Condition on Sobolev function on the boundary of a domain

Question

I encountered the following kind of a condition when studying boundary value problems:

Let $\Omega \subset \mathbb{R}^2$ be a domain. Let $\varphi \in H^1 (\Omega)$ be such that $\varphi \ge 0 $ on the boundary $\partial \Omega$ in $H^1 (\Omega)$.

As far as I know, "in $H^1 (\Omega)$" means the same as "almost everywhere". However, the boundary is 1-dimensional object so its 2-dimensional Lebesgue measure is equal to zero. So isn't the condition $\varphi \ge 0$ anyways satisfied?

Answer 1

Surely that kind of condition must be explained somewhere in the documents you are reading, because as you pointed out it makes no sense as it stands. These conditions always play a role after integration.

In your case, my guess is that $\phi\ge 0$ at $\partial \Omega$ should mean that $$\tag{1} \int_{\Omega} \phi(x)\partial_{x_j} \psi(x)\, dx\ge -\int_{\Omega}\partial_{x_j}\phi(x)\psi(x)\, dx, $$ for all $\psi\in C^\infty(\Omega)$ such that $\psi(x)\ge 0$ for $x\in\partial\Omega$. Note that the latter condition does make sense, as $\psi$ is a smooth function. Moreover, if $\phi$ is also smooth, then (1) is equivalent to $\phi(x)\ge0$ for all $x\in\partial \Omega$, since in that case $$ \int_{\Omega} \phi(x)\partial_{x_j} \psi(x)\, dx = -\int_{\Omega}\partial_{x_j}\phi(x)\psi(x)\, dx +\int_{\partial \Omega} \phi(x)\psi(x)\, dS, $$ and the latter integral is nonnegative.

However, you must always check your own context. This answer is just my guess.

Answer 2

Sometimes, such an inequality is understood in the following sense: $\phi\ge0$ on some set $E$ in the sense of $H^1$, if there is a sequence of smooth functions $\phi_n$ converging to $\phi$ in $H^1$ and satisfying $\phi_n\ge0$ on $E$ for all $n$. See for instance the book by Kinderlehrer & Stampacchia (Section II.5). This immediately connects to the theory of capacities of sets.