On Beurling’s boundary differential relation

We prove that for an arbitrary positive continuous function Φ on the complex plane ? there exists an injective disc algebra function φ and n ∈ ? such that ξ ? φ(ξ ⁿ ) solves Beurling’s boundary differential relation |f′(ξ)| = Φ(f(ξ)) on ?Δ. Moreover, if the growth of Φ is sublinear, the existence of univalent solutions of Beurling’s boundary differential relation is shown.     

The extended Bitsadze-Lavrent’ev-Tricomi boundary value problem

F. G. Tricomi ([5], [6]) originated the theory of boundary of value problems for mixed type equations by establishing the first mixed type equation known asthe Tricomi equation \(y \cdot u_{xx} + u_{yy} = 0\) which is hyperbolic fory<0, elliptic fory>0, and parabolic fory=0 and then observed that this equation could be applied in Aerodynamics and in general in Fluid Dynamics (transonic flows). See: M. Cribario [1], G. Fichera [2], and our doctoral dissertation [4]. Then M. A. Lavrent’ev and A. V. Bitsadze [3] established together a new mixed type boundary value problem for the equation \(\operatorname{sgn} (y) \cdot u_{xx} + u_{yy} = 0\) where sgn (y)=1 fory>0, =?1 fory<0, fory=0, which involved thediscontinuous coefficient K=sgn (y) ofu _xx while in the case of Tricomi equation the corresponding coefficientT=y wascontinuous. In this paper we establish another mixed type boundary value problem forthe extended Bitsadze-Lavrent’ev-Tricomi equation \(L u = \operatorname{sgn} (y) \cdot u_{xx} + \operatorname{sgn} (x) \cdot u_{yy} + r (x,y) \cdot u = f (x,y)\) where both coefficientsK=sgn (y),M=sgn (x) ofu _xx ,u _yy , respectively are discontinous,r=r (x, y) is once continuously differentiable,f=f (x, y) continuous, and then we prove a uniqueness theorem for quasi-regular solutions.     

Persistency of wellposedness of Ventcel’s boundary value problem under shape deformations

Ventcel boundary conditions are second order differential conditions that appear in asymptotic models. Like Robin boundary conditions, they lead to well-posed variational problems under a sign condition of the coefficient. This is achieved when physical situations are considered. Nevertheless, situations where this condition is violated appeared in several recent works where absorbing boundary conditions or equivalent boundary conditions on rough surfaces are sought for numerical purposes. The well-posedness of such problems was recently investigated: up to a countable set of parameters, existence and uniqueness of the solution for the Ventcel boundary value problem holds without the sign condition. However, the values to be avoided depend on the domain where the boundary value problem is set. In this work, we address the question of the persistency of the solvability of the boundary value problem under domain deformation.     

Numerical methods for determining the inhomogeneity boundary in a boundary value problem for Laplace’s equation in a piecewise homogeneous medium

A boundary value problem for Laplace’s equation in a bounded two-dimensional domain filled with a piecewise homogeneous medium is considered. The boundary of the inhomogeneity is assumed to be unknown. The inverse problem of determining the inhomogeneity boundary and the solution of the equation given the solution and its normal derivative on the boundary of the domain is discussed. Numerical methods are proposed for solving the inverse problem, and the results of numerical experiments are presented.     

Beurling’s Phenomenon in Two Variables

In the Hardy space over the unit disk H²(D), every shift-invariant subspace M is of the form H²(D)) for some inner function by Beurlings theorem, and the reproducing kernel of M is . The fact that is inner implies that is subharmonic and has boundary value 1 almost everywhere on T. In the two variable space H²(D²), things are far more complicated and there is no similar characterization of invariant subspaces M in terms of inner functions. However, we will show in this paper an analogous phenomenon in terms of reproducing kernels, namely, is subharmonic and has boundary value 1 almost everywhere on T². The proof uses an index theorem obtained recently.     

On conformal capacity and Teichmüller’s modulus problem in space

We solve an extremal problem for the conformal capacity of certain space condensers. The extremal condenser is conformally equivalent to Teichmüller’s ring. As an application, we give a dimension-free estimate for the minimal conformal capacity of the condensers with platesE, F such thata, b ∈ E,c, d ∈ F, wherea, b, c, d are given points in

Existence of positive solutions for periodic boundary value problem with sign-changing Green’s function

We prove the existence of positive solutions for the boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} y^{\prime \prime }+m^{2}y=\lambda g(t)f(y), &{}\quad 0\le t\le 2\pi , \\ y(0)=y(2\pi ), &{}\quad y^{\prime }(0)=y^{\prime }(2\pi ), \end{array} \right. \end{aligned}$$

for certain range of the parameter \(\lambda >0\), where \(m\in (1/2,1/2+\varepsilon )\) with \(\varepsilon >0\) small, and f is superlinear or sublinear at \(\infty \) with no sign-conditions at 0 assumed.

