Optimal control for a parabolic problem in a domain with highly oscillating boundary

In this article we study the homogenization of an optimal control problem for a parabolic equation in a domain with highly oscillating boundary. We identify the limit problem, which is an optimal control problem for the homogenized equation and with a different cost functional.     

Asymptotics for eigenelements of Laplacian in domain with oscillating boundary: multiple eigenvalues

We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary. We consider the case where the eigenvalue of the limit problem is multiple. We construct the leading terms of the asymptotic expansions for the eigenelements and verify the asymptotics.     

On the boundedness of solutions for degenerate nonlinear elliptic equations

We establish the boundedness of solutions of Dirichlet Problem for a class of degenerate nonlinear elliptic equations. To prove the result we follow a modification of Moser's method.     

Schauder estimates for solutions to boundary value problems for second order elliptic systems in polyhedral domains

Boundary value problems for second order elliptic differential equations and systems in a polyhedral domain are considered. The authors prove Schauder estimates and obtain regularity assertions for the solutions.     

Unilateral problems for the wave equation with degenerate and localized nonlinear damping: well‐posedness and non‐stability results

Unilateral problems related to the wave model subject to degenerate and localized nonlinear damping on a compact Riemannian manifold are considered. Our results are new and concern two main issues: (a) to prove the global well‐posedness of the variational problem; (b) to establish that the corresponding energy functional is not (uniformly) stable to equilibrium in general, namely, the energy does not converge to zero on the trajectory of every solution, even if a full linear damping is taken in place.     

Dirichlet principle using computers

In this article we shall give practical and numerical solutions of the Laplace equation on multidimensional spaces and show the numerical experiments by using computers. Our method is based on the Dirichlet principle by combinations with generalized inverses, Tikhonov's regularization and the theory of reproducing kernels.     

A convergence analysis and applications for the Newton-Kantorovich method in K-normed spaces

We study the local and semilocal convergence of the Newton-Kantorovich method to a solution of a nonlinear operator equation on aK-normed space setting. Using more precise majorizing sequences than before we show that in the semilocal case finer error bounds can be determined on the distances involved and an at least as precise information on the location of the solution as in earlier results. In the local case we show that a larger radius of convergence can be obtained.     

Effective conductivity of composite materials with random positions of cylindrical inclusions: finite number inclusions in the cell

The effective conductivity of composite materials with random position n ²,n?∈?N, and the cylindrical identical inclusions inside periodic cells is considered. We compare the results for symmetric and nonsymmetric cases of location of the inclusions in the cells and find that a symmetric structure provides a minimum for the effective conductivity among all the structures having n ² inclusions of such conductivity and sizes.