, J

, M for all (t, s) ? ? such that (x 0 ? t, y 0 ? s) ? supp(J)

. =-m-for-x-?,

, Now repeat all the computations with z = x 0 + b ? a instead of

, 0 and achieves a global maximum at (x 0 , y 0 ) ??. If (x 0 , y 0 ) ? R × ?, then the previous argument holds and u is constant on R × {y 0 }. If (x 0 , y 0 ) ? R × ??, then we have the following alternative -Either R×? J(x 0 ?t, y 0 ?s)(u(t, s)?u(x 0 , y 0 ))dtds = 0 and then the previous argument holds

, Applying the Hopf Lemma to Mu = ?u + ?(y)u x , we obtain a contradiction since

, The multidimentional Maximum Principle holds for Kernel of the form J(x, y, s) = k(x)k(y, s) with -k ? L 1 (R) is a positive continuous kernel such that [?b, ?a] ? [a, b] ? supp(k) for some 0 ? a < b . -k(y, s) is a positive continuous kernel, which satisfy the following properties : ? y ?? ? s y ??

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