Stability estimates for resolvents, eigenvalues and eigenfunctions of elliptic operators on variable domains

We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets $\phi (\Omega)$ parametrized by Lipschitz homeomorphisms $\phi$ defined on a fixed reference domain $\Omega$. Given two open sets $\phi (\Omega)$, $\tilde \phi (\Omega)$ we estimate the variation of resolvents, eigenvalues and eigenfunctions via the Sobolev norm $\|\tilde \phi -\phi \|_{W^{1,p}(\Omega)}$ for finite values of $p$, under natural summability conditions on eigenfunctions and their gradients. We prove that such conditions are satisfied for a wide class of operators and open sets, including open sets with Lipschitz continuous boundaries. We apply these estimates to control the variation of the eigenvalues and eigenfunctions via the measure of the symmetric difference of the open sets. We also discuss an application to the stability of solutions to the Poisson problem.Comment: 34 pages. Minor changes in the introduction and the refercenes. Published in: Around the research of Vladimir Maz'ya II, pp23--60, Int. Math. Ser. (N.Y.), vol. 12, Springer, New York 201

On the role of longitudinal momenta in high energy hadron-hadron scattering

We demonstrate a new method for the calculation of inelastic scattering cross-section, which in contrary to the Regge-based methods takes into account the energy momentum conservation law. By virtue of this method it was shown that the main contribution to integral expressing inelastic scattering cross-sections comes not from the multi-Regge domain. In particular this leads to the fact that accounting of longitudinal momenta contribution to virtualities is sufficient and results in the new mechanism of cross-section growth. The necessity of taking into account the large number of interference contributions is shown and the approximate method for this purpose is developed. By considering the interference contributions from a single fitting constant achieved a qualitative agreement of the total and inelastic cross sections with experimental data.Comment: 38 pages, 19 figures (A misspelled author's name corrected

Низкозатратные технологии аквакультуры для мелиоративных водоемов

Studies on reclamation ponds of Pripyat Polesie showed their suitability to conduct feeding, cage and integrated fish farming. Results of the studies indicate the appropriateness to use as seeding pike yearlings, two-year carp, grass carp and bighead carp for aquaculture feeding. Growing of sturgeon in cages, using as feed supplements Chlorella suspension, showed that the addition to the feed mixture of 10 % suspension of chlorella on the mass of the food provided to increase the average weight of Sterlet and Lena sturgeon by 19,9 and 25,1 % respectively. This technology saves up 4.3 € (7,5 %) of the 57 € spent on feeding the fish per day. The positive results showed during production of marketable fish in conjunction with the Muscovy ducks. The live weight of ducks that received during the initial period of growing the suspension of chlorella was 9,2 % higher than in the control group