However, in the case that the manifold has a non-empty boundary, propagation becomes really complicated.
In this construction, let ( M, g ) be a compact 2-dimensional manifold with non-empty boundary, let x 0 ∈ M and consider the manifold M ~ = M { x 0 } with the metric g ~ i j ( x ) = 1 d M ( x, x 0 ) 2 g i j ( x ), where d M ( x, x 0 ) is the distance between x and x 0 on ( M, g ).
For the reflected diffusion generated by L="Δ−∇V⋅∇ on a connected and complete Riemannian manifold M with empty or convex boundary, we establish some sharp estimates of supx∈M|∇G|(x) of the Poisson equation −LG= g in terms of the dimension, the diameter and the lower bound of curvature.
This discovery was the subject of several applications both in linear and nonlinear analysis (integral equations with singular kernels, differential equations defined on unbounded domains, neutral differential equations, differential operators having non-empty essential spectra, boundary value problems in Banach spaces and others).
Let be an -dimensional noncompact Riemannian -manifold with piecewise smooth boundary (possibly empty).
Let be a noncompact Riemannian manifold with the boundary (possibly empty).
In this paper we consider eigenvalues of the Dirichlet biharmonic operator on compact Riemannian manifolds with boundary (possibly empty) and prove a general inequality for them.
First of all, we give a general inequality for eigenvalues of the eigenvalue problem of elliptic operators in weighted divergence form on compact smooth metric measure space with boundary (possibly empty).
Spaces and constitute a space of -chains whose boundaries are empty, the relative -cycles.
It is an open problem in dimension n ≥ 3 that if the Dirichlet to Neumann map for the conductivity or potential is measured on an open non-empty subset of the boundary for Dirichlet data supported in that set we can determine uniquely the potential.
The set of active constraints is, and is non-empty only on the boundary of.
