arXiv subjects
Publications and source records attributed to Vladimir Gol'dshtein.
We prove the following version of Poincare duality for reduced $L_{q,p}$-cohomology: For any $1<q,p<\infty$, the $L_{q,p}$-cohomology of a Riemannian manifold is in duality with the interior $L_{p',q'}-cohomology for $1/p+1/p'=1$, $1/q+1/q'=1$.
We prove a non-vanishing result for the $L_{q,p}$-cohomology of complete simply-connected Riemannian manifolds with pinched negative curvature.
We study some relation between some geometrically defined classes of diffeomorphisms between manifolds and the $L_{q,p}$-cohomology of these manifolds. Some applications to vanishing and non vanishing results in $L_{q,p}$-cohomology are given.
We give sufficient conditions for the naturallity of the exterior differential under Sobolev mappings. In other words we study the validity of the equation $d f^* α= f^* dα$ for a smooth form $α$ and a Sobolev map $f$.
We show that every closed L_infty,loc - form on R^n is exact. Differential is understood in the sense of currents. The proof does not use any explicit geometric constructions. De Rham theorem follows.
We introduce the notion of a conformal de Rham complex of a Riemannian manifold. This is a graded differential Banach algebra and it is invariant under quasiconformal maps, in particular the associated cohomology is a new quasiconformal invariant.
We study the Robin boundary-value problem for bounded domains with isolated singularities. Because for such domains trace spaces of space $H^1(D)$ on its boundaries are weighted Sobolev spaces $L^{2, ξ}(\partial D)$ existence and uniqueness of corresponding Robin boundary-value problems depends on properties of embedding operators $I_1: H^{1}(D)\to L^{2}(D)$ and $I_{2}:H^{1}(D)\to L^{2,ξ}(\partial D)$ i.e. on type of singularities. We obtain an exact description of the weights $ξ$ for bounded domains with 'outside peaks' on its boundaries. This result allows us to formulate correctly the corresponding Robin boundary-value problems for elliptic operators.
We study the relation between Sobolev inequalities for differential forms on a Riemannian manifold $(M,g)$ and the $L_{q,p}$-cohomology of that manifold. The $L_{q,p}$-cohomology of $(M,g)$ is defined to be the quotient of the space of closed differential forms in $L^p(M)$ modulo the exact forms which are exterior differentials of forms in $L^q(M)$.
We give exact relations for certain types of the hierarchic fractal structures. In the blatant distinction from regular networks of the "small world" (SW) topology [1], regular fractal networks manifests the logarithmic dependence of the characteristic path length on number of vertices N which is typically just for random networks. Due to this fact, for the hierarchic networks there is no effect of an abrupt drop of the mean path length caused by the introduction into a regular structure only a few random edges. The clustering coefficient of the hierarchic structures is equal zero and in a process of randomization it slightly grows up to the random networks value k/N, where k is mean number of the nearest neighbors, in vivid contrast with the small world networks behavior. We considered also the problem of unification of two isolated SW and Caley tree networks in the limits of weak (through one point) and strong (many points) connection and deduced a specific algebra for the mean path length of the unified network in terms of the isolated networks path lengths. The sewing together of the tree-like networks may be performed in the one-bottom point or in the crown top to crown top manners. In such a way we come to the concept of bi-trees of two types having different topologies. We demonstrated the difference between two possible probabilistic models of introducing randomness: the model with substitution of a certain number of cut off links of the regular lattice by the equal number of edges stochastically distributed over the lattice (totally random model) and the artillery model, when one end of the broken link remains connected with the origin lattice.
New classes of non-smooth bounded domains D, for which the embedding operator from H^1(D) into L^2(D) is compact are introduced. Examples are given and applications to scattering by rough obstacles are mentioned.