On B-evolution theory and dynamic boundary conditions on a portion of the boundary

Abstract

In this paper we treat an elliptic boundary value problem of mixed type in which the boundary conditions on one of two disjoint boundaries are of a dynamic nature. The coefficients of the differential operators involved are dependent on the spatial as well as the time variable. We follow the approach of an abstract evolution problem of the form d(Bu(t))⊟/dt = A(t)u(t), Bu(0) = y in which A and B are unbounded, not necessarily closed operators, from a Banach space X to a Banach space Y, whose common domain is time-dependent. By treating the abstract problem with the aid of the theory of B-evolutions, it is shown that the boundary value problem is well-posed for arbitrary initial states in Y

Keywords:

• AMS(MOS): 35A05

Keywords:

• Time-dependent domain
• dynamic boundary conditions
• theory of B-evolutions