The study deals with the theory of interior capacities of condensers in a locally compact space, a condenser being treated here as a countable, locally finite collection of arbitrary sets with the sign $+1$ or $-1$ prescribed such that the closures of oppositely signed sets are mutually disjoint. We are motivated by the known fact that, in the noncompact case, the main minimum-problem of the theory is in general unsolvable, and this occurs even under very natural assumptions (e.g., for the Newtonian, Green, or Riesz kernels in $\mathbb R^n$, $n\geqslant2$, and closed condensers of finitely many plates). Therefore it was particularly interesting to find statements of variational problems dual to the main minimum-problem (and hence providing new equivalent definitions to the capacity), but now always solvable (e.g., even for nonclosed, unbounded condensers of infinitely many plates). For all positive definite kernels satisfying Fuglede's condition of consistency between the strong and vague ($={}$weak$*$) topologies, problems with the desired properties are posed and solved. Their solutions provide a natural generalization of the well-known notion of interior equilibrium measures associated with a set. We give a description of those solutions, establish statements on their uniqueness and continuity, and point out their characteristic properties. Such results are new even for classical kernels in $\mathbb R^n$, which is important in applications.

We determine all the caloric morphisms for rotation invariant (spherically symmetric) metrics, in the case where the space dimension is greater than two. We also treat the caloric morphisms on two dimensional spheres and hyperbolae.

In this paper, we study free boundary-value problems for $p$-harmonic maps on half simple spaces of Euclidean space, and obtain some Liouville-type theorems.

Our aim in this paper is to prove the boundedness of fractional integrals from the Herz-Morrey space with variable exponent $M\dot{K}^{\alpha ,\lambda}_{q_1, p_1(\, \cdot \, )}(\R^n)$ to $M\dot{K}^{\alpha ,\lambda}_{q_2, p_2(\, \cdot \, )}(\R^n)$.

An orientable $3$-orbifold is universal iff every closed, orientable $3$-manifold is the underlying space of an orbifold structure that is an orbifold-covering of it. The first known example of universal orbifold was $\textbf{B}_{4,4,4}=(S^{3}, B,4)$ where $B$ denotes the Borromean rings and all the isotropy groups are cyclic of order 4. The main result in this article is that the hyperbolic orbifold $\textbf{B}_{m,2p,2q}$ is universal for every $m\geq 3$, $p\geq 2$, $q\geq 2$.

We consider the abstract initial value problem for the system of evolution equations which describe motion of incompressible viscous and heat-conductive fluids in a bounded domain. It is difficulty of our problem that we do not neglect the viscous dissipation function in contrast to the Boussinesq approximation. This problem has uniquely a mild solution locally in time for general initial data, and globally in time for small initial data. Moreover, a mild solution of this problem can be a strong or classical solution under appropriate assumptions for initial data. We prove the above properties by the theory of analytic semigroups on Banach spaces.