Axisymmetrical vibrations of shells of revolution under abrupt loading

A frequency method is proposed for solving the problem of the vibrations of shells of revolution taking into account the energy dissipation under arbitrary force loading and on collision with a rigid obstacle. The Laplace transform is taken of the equation of the vibrations of a shell of revolution with non-zero initial conditions. For the inhomogeneous differential equation obtained, a variational method is used to solve the boundary-value problem, which consists of finding the Laplace-transformed boundary transverse and longitudinal forces and bending moments as functions of the boundary displacements. The equations of equilibrium of nodes, i.e. the corresponding equations of the finite-element method, are then compared, using results obtained earlier [1–4]. Amplitude-phase-frequency characteristics (APFCs) for the shell cross-sections selected are plotted. An inverse Laplace transformation is carried out using the clear relationship between the extreme points of the APFCs and the coefficients of the corresponding terms of the series in an expansion vibration modes [3]. In view of the fact that the proposed approach is approximate, numerical testing is used.     

Isolated boundary singularities of semilinear elliptic equations

Given a smooth domain ${\Omega\subset\mathbb{R}^N}$ such that ${0 \in \partial\Omega}$ and given a nonnegative smooth function ?? on ???, we study the behavior near 0 of positive solutions of ???u?=?u ^q in ?? such that u =? ?? on ???\{0}. We prove that if ${\frac{N+1}{N-1} < q < \frac{N+2}{N-2}}$ , then ${u(x)\leq C |x|^{-\frac{2}{q-1}}}$ and we compute the limit of ${|x|^{\frac{2}{q-1}} u(x)}$ as x ?? 0. We also investigate the case ${q= \frac{N+1}{N-1}}$ . The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.     

Nonlinear elliptic boundary value problem with equivalued surface in a domain with thin layer

This paper deals with certain kinds of boundary value problems with equivalued surface of nonlinear elliptic equations on a domain with thin layer. We introduce the concept of renormalized solution to this problem. Existence and uniqueness of renormalized solutions are given, and the limit behaviour of solutions is studied in this paper.     

Corner boundary layer in nonlinear singularly perturbed elliptic problems

The Dirichlet problem in a rectangle is considered for the elliptic equation ?²Δu = F(u, x, y, ?), where F(u, x, y, ?) is a nonlinear function of u. The method of corner boundary functions is applied to the problem. Assuming that the leading term of the corner part of the asymptotics exists, an asymptotic expansion of the solution is constructed and the remainder is estimated.     

Minimum error solutions of boundary layer equations

The minimum error solutions of boundary layer equations in the least square sense have been studied by employing the Euler-Lagrange equations. To test the method a class of problems,i.e., boundary layer on a flat plate, Hiemenz flow, boundary layer on a moving sheet and boundary layer in non-Newtonian fluids have been studied. The comparison of the results with approximate methods, like Karman-Pohlhuasen, local potential and other variational methods, shows that the present predictions are invariably better.     

SomeL 1 existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions

In this paper we study questions of existence, uniqueness, and continuous dependence for semilinear elliptic equations with nonlinear boundary conditions. In particular, we obtain results concerning the continuous dependence of the solutions on the nonlinearities in the problem, which in turn implies analogous results for a related parabolic problem. Such questions arise naturally in the study of potential theory, flow through porous media, and obstacle problems.Sponsored by the United States Army under Contract Nos. DAAG29-80-C-0041 and DAAL03-87-K-0043, and by the Air Force Office of Scientific Research under Contract No. AFOSR 84-0252 and by the National Science Foundation under Grant No. DMS-8505531. Research of the third author was carried out in part while visiting the Institute for Mathematics and Its Applications at the University of Minnesota.     

Renormalized solutions of elliptic equations with Robin boundary conditions

In the present paper, we consider elliptic equations with nonlinear and nonhomogeneous Robin boundary conditions of the type{-div(B(x, u)▽u) = f in ?,u = 0 on Γ_0,B(x, u)▽u·n→+γ(x)h(u) =g on Γ_1,where f and g are the element of L~1(?) and L~1(Γ_1), respectively. We define a notion of renormalized solution and we prove the existence of a solution. Under additional assumptions on the matrix field B we show that the renormalized solution is unique.     

Existence of solutions for quasilinear elliptic equations with jumping nonlinearities under the Neumann boundary condition

By variational methods, we prove the existence of a non-trivial solution for the quasilinear elliptic equations with jumping nonlinearities under the Neumann boundary condition. We also provide existence results for positive, negative and non-trivial multiple solutions. The studied equations contain the p-Laplacian problems as a special case.     

Approximate analytical solutions of a class of boundary layer equations over nonlinear stretching surface

Third order nonlinear ordinary differential equations, subject to appropriate boundary conditions arising in fluid dynamics, are solved using three different methods viz., the Dirichlet series, method of stretching of variables, and asymptotic function method. Similarity transformations are used to convert the governing partial differential equations into nonlinear ordinary differential equations. The numerical results obtained from the above methods for various problems are given in terms of skin friction. Our study revealed that the results obtained from these methods agree well with those of direct numerical simulation of ordinary differential equations. Also, these methods have advantages over pure numerical methods in obtaining derived quantities such as velocity profile accurately for various values of the parameters at a stretch.     

一类半线性椭圆型方程组边值问题的可解性

利用极大值原理和Holder,Poincare不等式,证明了一类半线性椭圆型方程组解的非负性和唯一性.在此基础上,又利用连续统理论证明了该边值问题有且仅有唯一的正解,推广了该边值问题可解性的结论.     

On the boundary values of the solutions of elliptic equations

In May 1978 Professor A. V. Balakrishnan invited me to report about boundary values of the solutions of elliptic equations in his seminar at the University of California, at UCLA. I thank him for this invitation. The present article is a summary of my report.