A simple,efficient, regioselective and one-pot preparation of N-hydroxy- and N–O-protected hydroxyindoles via cycloaddition of nitrosoarenes with alkynes. Synthetic scope,applications and novel by-products

The thermal reaction between nitrosoarenes and alkynes under alkylating conditions produces N-alkoxyindoles as the major products in moderate to good yields and excellent regioselectivity. Various electrophiles are used affording different N–O-protected hydroxyindoles in a multi-component fashion. Privileged acetylenic substrates used in reactions with substituted nitrosoarenes are arylalkynes or propiolates. Potentially bioactive compounds and other classes of highly functionalizable indole products were prepared. Reactions between o-carbomethoxy-nitrosoarenes and arylacetylenes provided tricyclic compounds containing an acylaziridine indoline skeleton.     

Exponential self-similar mixing and loss of regularity for continuity equations

We consider the mixing behavior of the solutions to the continuity equation associated with a divergence-free velocity field. In this Note, we sketch two explicit examples of exponential decay of the mixing scale of the solution, in case of Sobolev velocity fields, thus showing the optimality of known lower bounds. We also describe how to use such examples to construct solutions to the continuity equation with Sobolev but non-Lipschitz velocity field exhibiting instantaneous loss of any fractional Sobolev regularity.     

On the Hong–Krahn–Szego inequality for the p-Laplace operator

Given an open set Ω, we consider the problem of providing sharp lower bounds for λ ₂(Ω), i.e. its second Dirichlet eigenvalue of the p-Laplace operator. After presenting the nonlinear analogue of the Hong–Krahn–Szego inequality, asserting that the disjoint unions of two equal balls minimize λ ₂ among open sets of given measure, we improve this spectral inequality by means of a quantitative stability estimate. The extremal cases p = 1 and p = ∞ are considered as well.