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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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