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Absence de principe du maximum pour certaines équations paraboliques complexes. (French) Zbl 0960.35011

The authors show that the maximum principle, even in a weak form, does not hold, in general, for a class of parabolic operators with complex coefficients, when the dimension \(n\) is greater or equal to 5. More precisely, they consider the operator \(L=-\text{div} A\nabla\), where the matrix \(A(x)\) has complex measurable coefficients which are bounded and, for some \(\delta>0\), satisfy \(\text{Re} A(x)\xi\cdot{\bar\xi}\geq\delta|\xi|^2\), for a.e.\(x\in{\mathbb{R}}^n\), \(\forall\xi\in {\mathbb{C}}^n\). The main result states that if \(n\geq 5\) and \(T>0\), then there exists a matrix \(A(x)\) in the above class such that for every \(k>0\) there exists \(f\in L^2({\mathbb{R}}^n)\cap L^\infty({\mathbb{R}}^n)\), with \(\|f\|_\infty=1\), such that the solution \(u\) of the problem \(\partial u/ \partial t+ Lu=0\) in \(Q={\mathbb{R}}^n\times]0,T]\), \(u(x,0)=f(x)\) in \({\mathbb{R}}^n\), satisfies \(\sup_Q|u|\geq k\). The proof is based on an abstract result about semi-groups which permits to use a counterexample, given by V. G.Maz’ya, S. A. Nazarov and B. A. Plamenevskij [J. Soviet Math. 28, 726-734 (1985; Zbl 0562.35030)], about the non-regularity of the weak solutions of elliptic equations with complex coefficients.
Reviewer: V.Ferone (Napoli)

MSC:

35B50 Maximum principles in context of PDEs
35K15 Initial value problems for second-order parabolic equations

Citations:

Zbl 0562.35030
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