In this paper we introduce a numerical approach adapted to the minimizationof the eigenmodes of a membrane with respect to the domain. This method isbased on the combination of the Level Set method of S. Osher and J.A.Sethian with the relaxed approach. This algorithm enables both changing thetopology and working on a fixed regular grid.

The soil water movement modelgoverned by the initial-boundary value problem for a quasilinear1-D parabolic equation with nonlinear coefficients is considered.The generalized statement of the problem is formulated. Thesolvability of the problem is proved in a certain class offunctional spaces. The data assimilation problem for this model isanalysed. The numerical results are presented.

The control of the surface of water in a long canal bymeans of a wavemaker is investigated. The fluid motion is governed by the Korteweg-de Vries equation in Lagrangian coordinates.The null controllability of the elevation of the fluid surface is obtained thanks to a Carleman estimate and some weighted inequalities. The global uncontrollability is also established.

Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes and Zuazua, we study the null controllability of the heat equationin unbounded domains, typically $\mathbb R_+$ or  $\mathbb R^N$ . Considering an unbounded and disconnected control region of the form $\omega := \cup _n \omega _n$ , we prove two null controllability results:under some technical assumption on the control parts $\omega _n$ , we provethat every initial datum in some weighted L 2 space can be controlled to zero by usual control functions, and every initial datum in L 2(Ω) can be controlled to zero usingcontrol functions in a weighted L 2 space.At last we give several examples in which the control region has a finite measure and our null controllability results apply.

Refining the variational method introduced in Azé et al. [Nonlinear Anal. 49 (2002) 643-670], we givecharacterizationsof the existence of so-called global and local error bounds, for lowersemicontinuous functions defined on complete metric spaces. We thusprovide asystematic and synthetic approach to the subject, emphasizing the specialcaseof convex functions defined on arbitrary Banach spaces (refining theabstract partof Azé and Corvellec [SIAM J. Optim. 12 (2002) 913-927], and the characterization of the local metric regularityof closed-graph multifunctions between complete metric spaces.

We formulate an Hamilton-Jacobi partial differential equation H( x, D u(x))=0 on a n dimensional manifold M, with assumptions of convexity of H(x, .) and regularity of H (locally in a neighborhood of {H=0} in T*M); we define the “minsol solution” u, a generalized solution; to this end, we view T*Mas a symplectic manifold.The definition of “minsol solution” is suited to proving regularity results about u; in particular, we provein the first part that the closure of the set where u is not regular may be covered by a countable number of $n-1$ dimensional manifolds, but for a ${{\mathcal H}}^{n-1}$ negligeable subset.These results can be applied to the cutlocus of a C 2 submanifold of a Finsler manifold.