in the kind of inequality $$\label{young}|ab|\leq \epsilon a^2 + C_\epsilon b^2.$$

$L^p-$estimate for the uniform norm of solutions with null Dirichlet boundary condition {#LPestimate}
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In this section, we want to study, for some $p\geq2$, the $L^p-$ estimate for the uniform norm of the solution of (\[SPDEO\]). To get such estimate, we need stronger integrability conditions on the coefficients and the initial condition. To this end, we consider the following assumptions: for $\theta\in[0,1[$ and $p\geq2$:\
**Assumption (HI$\mathbf{2
p}$)**$$E\left(\left\|\xi\right\|_\infty^p+\left\|
f^0\right\|^2_{2,2;T}+\left\| |g^0 |\right\|^2_{2,2;T}+\left\| |h^0
|\right\|^2_{2,2;T}\right)<\infty .$$

**Assumption (HO$\mathbf{\infty p}$)** $$S'_0\in L^\infty(\Omega\times{\ensuremath{\mathcal O}})\ and\ E\left((\left\|
f'\right\|_{\infty,\infty;T})^p+(\left\|  |g'
|^2\right\|_{\infty,\infty;T})^{p/2}+(\left\|  |h'
|^2\right\|_{\infty,\infty;T})^{p/2}\right)<\infty .$$

To get the estimates we need, we apply Itô’s formula to $u-S'$, in order to take advantage of the fact that $S-S'$ is non-positive and that as $u$ is solution of (\[SPDEO\]) and $S'$ satisfies (\[obstacle\]), $u-S'$ satisfies $$\label{uminusS'}\left\{ \begin{split}&d(u_t-S'_t)=\partial_i  (a_{i,j}(x)\partial_j(u_t(x)-S'_t(x)))dt+(f(t,x,u_t(x),\nabla
u_t(x))-f'(t,x))dt \\&+\partial_i(g_i(t,x,u_t(x),\nabla
u_t(x))-g'_i(t,x))dt+(h_j(t,x,u_t(x),\nabla
u_t(x))-h'_j(t,x))dB^j_t\\&+\nu(x,dt), \\
 &(u-S')_0=\xi-S'_0\, ,\\&u-S'\geq S-S'\, . \end{split}\right.$$ that is why we introduce the following functions: $$\bar{f}(t,\omega,x,y,z)=f(t,\omega,x,y+S'_t,z+\nabla
S'_t)-f'(t,\omega,x)$$ $$\bar{g}(t,\omega,x,y,z)=g(t,\omega,x,y+S'_t,z+\nabla S'_t)-g'(t,\omega,x)$$ $$\bar{h}(t,\omega,x,y,z)=h(t,\omega,x,y+S'_t,z+\nabla S'_t)-h'(t,\omega,x).$$ Let us remark that the Skohorod condition for $u-S'$ is satisfied since $$\int_0^T\int_{{\ensuremath{\mathcal O}}} (u_s (x)-S'_s (x))-(S_s (x)-S'_s (x))\nu (ds ,dx)=\int_0^T\int_{{\ensuremath{\mathcal O}}} (u_s (x)-S_s (x))\nu (ds ,dx)=0.$$ It is obvious that $\bar{f}$, $\bar{g}$ and $\bar{h}$ satisfy the 