Extremal solutions of boundary value problems

To prove the existence of a solution of a two-point boundary value problem for an nth-order operator equation by the a priori estimate method, we study extremal solutions of auxiliary boundary value problems for an nth-order differential equation with simplest right-hand side, which have a unique solution under certain restrictions on the boundary conditions.     

Boussinesq–Cerruti problems for the hexagonally symmetric system of elasticity in n–dimensional half-space

The Poisson matrices of the analoga to the Boussinesq–Cerruti boundary value problems for the operator of transversely isotropic elastostatics in n–dimensional half-space are computed by Fourier transformation and given in explicit form. (Received: May 4, 2004; revised: January 30, 2006)     

A Mixed Boundary Value Problem for Polyanalytic Function of Order n in the Sobolev Space Wn,p (D)

We discuss the construction of a polyanalytic function Φ of order n on a simple bounded domain D. The function satisfies n prescribed generalized Riemann-Hilbert boundary conditions on the boundary ?D and n generalized jump conditions on a simple closed smooth contour γ contained in D. The boundary conditions are transformed into n classical Riemann-Hilbert problems and the n jump conditions into n Riemann problems of conjugation for some 2n holomorphic functions. These transformed problems are solved using the standard methods from the literature.     

On boundary value problems for an <Emphasis Type="Italic">n</Emphasis>th-order equation

For an nth-order differential equation with right-hand side satisfying the Carathéodory conditions, we consider a two-point boundary value problem with boundary conditions of which the first two have a general form and the remaining conditions are simplest. We show that the existence of a lower and an upper function and the compactness condition imply the solvability of this boundary value problem.     

Well-Posed Boundary Value Problems for Integrable Evolution Equations on a Finite Interval

We consider boundary value problems posed on an interval [0,L] for an arbitrary linear evolution equation in one space dimension with spatial derivatives of order n. We characterize a class of such problems that admit a unique solution and are well posed in this sense. Such well-posed boundary value problems are obtained by prescribing N conditions at x=0 and n–N conditions at x=L, where N depends on n and on the sign of the highest-degree coefficient _n in the dispersion relation of the equation. For the problems in this class, we give a spectrally decomposed integral representation of the solution; moreover, we show that these are the only problems that admit such a representation. These results can be used to establish the well-posedness, at least locally in time, of some physically relevant nonlinear evolution equations in one space dimension.     

How to change a boundary condition in a problem to make the problem have a prescribed spectrum

We consider a boundary value problem for an ordinary differential equation of order n with a spectral parameter in n boundary conditions. We suggest a method for changing one of the boundary conditions so as to make the problem have a prescribed spectrum.     

Abstract wave equations with acoustic boundary conditions

We define an abstract setting to treat wave equations equipped with time‐dependent acoustic boundary conditions on bounded domains of R ⁿ . We prove a well‐posedness result and develop a spectral theory which also allows to prove a conjecture proposed in [13]. Concrete problems are also discussed. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of nontrivial solutions for discrete elliptic boundary value problems

In this article, the existence of nontrivial solutions for the discrete elliptic boundary value problems is considered by using the extremum principle. Such system admits at least 2ⁿ nontrivial solutions when the nonlinear term is superlinear or sublinear. An explanation example is also given. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

New boundary element formulas for the biharmonic equation

We propose two new boundary integral equation formulas for the biharmonic equation with the Dirichlet boundary data that arises from plate bending problems in ℝ². Two boundary conditions, u and ∂u/∂n, usually yield a 2 × 2 non-symmetric matrix system of integral equations. Our new formulas yield scalar integral equations that can be handled more efficiently for theoretical and numerical purposes. In this paper we supply complete ellipticity and solvability analyses of our new formulas. Numerical experiments for simple Galerkin methods are also provided. This revised version was published online in June 2006 with corrections to the Cover Date.     

Analysis of differential operators associated with boundary value problems

We consider boundary value problems in a two-component domain of the Euclidean space R ⁿ obtained by eliminating from R ⁿ the boundary G. Traces on both sides of G are defined without limit passages. In a Hilbert trace space, we introduce orthogonal projections, analogs of the Calderon projections, which are used for constructing operators whose continuous invertibility implies the solvability of the corresponding boundary value problems. For the resolvents we obtain representations similar to the Krein formula. For a symmetric differential operator we show that the constructed resolvents of boundary value problems correspond to closed (not necessarily self-adjoint) extensions of this operator in the sense of von Neumann. Bibliography: 9 titles. Dedicated to N. N. Uraltseva Translated from Problemy Matematicheskogo Analiza, 40, May 2009, pp. 7–48.     

Higher order multi‐point boundary value problems

We consider the boundary value problem where n ? 2 and m ? 1 are integers, t_j ∈ [0, 1] for j = 1, …, m, and f and g_i, i = 0, …, n ? 1, are continuous. We obtain sufficient conditions for the existence of a solution of the above problem based on the existence of lower and upper solutions. Explicit conditions are also found for the existence of a solution of the problem. The differential equation has dependence on all lower order derivatives of the unknown function, and the boundary conditions cover many multi‐point boundary conditions studied in the literature. Schauder’s fixed point theorem and appropriate Nagumo conditions are employed in the analysis. Examples are given to illustrate the results. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

