value problems on unbounded domains

The finite element (FE) solutions of a general elliptic equation ?div([a_ij] ??u) + u = f in an exterior domain Ω, which is the complement of a bounded subset of R ³, is considered. The most common approach to deal with exterior domain problems is truncating an unbounded subdomain Ω_∞, so that the remaining part Ω_B = Ω\Ω _∞ is bounded, and imposing an artificial boundary condition on the resulted artificial boundary Γ_a = Ω _∞ ∩ Ω _B. In this article, instead of discarding an unbounded subdomain Ω_∞ and introducing an artificial boundary condition, the unbounded domain is mapped to a unit ball by an auxiliary mapping. Then, a similar technique to the method of auxiliary mapping, introduced by Babu?ka and Oh for handling the domain singularities, is applied to obtain an accurate FE solution of this problem at low cost. This method thus does have neither artificial boundary nor any restrictions on f. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

On splitting theorems for CAT(0) spaces and compact geodesic spaces of non-positive curvature

In this paper, we show some splitting theorems for CAT(0) spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non-positive curvature. A CAT(0) group Γ is said to be rigid, if Γ determines its boundary up to homeomorphisms of a CAT(0) space on which Γ acts geometrically. C. Croke and B. Kleiner have constructed a non-rigid CAT(0) group. As an application of the splitting theorems for CAT(0) spaces, we obtain that if Γ₁ and Γ₂ are rigid CAT(0) groups then so is Γ₁ × Γ₂.     

Convergence of boundary optimal controls in mixed elliptic problems

We consider a steady-state heat conduction problem P_α withmixed boundary conditions for the Poisson equation in a bounded multidimensional domain Ω depending of a positive parameter α which represents the heat transfer coefficient on a portion Γ₁ of the boundary of Ω. We consider, for each α > 0, a cost function J_α and we formulate boundary optimal control problems with restrictions over the heat flux q on a complementary portion Γ₂ of the boundary of Ω. We obtain that the optimality conditions are given by a complementary free boundary problem in Γ₂ in terms of the adjoint state. We prove that the optimal control q and its corresponding system state u and adjoint state p for each α are strongly convergent to q_op, u and p in L²(Γ₂), H¹(Ω), and H¹(Ω) respectively when α → ∞. We also prove that these limit functions are respectively the optimal control, the system state and the adjoint state corresponding to another boundary optimal control problem with restrictions for the same Poisson equation with a different boundary condition on the portion Γ₁. We use the elliptic variational inequality theory in order to prove all the strong convergences. In this paper, we generalize the convergence result obtained in Ben Belgacem-El Fekih-Metoui, ESAIM:M2AN, 37 (2003), 833-850 by considering boundary optimal control problems with restrictions on the heat flux q defined on Γ₂ and the parameter α (which goes to infinity) is defined on Γ₁. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)