We consider second order elliptic divergence form systems with complex measurable coefficients$A$that are independent of the transversal coordinate, and prove that the set of$A$for which the boundary value problem with$L$_{2}Dirichlet or Neumann data is well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when$A$is either Hermitean, block or constant. Our methods apply to more general systems of partial differential equations and as an example we prove perturbation results for boundary value problems for differential forms.

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We introduce a notion of super-potential for positive closed currents of bidegree ($p$,$p$) on projective spaces. This gives a calculus on positive closed currents of arbitrary bidegree. We define in particular the intersection of such currents and the pull-back operator by meromorphic maps. One of the main tools is the introduction of structural discs in the space of positive closed currents which gives a “geometry” on that space. We apply the theory of super-potentials to construct Green currents for rational maps and to study equidistribution problems for holomorphic endomorphisms and for polynomial automorphisms.

We study bounded ancient solutions of the Navier–Stokes equations. These are solutions with bounded velocity defined in$R$^{$n$}× (−1, 0). In two space dimensions we prove that such solutions are either constant or of the form$u$($x$,$t$) =$b$($t$), depending on the exact definition of admissible solutions. The general 3-dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axisymmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions, and obtain extensions of results in a recent paper by Chen, Strain, Tsai and Yau [4].