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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.
MSC:
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
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Full Text: Euclid arXiv