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A uniqueness theorem for second order hyperbolic differential equations. (English) Zbl 0815.35063
This paper gives a generalization of an interesting result of unicity due to L. Robbiano [Commun. Partial Differ. Equations 16, No. 4/5, 789- 800 (1991; Zbl 0735.35086 )]. The author shows first by a simple argument that the constant which appears in Robbiano’s result is universal. Then a very careful examination of the choice of weights in the Carleman estimates leads to a better result, which is not yet optimal, but nearly so.
The generalization obtained (Theorem 1) is the following: If \(P = \sum_{| \alpha | \leq 2} a_ \alpha (x) D^ \alpha\) is a second order hyperbolic differential operator in an open set \(X \subset \mathbb{R}^{n+1}\) such that the leading coefficients are locally Lipschitz and all the coefficients of \(P\) are in \(L^ \infty_{\text{loc}}\) then, if there exists a timelike vector \(\theta\) such that the \(a_ \alpha\) are constant in the direction \(\theta\), and if we denote by \(p(x, \xi)\) the real valued principal symbol of \(P\) and by \(p(x, \xi, \eta)\) the corresponding symmetric bilinear form and let \(\partial^ b\) be the covector defined by \(p(x, \theta^ b, \cdot) = \langle \theta, \cdot \rangle\) then \(\overline N (\text{supp} u) \subset \{(x, \xi)\) \(0 \geq p(x, \theta^ b, \theta^ b) p(x, \xi, \xi) > (1 - k^ 2) p(x, \theta^ b, \xi)^ 2\}\) for any solution \(u\) of \(Pu = 0\) in \(X\), such that \(u \in H^{\text{loc}}_{(1)} (X)\). Here \(k\) is a universal constant, \(k \in [1, \sqrt {27/23}]\) and \(N (\text{supp} u)\) is the normal set of supp \(u\).
This result is the consequence of a rather implicit uniqueness theorem, which in turn results from good Carleman estimates. If the hypotheses of Theorem 1 are satisfied for \(k + 1\) linearly independent timelike vectors \(\theta^ 0, \dots, \theta^ k\), then \(\overline N (\text{supp} u)\) is contained in the set between the characteristic cone and its dilation by a scale factor in the directions orthogonal to \(\theta^ 0, \dots, \theta^ k\). For \(k = n\) one finds Holmgren’s unicity theorem.
The condition on the coefficients on a timelike direction are unavoidable, as counter examples show.

MSC:
35L15 Initial value problems for second-order hyperbolic equations
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References:
[1] Hörmander L., Linear prtial diffrential opertors (1963) · doi:10.1007/978-3-642-46175-0
[2] Hörmander L., The analysis of linear partial diffrential operators (1983)
[3] DOI: 10.1016/0022-0396(88)90026-5 · Zbl 0699.35162 · doi:10.1016/0022-0396(88)90026-5
[4] DOI: 10.1512/iumj.1972.22.22022 · Zbl 0227.35064 · doi:10.1512/iumj.1972.22.22022
[5] DOI: 10.1080/03605309108820778 · Zbl 0735.35086 · doi:10.1080/03605309108820778
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