if $g$ is a viscosity solution of (\[eikonal\]) and therefore $g$ is locally semiconcave in $\Omega$ and $u\in BV_{loc}(\Omega,\mathbb{S}^1)$.

Therefore, as a conclusion, one deduces the following equivalences for this particular problem $$\begin{array}{c}
\displaystyle\mbox{viscosity solution to (\ref{eikonal})}\quad\Longleftrightarrow\quad \mbox{ entropy condition (\ref{positive_limit}) }\\[5mm]
\Longleftrightarrow \mbox{ minimality of the entropy production (\ref{inf_energy})}\quad .
\end{array}$$

The paper is organized as follows: First, in section 2, we establish some technical preliminary results. Then in Section \[section\_blow\_up\] we will show, that the measure $$\int_{\mathbb{R}}m(x,t,a)\,da$$ has no points with strictly negative density, outside possibly a set of 1-dimensional measure $0$, i.e. we claim $$\label{uuu}
\lim_{r\rightarrow 0^+}\frac{1}{r}\int_{\mathbb{R}}m\left(B_{r}((x_{0},t_{0})),a\right)\,da\geq 0\quad\mbox{ for }{\mathcal H}^1\mbox{ a.e. }~ (x_{0},t_{0})\in\mathbb{R}\times(0,T).$$ In the last section, using an argument similar to the one used to prove the main result in [@ALR], we deduce that the non negativity condition (\[uuu\]) implies that $u$ is entropic.

Preliminary results {#preliminary}
-------------------

In this section we define a notion of weak entropy solutions (see Definition \[def\_entropy\_solution\]) of scalar conservation laws on domain of trapezoidal shape (see (\[definition\_boundary\]). Afterward we will prove Lemma \[properties\_of\_entropy\_solutions\], which roughly says that for that kind of entropy solutions the same properties hold as in the classical case. We will use this results in Section 