## Advances in Differential Equations

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

### The Friedrichs extension for elliptic wedge operators of second order

Thomas Krainer and Gerardo A. Mendoza

#### 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

**Mathematical Reviews number (MathSciNet)**

MR3738648

**Zentralblatt MATH identifier**

1380.58021

**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