---
date: 2022-04-05 23:42
modification date: Sunday 24th April 2022 18:32:42
title: "elliptic operator"
aliases: [elliptic operator, Sobolev space, Sobolev spaces]
---

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- Refs:
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# elliptic operator

![](attachments/Pasted%20image%2020220502175557.png)

![](attachments/Pasted%20image%2020220424183240.png)

A key fact related to [Elliptic regularity](Elliptic%20regularity):

![](attachments/Pasted%20image%2020220424183336.png)

Definition

![](attachments/Pasted%20image%2020220424183617.png)
![](attachments/Pasted%20image%2020220424183651.png)


# Elliptic regularity

![](attachments/Pasted%20image%2020220424183728.png)


# Misc

![attachments/Pasted image 20210613130822.png](attachments/Pasted%20image%2020210613130822.png)

![attachments/Pasted image 20210613130850.png](attachments/Pasted%20image%2020210613130850.png)

![attachments/Pasted image 20210613130958.png](attachments/Pasted%20image%2020210613130958.png)

![attachments/Pasted image 20210613131036.png](attachments/Pasted%20image%2020210613131036.png)

![attachments/Pasted image 20210613131148.png](attachments/Pasted%20image%2020210613131148.png)

![attachments/Pasted image 20210613131242.png](attachments/Pasted%20image%2020210613131242.png)

Require $r=s$ to be elliptic.

![attachments/Pasted image 20210613131447.png](attachments/Pasted%20image%2020210613131447.png)

![attachments/Pasted image 20210613131618.png](attachments/Pasted%20image%2020210613131618.png)

Why elliptic operators are of interest in gauge theory:

![attachments/Pasted image 20210613131808.png](attachments/Pasted%20image%2020210613131808.png)