zbMATH — the first resource for mathematics

The Friedrichs extension for elliptic wedge operators of second order. (English) Zbl 1380.58021
Summary: Let \(\mathcal M\) be a smooth compact manifold whose boundary is the total space of a fibration \(\mathcal N\rightarrow \mathcal Y\) with compact fibers, let \(E\rightarrow \mathcal M\) be a vector bundle. Let \[ A:C_c^\infty (\overset\circ{\mathcal M};E)\subset x^{-\nu} L^2_b(\mathcal {M;E)} x^{-\nu} L^2_b(\mathcal M;E) \eqno{(*)} \] be a second order elliptic semibounded wedge operator. Under certain mild natural conditions on the indicial and normal families of \(A\), the trace bundle of \(A\) relative to \(\nu\) splits as a direct sum \(\mathcal T=\mathcal T_F\oplus\mathcal T_{aF}\) and there is a natural map \(\mathfrak P :C^{\infty}(\mathcal Y;\mathcal T_F)\to C^{\infty}( \overset\circ{\mathcal M};E)\) such that \(C^{\infty}_{\mathcal T_F}(\mathcal M;E)=\mathfrak P (C^{\infty}(\mathcal Y;\mathcal T_F)) +\dot C^{\infty}(\mathcal M;E)\subset\mathcal D_{\max}(A)\). It is shown that the closure of \(A\) when given the domain \(C^{\infty}_{\mathcal T_F}(\mathcal M;E)\) is the Friedrichs extension of \((*)\) and that this extension is a Fredholm operator with compact resolvent. Also given are theorems pertaining the structure of the domain of the extension which completely characterize the regularity of its elements at the boundary.
58J32 Boundary value problems on manifolds
58J05 Elliptic equations on manifolds, general theory
35J47 Second-order elliptic systems
35J57 Boundary value problems for second-order elliptic systems
Full Text: Euclid arXiv