“Radiation condition and instability phenomenon at a corner interface between a dielectric and a negative material”
Mercoledì 31 Marzo 2021, ore 17:00 - Zoom - Xavier Claeys (Université Pierre et Marie Curie Laboratoire Jacques-Louis Lions, Paris, France)
In the present talk, we consider a 2D second order transmission problem involving a piece-wise constant coefficient in the principal part. This coefficient takes two values with opposite signs, which delimits an interface between two media. Although this problem may look elliptic at first glance, it is not, due to the sign change of the coeffcient. In the case where the interface contains a corner, which we will assume, the problem is not even Fredholm in a standard Sobolev setting if the contrast between the two media belongs to some critical interval. We will focus on this situation, showing that the problem recovers Fredholmness in a different functional setting, that relies on weighted Sobolev spaces and some outgoing radiation condition at the corner. We will also discuss recent results of asymptotic analysis that examine the effect of rounding the corner in such a context. We will present an exotic instability phenomenon appearing for our problem as the size for the rounded corner goes to zero.