Some pathological examples of solutions to a Beltrami equation

We construct an example of a bounded solution to a uniformly elliptic Beltrami equation that has no nontangential limit values almost everywhere on the boundary of the unit disk and also an example of a solution to such an equation that is not identically zero and has zero nontangential limit values almost everywhere on the boundary of the unit disk. These examples show that, in the general case of the Hardy spaces of solutions to a uniformly elliptic Beltrami equation (and to more general noncanonical first-order elliptic systems), the usual statement of boundary value problems used for holomorphic and generalized analytic functions is ill-posed.     

Derivative Estimates for the Solution of Hyperbolic Affine Sphere Equation

Considering the hyperbolic affine sphere equation in a smooth strictly convex bounded domain with zero boundary values, the sharp derivative estimates of any order for its convex solution are obtained.     

On bounded solutions of a nonlinear Schrödinger equation on the half-line

We consider the problem on nonzero solutions of the Schrödinger equation on the half-line with potential that implicitly depends on the wave function via a nonlinear ordinary differential equation of the second order under zero boundary conditions for the wave function and the condition that the potential is zero at the beginning of the interval and its derivative is zero at infinity. The problem is reduced to the analysis and investigation of solutions of the Cauchy problem for a system of two nonlinear second-order ordinary differential equations with initial conditions depending on two parameters. We show that if the solution of the Cauchy problem for some parameter values can be extended to the entire half-line, then there exists a nonzero solution of the original problem with finitely many zeros.     

On the determination of Dirichlet or transmission eigenvalues from far field data

We show that the Herglotz wave function with kernel the Tikhonov regularized solution of the far field equation becomes unbounded as the regularization parameter tends to zero iff the wavenumber k belongs to a discrete set of values. When the scatterer is such that the total field vanishes on the boundary, these values correspond to the square root of Dirichlet eigenvalues for ?Δ. When the scatterer is a nonabsorbing inhomogeneous medium these values correspond to so-called transmission eigenvalues.     

Universal boundary value problem for equations of mathematical physics

A new statement of a boundary value problem for partial differential equations is discussed. An arbitrary solution to a linear elliptic, hyperbolic, or parabolic second-order differential equation is considered in a given domain of Euclidean space without any constraints imposed on the boundary values of the solution or its derivatives. The following question is studied: What conditions should hold for the boundary values of a function and its normal derivative if this function is a solution to the linear differential equation under consideration? A linear integral equation is defined for the boundary values of a solution and its normal derivative; this equation is called a universal boundary value equation. A universal boundary value problem is a linear differential equation together with a universal boundary value equation. In this paper, the universal boundary value problem is studied for equations of mathematical physics such as the Laplace equation, wave equation, and heat equation. Applications of the analysis of the universal boundary value problem to problems of cosmology and quantum mechanics are pointed out.     

二阶拟线性椭圆型方程很弱解的唯一性

本文讨论了一个二阶拟线性椭圆型方程的很弱解u∈Wl1o,cr(Ω)的唯一性,边界条件为很弱边值,即在Ω\E上取零边界值,而E是一个满足capt(E)=0的闭集.文中应用了Hodge分解的方法构造检验函数.     

On the limiting behavior of Burger's equation

The limiting behavior as the viscosity goes to zero of the solution of the first boundary value problem for Burger's equation is considered. The method consists in identifying the solution of Burger's equation with the optimal control of an appropriate stochastic control problem.     

Momentum and heat transfer of the Falkner–Skan flow with algebraic decay: An analytical solution

In this paper, an analytical solution of the Falkner–Skan equation with mass transfer and wall movement is obtained for a special case, namely a velocity power index of ?1/3, with an algebraically decaying velocity profile. The solution is given in a closed form. Under different values of the mass transfer parameter, the wall can be fixed, moving in the same direction as the free stream, or opposite to the free stream (reversal flow near the wall). The thermal boundary layer solution is also presented with a closed form for a prescribed power-law wall temperature, which is expressed by the confluent hypergeometric function of the second kind. The temperature profiles and the wall temperature gradients are plotted. Interesting but complicated variation trends for certain combinations of controlling parameters are observed. Under certain parameter combinations, there exist singular points or poles for the wall temperature gradients, namely wall heat flux. With poles, the temperature profiles can cross the zero temperature line and become negative. From the results, it is also found empirically that under certain given values of the Prandtl number (Pr) and flow controlling parameter (b), the number of times for the temperature profiles crossing the zero line is the same as the number of poles when the wall temperature power index varies between zero and a given value n. The current result provides a new analytical solution for the Falkner–Skan equation with algebraic decay and greatly enriches the understanding of this important equation as well as the heat transfer characteristics for this flow.     

On an integral representation of the solution of the Laplace equation with mixed boundary conditions

We obtain an integral representation of the solution of the Laplace equation with three distinct boundary conditions. Depending on the statement of the problem, the homogeneous boundary value problem may have nontrivial solutions; in other cases, the solution of the homogeneous problem is zero. Note that the inhomogeneous problem is always solvable.     

