problem and so introduce what we call [*local regular measures*]{}.\
The paper is organized as follows: in section 2 we introduce notations and hypotheses. In section 3, we establish the $L^p-$estimate for uniform norm of the solution with null Dirichlet boundary condition. Section 4 is devoted to the main result: the maximum principle for local solutions whose proof is based on an Itô formula satisfied by the positive part of any local solution with lateral boundary condition, $M$. The last section is an Appendix in which we give the proofs of several lemmas.

Preliminaries
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$L^{p,q}-$space
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Let $\mathcal{O}\subset {{\ensuremath{\mathbb R}} }^d$ be an open bounded domain and $L^2(\mathcal{O})$ the set of square integrable functions with respect to the Lebesgue measure on $\mathcal{O}$, it is an Hilbert space equipped with the usual scalar product and norm as follows $$(u,v)=\int_\mathcal{O}u(x)v(x)dx,\qquad\parallel u\parallel=(\int_\mathcal{O}u^2(x)dx)^{1/2}.$$ In general, we shall extend the notation $$(u,v)=\int_\mathcal{O}u(x)v(x)dx,$$ where $u,\ v$ are measurable functions defined on ${\ensuremath{\mathcal O}}$ such that $uv\in L^1({\ensuremath{\mathcal O}})$.\
The first order Sobolev space of functions vanishing at the boundary will be denoted by $H_0^1({\ensuremath{\mathcal O}})$, its natural scalar product and norm are $$\left( u,v\right) _{H_0^1\left( {\cal O}\right) }=\left( u,v\right) +\int_{{\cal O}}\sum_{i=1}^d\left( \partial _iu\left( x\right) \right)
