The edge algebra structure of boundary value problems
The aim of this paper is to treat boundary value problems in the frame of the edge pseudo-differential calculus with constant discrete asymptotics [cf. \textit{B.-W. Schulze}, `Boundary value problems and singular pseudo-differential operators', Wiley, Chichester (1998; Zbl 0907.35146)]. Basic question is the following. Let \(A\) be a given classical pseudo-differential operator on the double \(2X\) of a compact smooth manifold \(X\) with boundary \(Y\), and consider \(B= r^+Ae^+\) where \(r^+\), \(e^+\) are, respectively, the restriction to \(X\) and the extension by zero from \(X\) to \(2X\). Then, if \(A\) has not the transmission property, the operator \(B: C^\infty_0(\text{int }X)\to C^\infty(\text{int }X)\) has not an extension to the standard Sobolev spaces. Generally speaking, one should like to identify an algebra containing \(B\), with good continuity properties on suitable function spaces. This is achieved by the authors passing to the scale of the weighted Sobolev spaces is employed on manifolds with edges, \(Y=\partial X\) being the edge and \(\overline{\mathbb{R}}_+\) the inner normal, model cone of the wedge \(\overline{\mathbb{R}}_+\times Y\). In this frame, the authors obtain a symbolic calculus, extending preceding results in literature [see \textit{L. Boutet de Monvel}, Acta Math. 126, 11-51 (1971; Zbl 0206.39401), \textit{G. I. Eskin}, `Boundary value problems for elliptic pseudodifferential equations', Math. Monogr. 52, Am. Math. Soc., Providence, RI (1980; Zbl 0458.35002)].
scientific article; zbMATH DE number 1901457
scientific article