The Width-Volume Inequality

We prove that a bounded open set U in has k-width less than C(n) Volume(U) ^k/n . Using this estimate, we give lower bounds for the k-dilation of degree 1 maps between certain domains in . In particular, we estimate the smallest (n – 1)-dilation of any degree 1 map between two n-dimensional rectangles. For any pair of rectangles, our estimate is accurate up to a dimensional constant C(n). We give examples in which the (n – 1)-dilation of the linear map is bigger than the optimal value by an arbitrarily large factor. Received: January 2006, Revision: May 2006, Accepted: June 2006     

Partial Fourier approximation of the Lamé equations in axisymmetric domains

In this paper, we study the partial Fourier method for treating the Lamé equations in three‐dimensional axisymmetric domains subjected to non‐axisymmetric loads. We consider the mixed boundary value problem of the linear theory of elasticity with the displacement û , the body force f̂ ϵ (L₂)³ and homogeneous Dirichlet and Neumann boundary conditions. The partial Fourier decomposition reduces, without any error, the three‐dimensional boundary value problem to an infinite sequence of two‐dimensional boundary value problems, whose solutions û _n (n = 0, 1, 2,…) are the Fourier coefficients of û . This process of dimension reduction is described, and appropriate function spaces are given to characterize the reduced problems in two dimensions. The trace properties of these spaces on the rotational axis and some properties of the Fourier coefficients û _n are proved, which are important for further numerical treatment, e.g. by the finite‐element method. Moreover, generalized completeness relations are described for the variational equation, the stresses and the strains. The properties of the resulting system of two‐dimensional problems are characterized. Particularly, a priori estimates of the Fourier coefficients û _n and of the error of the partial Fourier approximation are given. Copyright © 1999 John Wiley & Sons, Ltd.     

Decomposition of complete graphs into 5‐cubes

Necessary conditions for the complete graph on n vertices to have a decomposition into 5‐cubes are that 5 divides n ? 1 and 80 divides n(n ? 1)/2. These are known to be sufficient when n is odd. We prove them also sufficient for n even, thus completing the spectrum problem for the 5‐cube and lending further weight to a long‐standing conjecture of Kotzig. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 159–166, 2006     

Deterministic equivalent for the Allen-Cahn energy of a scaling law in the Ising model

For the Allen-Cahn functional we study the following problem: for which prescribed amount m of volume is there the appearence of a droplet of one phase inside the other? Under a suitable assumption on the domain we show that the breaking of symmetry occurs at the same value of m as for the limit of the sharp interface energy. We also prove that there exists a threshold for m of order $\varepsilon^\frac{n}{n+1}For the Allen-Cahn functional we study the following problem: for which prescribed amount m of volume is there the appearence of a droplet of one phase inside the other? Under a suitable assumption on the domain we show that the breaking of symmetry occurs at the same value of m as for the limit of the sharp interface energy. We also prove that there exists a threshold for m of order so that either there is the appearence of the droplet or there is no breaking of symmetry.     

On Representation Formulas and Radiation Conditions

Solutions of linear elliptic partial differential equations in unbounded domains can be represented by boundary potentials if they satisfy certain conditions at infinity. These radiation conditions depend on the fundamental solution chosen for the integral representation. We prove some basic results about radiation conditions in a rather general framework. Fundamental solutions G are considered that are defined only on the complement of a compact set. It turns out, however, and we present examples for this, that the more interesting results only hold if G is defined on all of ℝⁿ or if it is a Green function for an exterior boundary value problem. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

The perturbed Sparre Andersen model with a threshold dividend strategy

In this paper, we consider a Sparre Andersen model perturbed by diffusion with generalized Erlang(n)-distributed inter-claim times and a threshold dividend strategy. Integro-differential equations with certain boundary conditions for the moment-generation function and the mth moment of the present value of all dividends until ruin are derived. We also derive integro-differential equations with boundary conditions for the Gerber–Shiu functions. The special case where the inter-claim times are Erlang(2) distributed and the claim size distribution is exponential is considered in some details.     

Generalized Stokes resolvent equations in an infinite layer with mixed boundary conditions

In this paper we prove unique solvability of the generalized Stokes resolvent equations in an infinite layer Ω₀ = ℝ^{n –1} × (–1, 1), n ≥ 2, in L^q ‐Sobolev spaces, 1 < q < ∞, with slip boundary condition of on the “upper boundary” ∂Ω⁺₀ = ℝ^{n –1} × {1} and non‐slip boundary condition on the “lower boundary” ∂Ω^–₀ = ℝ^{n –1} × {–1}. The solution operator to the Stokes system will be expressed with the aid of the solution operators of the Laplace resolvent equation and a Mikhlin multiplier operator acting on the boundary. The present result is the first step to establish an L^q ‐theory for the free boundary value problem studied by Beale [9] and Sylvester [22] in L ²‐spaces. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Lp–Lq maximal regularity for Stokes equations with Robin boundary condition in a bounded domain

We obtain the L_p–L_q maximal regularity of the Stokes equations with Robin boundary condition in a bounded domain in ?ⁿ (n?2). The Robin condition consists of two conditions: v ? u=0 and αu+β(T(u, p)v – 〈T(u, p)v, v〉v)=h on the boundary of the domain with α, β?0 and α+β=1, where u and p denote a velocity vector and a pressure, T(u, p) the stress tensor for the Stokes flow and v the unit outer normal to the boundary of the domain. It presents the slip condition when β=1 and non‐slip one when α=1, respectively. The slip condition is appropriate for problems that involve free boundaries. Copyright © 2006 John Wiley & Sons, Ltd.