# Strong Maximum Principle for very weak supersolutions of Laplacian operator

I know classical strong maximum principles for supersolutions of Laplacian operator, which says:

Suppose $$u \in C^2(\Omega)\cap C(\overline{\Omega})$$ satisfies $$-\Delta u \geq 0$$ in $$\Omega$$.

If $$u \geq 0$$ on $$\partial \Omega$$ then $$u \geq 0$$ in $$\Omega$$. In fact, either $$u > 0$$ in $$\Omega$$ or $$u \equiv 0$$ in $$\Omega$$.

I am reading an article, there author use this version of strong maximum principle:

Assume $$u \in L_{\mathrm{loc}}^1(\Omega)$$ is a very weak supersolutions of $$- \Delta u =0$$ in the sense of distributions, means for every non-negative $$\phi \in C_0^{\infty}(\Omega)$$ we have $$\int_{\Omega} -u \, \Delta \phi \, \mathrm{d}x \geq 0$$

Also assume $$u \geq 0$$ on $$\partial \Omega$$ and $$u \not\equiv 0$$. Then
$$u > 0 \quad \mathrm{in} \,\, \Omega$$

There is no reference for it, and I could not find any proof for it. Does anyone know a proof for it, or a reference for it's proof.

Thanks.

As such the statement does not make sense: if $u \in L^1_\mathrm{loc} (\Omega)$, its restriction on $\partial \Omega$ is not well-defined.
If $u \ge 0$ on $\Omega \setminus K$, where $K \subset \Omega$ is a compact subset and $-\Delta u \ge 0$ in the sense of distributions and if $\Omega$ is connected, then either $u = 0$ in $\Omega$ or $u > 0$ almost everywhere in $\Omega$. First you apply the weak maximum principle to a convolution of $u$ with a radial mollifier on a suitable subdomain of $\Omega$. This shows that $u \ge 0$. You then apply the mean value property for superharmonic functions to conclude.