On the Nirenberg-Treves condition for local solvability

ON THE LOCAL NIRENBERG PROBLEM FOR σk-TYPE CURVATURES

Solvability condition for the moving contact line.

We consider the motion of a contact line between a fluid, gas, and solid, as it occurs when a drop advances over a solid surface. This motion is controlled by a microscopic length scale near the contact line, such as a slip length or the precursor thickness. The capillary profile inside the drop is linked to the contact line through an intermediate region which is characterized by an interface ...

Cutting the Loss of Derivatives for Solvability under Condition (ψ)

For a principal type pseudodifferential operator, we prove that condition (ψ) implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker’s paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from ǫ + 3/2 for any ǫ > 0 (Dencker’s most recent result) to 3/2 ...

On the Solvability in Local Extensions

The aim of this paper is the study of the solvability of finite extensions of local fields in the general case that is when the residue field is not necessarily perfect. 1 Perfectness of residue fields in ramification theory In case of a perfect residue field or at least when the residue extension is assumed to be separable (ie. classical case) the ramification theory is well studied for severa...

On the Local Solvability of Darboux’s Equation

We reduce the question of local nonsolvability of the Darboux equation, and hence of the isometric embedding problem for surfaces, to the local nonsolvability of a simple linear equation whose type is explicitly determined by the Gaussian curvature. Let (M, g) be a two-dimensional Riemannian manifold. A well-known problem is to ask, when can one realize this locally as a small piece of a surfac...