MathOverflow Asked by Yongmin Park on December 20, 2020

In Gilbarg-Trudinger‘s section on the maximum principle for weak solutions, the sup of a boundary of a Sobolev function defined as follows:

Let $Omega$ be a bounded domain in $mathbb{R}^n$. For $u, v in H^1(Omega)$, $u leq v$ on $partial Omega $ if $(u-v)^+ in H^1_0 (Omega)$. Define

$$sup_{partial Omega} u = inf ; { k in mathbb{R} ; | ; u leq k text{ on } partial Omega }, quad u in H^1(Omega).$$

Here, of course, $H^1(Omega) = W^{1,2}(Omega)$ and $H^1_0(Omega) = W^{1,2}_0(Omega)$.

My simple question is whether the following inequalities are valid.

Let $u, v in H^1(Omega)$.

- $sup_{partial Omega} (u+v) leq sup_{partial Omega} u + sup_{partial Omega} v.$
- For $k in mathbb R$ with $sup_{partial Omega} u leq k$, $u leq k$ on $partial Omega$.
- If $u leq v$ on $partial Omega$, then $sup_{partial Omega} u leq sup_{partial Omega} v.$

I tried to prove these inequalities by the definition, but it does not seem straightforward. However, if we assume that the domain is at least Lipschitz, then we can use the trace operator. Considering this post on Math.SE, I can produce a quite straightforward proof for these inequalities since the sup over the boundary is the usual essential sup over the boundary.

I initially guessed that the inequalities can be showed even if we do not require any regularity on the boundary, but now I am not sure. Is there any proof without using the trace operator?

Thanks!

P.S I posted the same question on Math.SE. (link)

2 Asked on November 3, 2021 by james_t

calculus of variations fa functional analysis integral operators integral transforms integration

1 Asked on November 3, 2021

1 Asked on November 3, 2021

differential equations fa functional analysis fixed point theorems periodic functions

2 Asked on November 3, 2021 by user158636

ag algebraic geometry algebraic surfaces arithmetic geometry

0 Asked on November 3, 2021

0 Asked on November 3, 2021 by qixiao

1 Asked on November 3, 2021

0 Asked on March 4, 2021 by matt-roberts

5 Asked on February 28, 2021 by wanderer

0 Asked on February 25, 2021 by user_501

4 Asked on February 25, 2021 by alexandre-eremenko

1 Asked on February 23, 2021 by rori

3 Asked on February 22, 2021 by penelope-benenati

co combinatorics combinatorial optimization mg metric geometry pr probability

1 Asked on February 21, 2021 by user163784

3 Asked on February 21, 2021 by madeleine-birchfield

axioms ct category theory foundations groupoids higher category theory

0 Asked on February 21, 2021 by ivo-terek

0 Asked on February 21, 2021 by vidit-nanda

ct category theory equivariant homotopy group actions higher category theory

2 Asked on February 20, 2021 by edwin-beggs

1 Asked on February 20, 2021 by sascha

almost periodic function ca classical analysis and odes ergodic theory fourier analysis real analysis

1 Asked on February 18, 2021 by mikhail-tikhomirov

Get help from others!

Recent Questions

Recent Answers

- haakon.io on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?
- Jon Church on Why fry rice before boiling?
- Joshua Engel on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?

© 2022 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, MenuIva, UKBizDB, Menu Kuliner, Sharing RPP