Bifurcations in the Generalized Korteweg–de Vries Equation

We study the generalized Korteweg–de Vries (KdV) equation and the Korteweg–de Vries–Burgers (KdVB) equation with periodic in the spatial variable boundary conditions. For various values of parameters, in a sufficiently small neighborhood of the zero equilibrium state we construct asymptotics of periodic solutions and invariant tori. Separately we consider the case when the stability spectrum of the zero solution contains a countable number of roots of the characteristic equation. In this case we state a special nonlinear boundary-value problem which plays the role of a normal form and determines the dynamics of the initial problem.     

Stabilization of the solution to a heat equation in the exterior of a sphere with boundary control

The problem of boundary control stabilization of the solution to the heat equation defined in the exterior of a sphere is studied in the paper. The boundary control function stabilizing the solution to zero with the rate 1/t^k is constructed for any k > 0.     

On The Parametric Model of Western Boundary Outflow

A simple model equation for western boundary outflow in the Stommel model of the large scale ocean circulation is obtained by evaluating the potential vorticity equation at the western boundary. A series solution to this model equation demonstrates similar behavior to the boundary layer solution of the potential vorticity equation, in particular that “resonances” are present at a discrete series of parameter values which necessitate the addition of logarithms to the series; these resonances occur because the model equation has a logarithmic branch point at these values.     

Helmholtz equation outside an open arc in a plane with a mixed boundary condition

A boundary value problem for the Helmholtz equation outside an open arc in a plane is studied with mixed boundary conditions. In doing so, the Dirichlet condition is specified on one side of the open arc and the boundary condition of the third kind is specified on the other side of the open arc. We consider non-propagative Helmholtz equation, real-valued solutions of which satisfy maximum principle. By using the potential theory the boundary value problem is reduced to a system of singular integral equations with additional conditions. By regularization and subsequent transformations, this system is reduced to a vector Fredholm equation of the second kind and index zero. It is proved that the obtained vector Fredholm equation is uniquely solvable. Therefore the integral representation for a solution of the original boundary value problem is obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global well-posedness for the microscopic FENE model with a sharp boundary condition

We prove global well-posedness for the microscopic FENE model under a sharp boundary requirement. The well-posedness of the FENE model that consists of the incompressible Navier–Stokes equation and the Fokker–Planck equation has been studied intensively, mostly with the zero flux boundary condition. In this article, we show that for the well-posedness of the microscopic FENE model ($b>2$) the least boundary requirement is that the distribution near boundary needs to approach zero faster than the distance function. Under this condition, it is shown that there exists a unique weak solution in a weighted Sobolev space. Moreover, such a condition still ensures that the distribution is a probability density. The sharpness of this boundary requirement is shown by a construction of infinitely many solutions when the distribution approaches zero no faster than the distance function.     

An application of braid group theory to the finite time dead-core rate

We consider the dead-core problem for the semilinear heat equation with strong absorption and with positive boundary values in a ball. We investigate the dead-core rate, i.e. the rate at which the solution reaches its first zero. We first show, as in the one-dimensional case, that the dead-core rate is always faster than the self-similar rate. By using some special solutions and the braid group theory, we then derive the exact dead-core rates for a large class of initial data.     

Investigation of Motion Stability for Elastic Cylindrical Bodies in Two Measures

We obtain sufficient conditions for stability, in two specific measures, of the zero solution of the linearized equation of motion of isotropic elastic bodies subjected to the action of a force load under kinematic, dynamic, and mixed boundary conditions. For cylinder bodies, the establishment of the conditions for stability is reduced to solving the corresponding problem of variational calculus with fixed edges. We find the critical values of parameters in the particular case of loading of a cylindrical body under various conditions of fixing.     

Approximate solution to a boundary value problem for the Lyapunov differential equation with a parameter

A special boundary value problem is studied for the Lyapunov differential equation which is used for investigation of the asymptotic properties of solutions to systems of periodic differential equations with a parameter. An algorithm is proposed for constructing an approximate solution to this boundary value problem, and conditions on the parameter are found under which the zero solution to the system is asymptotically stable.     

Necessary and sufficient condition for the generalized solution of a mixed problem for the wave equation to belong to the class L p for p ≥ 1

In terms of requirements imposed on the boundary function, we obtain a necessary and sufficient condition for the generalized solution of a mixed problem for the wave equation with zero initial conditions and with boundary conditions of the first kind to belong to the class L _p .     

The modified jump problem for the Laplace equation and singularities at the tips

The boundary value problem for the Laplace equation outside several cuts in a plane is studied. The jump of the solution of the Laplace equation and the boundary condition containing the jump of its normal derivative are specified of the cuts. The unique solution of this problem is obtained. The problem is reduced to the uniquely solvable Fredholm equation of the second kind and index zero. The singularities at the ends of the cuts are investigated.     

Invariant Manifolds for the Burgers Equation Defined on a Semiaxis

Stable nonlocal invariant manifolds for the Burgers equation defined on R₊ are constructed. One problem of a boundary control stabilizing the solution of this equation to zero is also studied. Results of numerical experiments are presented.