will also consider $\mathcal{W}_T=\{\varphi\in\mathcal{W};\varphi(T)=0\}$, $\mathcal{W}^+=\{\varphi\in\mathcal{W};\varphi\geq0\}$, $\mathcal{W}_T^+=\mathcal{W}_T\cap\mathcal{W}^+$.

An element $v\in \mathcal{K}$ is said to be a [**parabolic potential**]{} if it satisfies: $$\forall\varphi\in\mathcal{W}_T^+,\
\int_0^T-(\frac{\partial\varphi_t}{\partial
t},v_t)dt+\int_0^T\mathcal{E}(\varphi_t,v_t)dt\geq0.$$ We denote by $\mathcal{P}$ the set of all parabolic potentials.

The next representation property is crucial:

(Proposition 1.1 in [@PIERRE])\[presentation\] Let $v\in\mathcal{P}$, then there exists a unique positive Radon measure on $[0,T[\times\mathcal{O}$, denoted by $\nu^v$, such that: $$\forall\varphi\in\mathcal{W}_T\cap\mathcal{C},\ \int_0^T(-\frac{\partial\varphi_t}{\partial t},v_t)dt+\int_0^T\mathcal{E}(\varphi_t,v_t)dt=\int_0^T\int_\mathcal{O}\varphi(t,x)d\nu^v.$$ Moreover, $v$ admits a right-continuous (resp. left-continuous) version $\hat{v} \ (\makebox{resp. } \bar{v}): [0,T]\mapsto L^2
({\ensuremath{\mathcal O}})$ .\
Such a Radon measure, $\nu^v$ is called [**a regular measure**]{} and we write: $$\nu^v =\frac{\partial v}{\partial t}+Av .$$

Let $K\subset [0,T[\times\mathcal{O}$ be compact, $v\in\mathcal{P}$ is said to be *$\nu-$superior* than 1 on $K$, if there exists a sequence $v_n\in\mathcal{P}$ with $v_n\geq1\ a.e.$ on a neighborhood of $K$ converging to $v$ in $L^2([0,T];H_0^1(\mathcal{O}))$.

We denote:$$\mathscr{S}_K=\{v\in\mathcal{P};\ v\ is\ \nu-superior\
to\ 1\ on\ K\}.$$

(Proposition 2.1 in [@PIERRE]) Let $K\subset [0,T[\times\mathcal{O}$ compact, then $\mathscr{S}_K$ admits a smallest $v_K\in\mathcal{P}$ and the measure $\nu^v_K$ whose support is in $K$ satisfies $$\int_0^T\int_\mathcal{O}d\nu^v_K=\inf_{v\in\mathcal{P}}\{\int_0^T\int_\mathcal{O}d\nu^v;\ v\in\mathscr{S}_K\}.$$

(Parabolic Capacity)

-   Let $K\subset [0,T[\times\mathcal{O}$ be compact, we define $cap(K)=\int_0^T\int_\mathcal{O}d\nu^v_K$;

-   let $O\subset
    [0,T[\times\mathcal{O}$ be open, we define $cap(O)=\sup\{cap(K);\
    K\subset O\ compact\}$;

-   for any borelian $E\subset [0,T[\times\mathcal{O}$, we define $cap(E)=\inf\{cap(O);\
    