Weak observability estimates for 1-D wave equations with rough coefficients

Fanelli, Francesco; Zuazua, Enrique

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 2, pp. 245-277.

Dans cet article on démontre des inégalités d'observabilité pour l'équation des ondes à coefficients non-Lipschitzien en une dimension d'espace. Pour des coefficients qui sont dans la classe de Zygmund, on prouve une estimation « classique », qui étend le résultat bien connu d'observabilité dans l'espace d'énergie pour des coefficients à variation bornée. Au contraire, quand les coefficients sont log-Lipschitz ou log-Zygmund, on prouve des estimations d'observabilité « avec perte de dérivées » : pour contrôler l'énergie totale des solutions, il faut mesurer des normes de Sobolev d'ordre plus élevé au bord de l'intervalle. Ce dernier résultat représente le cas intermédiaire entre le cas des coefficients Lipschitz (ou Zygmund), où les estimations d'observabilité sont satisfaites dans l'espace d'énergie, et celui des coefficients Hölder, où elles échouent à n'importe quel ordre (comme prouvé dans [9]) à cause d'une perte infinie de dérivées. On établit aussi une relation optimale entre le module de continuité des coefficients et la perte de dérivées dans les inégalités d'observabilité. En particulier, on démontre que, quelle que soit l'hypothèse plus faible que celle de log-Lipschitz (pas seulement celle de Hölder, par exemple), les estimations ne sont pas valables en général, tandis que pour toute condition intermédiaire entre celle de Lipschitz et log-Lipschitz on a une inégalité avec une perte d'un nombre fini de dérivées. Cette classification a un équivalent aussi au niveau de la variation seconde des coefficients.

In this paper we prove observability estimates for 1-dimensional wave equations with non-Lipschitz coefficients. For coefficients in the Zygmund class we prove a “classical” estimate, which extends the well-known observability results in the energy space for BV regularity. When the coefficients are instead log-Lipschitz or log-Zygmund, we prove observability estimates “with loss of derivatives”: in order to estimate the total energy of the solutions, we need measurements on some higher order Sobolev norms at the boundary. This last result represents the intermediate step between the Lipschitz (or Zygmund) case, when observability estimates hold in the energy space, and the Hölder one, when they fail at any finite order (as proved in [9]) due to an infinite loss of derivatives. We also establish a sharp relation between the modulus of continuity of the coefficients and the loss of derivatives in the observability estimates. In particular, we will show that under any condition which is weaker than the log-Lipschitz one (not only Hölder, for instance), observability estimates fail in general, while in the intermediate instance between the Lipschitz and the log-Lipschitz ones they can hold only admitting a loss of a finite number of derivatives. This classification has an exact counterpart when considering also the second variation of the coefficients.

Classification : 93B07, 35L05, 35Q93
Mots clés : 1-D wave equation, Non-Lipschitz coefficients, Zygmund regularity, Observability estimates, Loss of derivatives

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