On a one-parameter continuous family of pairs of complementary boundary conditions
Volume 266 / 2022
Abstract
Let $C[0,\infty ]$ be the space of continuous functions on the right half-axis $\mathbb {R}^+$ with finite limits at $\infty $, and let $C[-\infty ,\infty ]$ be the space of continuous functions on the entire $\mathbb {R}$ that have finite limits at both $-\infty $ and $\infty $. It has been known for some time that classical Feller–Wentzell boundary conditions for the Laplace operator in $C[0,\infty ]$ are in one-to-one correspondence with certain subspaces of continuous functions on $\mathbb {R}$ that are invariant under the basic cosine family and the heat semigroup. In particular, the Robin boundary condition $$f’(0)=\gamma f(0),$$ where $\gamma \ge 0$ is a parameter, is linked with the subspace $C_R^\gamma \subset C[-\infty ,\infty ]$ of those $f$ that satisfy $f(-x) = f(x) - 2\gamma \int _0^x {\rm e} ^{-\gamma (x-y)} f(y) \, \mathrm {d} y$ for $x\ge 0$. In this paper we find a natural operator $P_\gamma $ that projects $C[-\infty ,\infty ]$ onto $C_R^\gamma $ and with its help prove a surprising result saying that, for $\gamma \gt 0$, $C_R^\gamma $ is complemented by the subspace $C_F^\gamma \subset C[-\infty ,\infty ]$ linked with the particular case of Feller–Wentzell boundary conditions describing slowly reflecting boundary (or sticky boundary), that is, with the condition $$f”(0)=\gamma f’(0).$$
