×

The spectral radius of the classical layer potentials on convex domains. (English) Zbl 0820.35025

Dahlberg, B. (ed.) et al., Partial differential equations with minimal smoothness and applications. Proceedings of an IMA Participating Institutions (PI) conference, held Chicago, IL, USA, March 21-25, 1990. New York etc.: Springer-Verlag. IMA Vol. Math. Appl. 42, 129-137 (1992).
Let \(D\) denote a bounded Lipschitz domain in \(\mathbb{R}^ n\), \(\omega_ n\) the area of the unit sphere in \(\mathbb{R}^ n\), and \[ Kf(P) = \lim_{\varepsilon \to 0} {1 \over \omega_ n} \int_{| P - Q | > \varepsilon} {N_ Q \circ (Q - P) \over | P - Q |^ n} f(Q)d \sigma (Q), \] the limit understood in the sense of \(L^ 2 (\partial D,d \sigma)\). Let \(K^*\) be the adjoint of the operator \(K\). Our main result is Theorem: If \(D\) is a bounded Lipschitz and convex domain in \(\mathbb{R}^ n\), \(n \geq 2\), then the spectral radius of \(K^*\) on \(L^ 2_ 0 (\partial D, d\sigma):= \{f \in L^ 2 (\partial D,d \sigma) : \int_{\partial D} fd \sigma = 0\}\) is \(< 1/2\). In particular \((- {1\over2}I + K^*)^{-1} = - 2 \sum^ \infty_{j=0} 2^ j K^{*j}\), where the series converges absolutely in the operator norm on \(L^ 2_ 0 (\partial D,d \sigma)\).
For the entire collection see [Zbl 0762.00004].

MSC:

35C15 Integral representations of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
PDFBibTeX XMLCite