In Chap. 7 we give a simple and direct proof of Theorem 6.3 by making use of the theory of positive mappings in ordered Banach spaces (Theorem 7.14). The proof of Theorem 6.3 is divided into three sections, Sects. 7.1, 7.2, and 7.3. In Sect. 7.1 we prove an existence and uniqueness theorem for the linear Robin problem \(\displaystyle \begin {cases} Av = g &\mbox{in}\ D, \\ B{\boldsymbol \gamma }v = \varphi &\mbox{on}\ \partial D \end {cases} \) in the framework of Hölder spaces (Theorem 7.1). Namely, Theorem 1.2 will be proved as Theorem 7.1 in Sect. 7.1. In the proof of Theorem 7.1 we make use of a special reduction to the boundary via pseudo-differential operators. We show that the linear Robin problem (6.5) can be reduced to the study of a pseudo-differential operator on the boundary (Proposition 7.2). This is a modern version of the classical Fredholm integral equation. The Poisson operator \({\mathcal P}\) , the Dirichlet-to-Neumann operator \(\Pi \) (formula (7.3)), and the Fredholm boundary operator \(T = a(x^{\prime })\,\Pi + b(x^{\prime })\) (formula (7.4)) are expressed in terms of pseudo-differential operators in Table 7.1. In Sect. 7.4, we make use of the function \(\phi (x) = K1(x)\) in order to introduce a new ordered Banach space \(C_{\phi }(\overline {D}) (\subset C(\overline {D}))\) with a positive cone \(P_{\phi }\) so that the resolvent \(\displaystyle \begin{array}{@{}rcl@{}} \begin {array}{lll} K \colon &\,C(\overline {D}) \longrightarrow C_{\phi }(\overline {D}) \\ &\,\qquad g \longmapsto v\end {array} \end{array} \) of the linear Robin problem \(\displaystyle \begin {cases} Av = g &\mbox{in} D,\\ B{\boldsymbol \gamma }v = 0 &\mbox{on}\ \partial D \end {cases} \) is compact and strongly positive (see Proposition 7.13). Section 7.5 is devoted to the proof of Theorem 6.3 . Theorem 6.3 follows by combining Theorems 7.7 and 7.14.

Spectral Analysis of the Closed Realization \(\mathfrak {A}\)

章节
发表日期：

2025年

查看全文

Kazuaki Taira

摘要

In Chap. 7 we give a simple and direct proof of Theorem 6.3 by making use of the theory of positive mappings in ordered Banach spaces (Theorem 7.14). The proof of Theorem 6.3 is divided into three sections, Sects. 7.1, 7.2, and 7.3. In Sect. 7.1 we prove an existence and uniqueness theorem for the linear Robin problem \(\displaystyle \begin {cases} Av = g &\mbox{in}\ D, \\ B{\boldsymbol \gamma }v = \varphi &\mbox{on}\ \partial D \end {cases} \) in the framework of Hölder spaces (Theorem 7.1). Namely, Theorem 1.2 will be proved as Theorem 7.1 in Sect. 7.1. In the proof of Theorem 7.1 we make use of a special reduction to the boundary via pseudo-differential operators. We show that the linear Robin problem (6.5) can be reduced to the study of a pseudo-differential operator on the boundary (Proposition 7.2). This is a modern version of the classical Fredholm integral equation. The Poisson operator \({\mathcal P}\) , the Dirichlet-to-Neumann operator \(\Pi \) (formula (7.3)), and the Fredholm boundary operator \(T = a(x^{\prime })\,\Pi + b(x^{\prime })\) (formula (7.4)) are expressed in terms of pseudo-differential operators in Table 7.1. In Sect. 7.4, we make use of the function \(\phi (x) = K1(x)\) in order to introduce a new ordered Banach space \(C_{\phi }(\overline {D}) (\subset C(\overline {D}))\) with a positive cone \(P_{\phi }\) so that the resolvent \(\displaystyle \begin{array}{@{}rcl@{}} \begin {array}{lll} K \colon &\,C(\overline {D}) \longrightarrow C_{\phi }(\overline {D}) \\ &\,\qquad g \longmapsto v\end {array} \end{array} \) of the linear Robin problem \(\displaystyle \begin {cases} Av = g &\mbox{in} D,\\ B{\boldsymbol \gamma }v = 0 &\mbox{on}\ \partial D \end {cases} \) is compact and strongly positive (see Proposition 7.13). Section 7.5 is devoted to the proof of Theorem 6.3 . Theorem 6.3 follows by combining Theorems 7.7 and 7.14.