We study the behavior of weak solutions to a time-harmonic fluid-structure interaction problem in the exterior of a thin, elastic domain, in the limit of vanishing thickness. Boundary integral operators are used to reduce the exterior problem. Formal asymptotic expansions are developed and justified. The asymptotic procedure is simple, and involves the solution of a reduced exterior problem. Exact representation formulae for the elastic displacements in the thin domain are also provided.

We prove the continuity of bounded, weak solutions of the singular parabolic equation $ \beta(u)_t=Lu, $ where $Lu$ is a second-order, uniformly elliptic operator in divergence form with bounded and measurable coefficients and $\beta(\cdot)$ is a maximal monotone graph in ${\bf R}\times {\bf R}$ exhibiting an arbitrary but finite number of jumps.

We deal with the following nonlinear elliptic problem: $$ \begin{cases} - \hbox{div} \ {{\bf a}}( x , u, \nabla u) + b( x , u , \nabla u) = f & \ \hbox{ in } \ \Omega \\ u = 0 & \ \hbox{ on} \ \partial \Omega, \end{cases} $$ where $\Omega$ is a bounded, open set in $\mathbb R^N$, $f \in L^1 ( \Omega )$, $- \hbox{div} \ {{\bf a}}(x, u, \nabla u) $ defines an operator satisfying Leray---Lions-type conditions, and the lower-order term satisfies natural growth conditions and some other properties; we point out that these properties do not include a sign assumption (see in (2) below our model example). We prove existence of an entropy solution for this problem (see Definition 2.2 below), and we show that, under a natural monotonicity hypothesis, there exists a smallest entropy solution.