Advances in Differential Equations

The Friedrichs extension for elliptic wedge operators of second order

Thomas Krainer and Gerardo A. Mendoza

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Abstract

Let ${\mathcal M}$ be a smooth compact manifold whose boundary is the total space of a fibration ${\mathcal N}\to {\mathcal Y}$ with compact fibers, let $E\to{\mathcal M}$ be a vector bundle. Let \begin{equation} A:C_c^\infty( \overset{\,\,\circ} {\mathcal M};E)\subset x^{-\nu} L^2_b({\mathcal M};E)\to x^{-\nu} L^2_b({\mathcal M};E) $ \tag*{(†)} \end{equation} 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 ${\mathscr T}={\mathscr T}_F\oplus{\mathscr T}_{aF}$ and there is a natural map ${\mathfrak P} :C^\infty({\mathcal Y};{\mathscr T}_F)\to C^\infty( \overset{\,\,\circ} {\mathcal M};E)$ such that $C^\infty_{{\mathscr T}_F}({\mathcal M};E)={\mathfrak P} (C^\infty({\mathcal Y};{\mathscr T}_F)) +\dot C^\infty({\mathcal M};E)\subset {\mathcal D}_{\max}(A)$. It is shown that the closure of $A$ when given the domain $C^\infty_{{\mathscr T}_F}({\mathcal M};E)$ 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.

Article information

Source
Adv. Differential Equations Volume 23, Number 3/4 (2018), 295-328.

Dates
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ade/1513652448

Subjects
Primary: 58J32: Boundary value problems on manifolds 58J05: Elliptic equations on manifolds, general theory [See also 35-XX] 35J47: Second-order elliptic systems 35J57: Boundary value problems for second-order elliptic systems

Citation

Krainer, Thomas; Mendoza, Gerardo A. The Friedrichs extension for elliptic wedge operators of second order. Adv. Differential Equations 23 (2018), no. 3/4, 295--328. https://projecteuclid.org/euclid.ade/1513652448


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