Study of the Limit of Transmission Problems in a Thin Layer by the Sum Theory of Linear Operators
Università degli Studi di Bologna
Dipartimento di Matematica | Laboratoire de Mathématiques Appliquées
Université du Havre |
We consider a family
, where
is a small positive parameter, of
singular elliptic transmission problems in the juxtaposition
of two bodies, the cylindric medium
and the thin layer
. It is assumed that the coefficient in
is
. Such problems
model for instance heat propagation between the body
, the layer
(when supposed with infinite conductivity), and the ambient space. After
performing a rescaling in the thin layer to transform the problem in the fixed
domain
, it is shown that the sum of operators’ method by Da
Prato and Grisvard works and gives an existence and uniqueness result in the
framework
spaces,
. We deduce that the family of solutions
converges in
to a function
in the case of second member in
and
converges in
for a second member in
, (
). We then
prove that the restriction of the limit
to
is in fact the solution
to an elliptic problem on
with a boundary condition of Ventcel’s
type and it has an optimal regularity.
Key words: sums of linear operators, elliptic problems, interpolation spaces, Ventcel’s
problem.
2000 Mathematics Subject Classification: 34K06, 34K10, 34K30.