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On sup over boundaries of Sobolev functions

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

  1. $sup_{partial Omega} (u+v) leq sup_{partial Omega} u + sup_{partial Omega} v.$
  2. For $k in mathbb R$ with $sup_{partial Omega} u leq k$, $u leq k$ on $partial Omega$.
  3. 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)

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