Exterior Robin Problem

Abstract

The exterior Robin problem consists in finding u such that

$$\displaystyle \begin {array}{rl} &Au(x)=0,\quad x\in S^-,\\ &\big (T-\sigma (x)\big )u(x)=\mathscr {R}(x),\quad x\in {\partial S},\\ &u\in \mathscr {A}^*, \end {array} $$

where the symmetric, positive definite, continuous 3 × 3 matrix function σ and the vector function $\mathscr {R}\in C^{0,\alpha }({\partial S})$, α ∈ (0, 1), are prescribed.