Notes on extremal and tame valued fields

We extend the characterization of extremal valued fields given in [2] to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finite p-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in [8] for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every aleph_1-saturated valued field the valuation is a composition of extremal valuations of rank 1

The existential theory of equicharacteristic henselian valued fields

We study the existential (and parts of the universal-existential) theory of equicharacteristic henselian valued fields. We prove, among other things, an existential Ax-Kochen-Ershov principle, which roughly says that the existential theory of an equicharacteristic henselian valued field (of arbitrary characteristic) is determined by the existential theory of the residue field; in particular, it is independent of the value group. As an immediate corollary, we get an unconditional proof of the decidability of the existential theory of Fq((t))

The inhomogeneous Dirichlet Problem for natural operators on manifolds

We shall discuss the inhomogeneous Dirichlet problem for: $f(x,u, Du, D^2u) = \psi(x)$ where $f$ is a "natural" differential operator, with a restricted domain $F$, on a manifold $X$. By "natural" we mean operators that arise intrinsically from a given geometry on $X$. An important point is that the equation need not be convex and can be highly degenerate. Furthermore, the inhomogeneous term can take values at the boundary of the restricted domain $F$ of the operator $f$. A simple example is the real Monge-Amp\`ere operator ${\rm det}({\rm Hess}\,u) = \psi(x)$ on a riemannian manifold $X$, where ${\rm Hess}$ is the riemannian Hessian, the restricted domain is $F = \{{\rm Hess} \geq 0\}$, and $\psi$ is continuous with $\psi\geq0$. A main new tool is the idea of local jet-equivalence, which gives rise to local weak comparison, and then to comparison under a natural and necessary global assumption. The main theorem applies to pairs $(F,f)$, which are locally jet-equivalent to a given constant coefficient pair $({\bf F}, {\bf f})$. This covers a large family of geometric equations on manifolds: orthogonally invariant operators on a riemannian manifold, G-invariant operators on manifolds with G-structure, operators on almost complex manifolds, and operators, such as the Lagrangian Monge-Amp\`ere operator, on symplectic manifolds. It also applies to all branches of these operators. Complete existence and uniqueness results are established with existence requiring the same boundary assumptions as in the homogeneous case [10]. We also have results where the inhomogeneous term $\psi$ is a delta function.Comment: Some minor addition