The Inverse Problem of the Calculus of Variations for Sixth- and Eighth-order Scalar Ordinary Differential Equations

On the equation manifold of the 2nth-order scalar ordinary differential equation, n3,

Convex Separable Minimization Subject to Bounded Variables

A minimization problem with convex and separable objective function subject to a separable convex inequality constraint and bounded variables is considered. A necessary and sufficient condition is proved for a feasible solution to be an optimal solution to this problem. Convex minimization problems subject to linear equality/linear inequality constraint, and bounds on the variables are also considered. A necessary and sufficient condition and a sufficient condition, respectively, are proved for a feasible solution to be an optimal solution to these two problems. Algorithms of polynomial complexity for solving the three problems are suggested and their convergence is proved. Some important forms of convex functions and computational results are given in the Appendix.     

A free boundary problem in an absorbing porous material with saturation dependent permeability

We prove a well posedness result for a free boundary problem describing the filtration of an incompressible viscous fluid in a porous medium containing water absorbing granules.?The location of the wetting front (the free boundary) is described by a Stefan like problem for a parabolic equation which is coupled with an hyperbolic equation describing the absorption kinetic of the granules. Received December 1999     

A note on triangular derivations of

For a field of characteristic zero, and for each integer , we construct a triangular derivation of whose ring of constants, though finitely generated over , cannot be generated by fewer than elements.

     

Whitney's extension problem for multivariate -functions

We prove that the trace of the space to an arbitrary closed subset is characterized by the following ``finiteness' property. A function belongs to the trace space if and only if the restriction to an arbitrary subset consisting of at most can be extended to a function such that

The constant is sharp.

The proof is based on a Lipschitz selection result which is interesting in its own right.

     

On the Influence of the Lowest Terms on Dirichlet—Type Problems for Elliptic Systems Degenerate at a Point

We consider Dirichlet–type problems for weakly connected systems of elliptic equations of second order with a strong degeneracy at an inner point of the domain, when, in a neighborhood of this point, the asymptotics of a solution is additionally given. The form of the given asymptotics essentially depends on the properties of the coefficients at the first–order derivatives. We prove the existence and uniqueness of solutions of the problems considered in Hölder function classes.     

On the Dirichlet Problem with an Asymptotic Condition for an Elliptic System Strongly Degenerate at a Point

We consider a Dirichlettype problem for a system of elliptic equations of second order with a strong degeneracy at an inner point of the domain, when, in a neighborhood of this point, the principal term of the asymptotics of a solution is additionally given. We prove the existence and uniqueness of a solution of the problem considered in a weighted class of Hölder vector functions.     

A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges

The solution of the Stokes problem in three-dimensional domainswith edges has anisotropic singular behaviour which is treatednumerically by using anisotropic finite element meshes. Thevelocity is approximated by Crouzeix–Raviart (nonconformingP₁ ) elements and the pressure by piecewise constants. Thismethod is stable for general meshes (without minimal or maximalangle condition). Denoting by N_e the number of elements in themesh, the interpolation and consistency errors are of the optimalorder h N_e^–1/3 which is proved for tensor product meshes.As a by-product, we analyse also nonconforming prismatic elementswith P₁ [oplus ] span {x₃²} as the local space for the velocitywhere x₃ is the direction of the edge.     

Statistical Inference in a Linear Model for Spatially Located Sensors and Random Input

In the paper we consider a random linear model for observations provided by spatially located sensors measuring signals coming from one source. For this model a set of sufficient and complete statistics are found, and it is shown that the maximum likelihood estimators of unknown parameters (characteristics of the source) are functions of those statistics. The problem of nonnegative estimators of variance components of the model is shortly discussed. Comparisons of the mean squared errors of several estimators are given. Numerical example concerning hunting for defects in solar cells is considered in details.     

On Logarithmic Smoothing of the Maximum Function

We consider the maximum function f resulting from a finite number of smooth functions. The logarithmic barrier function of the epigraph of f gives rise to a smooth approximation g of f itself, where >0 denotes the approximation parameter. The one-parametric family g converges – relative to a compact subset – uniformly to the function f as tends to zero. Under nondegeneracy assumptions we show that the stationary points of g and f correspond to each other, and that their respective Morse indices coincide. The latter correspondence is obtained by establishing smooth curves x() of stationary points for g , where each x() converges to the corresponding stationary point of f as tends to zero. In case of a strongly unique local minimizer, we show that the nondegeneracy assumption may be relaxed in order to obtain a smooth curve x().