     

A boundary value problem for Poisson’s two-media equation inL p /2 -spaces

In the paper we study a binding boundary value problem for two media for Poisson's equation μΔu=f(x) with solutions in the class , 1<p<∞, with the corresponding seminorm, where

A boundary value problem for hypermonogenic functions in Clifford analysis

This paper deals with a boundary value problem for hypermonogenic functions in Clifford analysis. Firstly we discuss integrals of quasi-Cauchy's type and get the Plemelj formula for hypermonogenic functions in Clifford analysis, and then we address Riemman boundary value problem for hypermonogenic functions.     

Nonlinear Stefan problem with convective boundary condition in Storm’s materials

We consider a nonlinear one-dimensional Stefan problem for a semi-infinite material x > 0, with phase change temperature T_f. We assume that the heat capacity and the thermal conductivity satisfy a Storm’s condition, and we assume a convective boundary condition at the fixed face x = 0. A unique explicit solution of similarity type is obtained. Moreover, asymptotic behavior of the solution when \({h\rightarrow + \infty}\) is studied.     

Approximate grid solution of a nonlocal boundary value problem for Laplace’s equation on a rectangle

A nonlocal boundary value problem for Laplace’s equation on a rectangle is considered. Dirichlet boundary conditions are set on three sides of the rectangle, while the boundary values on the fourth side are sought using the condition that they are equal to the trace of the solution on the parallel midline of the rectangle. A simple proof of the existence and uniqueness of a solution to this problem is given. Assuming that the boundary values given on three sides have a second derivative satisfying a Hölder condition, a finite difference method is proposed that produces a uniform approximation (on a square mesh) of the solution to the problem with second order accuracy in space. The method can be used to find an approximate solution of a similar nonlocal boundary value problem for Poisson’s equation.     

An application of Levy-Caccioppoli’s global inversion theorem to a boundary value problem for the Soret effect

The Levy-Caccioppoli’s global inversion theorem is used to prove existence and uniqueness for a problem of heat and mass transfer. The relevant boundary value problem is first transformed in a suitable two-point problem for a first order differential equation.     

Heisenberg’s uncertainty principle in the sense of Beurling

We shed new light on Heisenberg??s uncertainty principle in the sense of Beurling, by offering a fundamentally different proof which allows us to weaken the assumptions rather substantially. The new formulation is pretty much optimal, as can be seen from examples. Our arguments involve Fourier and Mellin transforms. We also introduce a version which applies to two given functions. Finally, we show how our approach applies in the higher dimensional setting.     

Waring’s problem in prime numbers

The existence of a constant V(n) such that any sufficiently large natural number can be represented as a sum of nth degrees of primes in total quantity not exceeding this constant is proved.     

Parametric stability in Robe’s problem

In this work we have given a Hamiltonian formulation to Robe’s problem, obtaining again the classic results. We have computed the resonances existing in the circular case and obtained some information with regard to the linear stability of the central equilibrium of Robe’s problem in the elliptic case. In some critical cases we have constructed, in the parameter plane, the boundary curves that separate the regions of stability and instability.     

Boundary integral operators and boundary value problems for Laplace’s equation

In this paper, we define boundary single and double layer potentials for Laplace’s equation in certain bounded domains with d-Ahlfors regular boundary, considerably more general than Lipschitz domains. We show that these layer potentials are invertible as mappings between certain Besov spaces and thus obtain layer potential solutions to the regularity, Neumann, and Dirichlet problems with boundary data in these spaces.     

Irregular boundary value problem for the Sturm–Liouville operator

We consider an eigenvalue problem for the Sturm–Liouville operator with nonclassical asymptotics of the spectrum. We prove that this problem, which has a complete system of root functions, is not almost regular (Stone-regular) but its Green function has a polynomial order of growth in the spectral parameter.     

Plateau’s problem in Finsler 3-space

We explore a connection between the Finslerian area functional based on the Busemann–Hausdorff-volume form, and well-investigated Cartan functionals to solve Plateau’s problem in Finsler 3-space, and prove higher regularity of solutions. Free and semi-free geometric boundary value problems, as well as the Douglas problem in Finsler space can be dealt with in the same way. We also provide a simple isoperimetric inequality for minimal surfaces in Finsler spaces.