Nonlinear transmission problems in bounded domains of Rⁿ

Abstract

We study a semilinear elliptic problem in a bounded domain of Rⁿ(n2) with nonlinear transmission conditions on a submanifold of the domain. This problem is related to electromagnetic wave propagation in fibers, where such conditions appear at the interface between the core and the cladding of the fiber. We present a variational formulation of this problem and obtain a weak solution by means of the mountain pass lemma. By constructing suitable trace spaces, we show that this solution satisfies the required interface conditions in a generalized sense.

Keywords:

• semilinear elliptic equation
• nonlinear transmission condition
• variational methods
• electromagnetic waves in fibres