The paper gives a somewhat surprising estimate in the Holder space $C^{2\alpha}$ for the pressure in terms of the Holder norm $C^\alpha$ of the velocity. Here, the velocity and pressure are related by either the incompressible Euler or NS equation. I came up with this estimate in 2010. I told several people about it and I spent some time looking for an interesting application which I couldn't find. The result was never sent for publication but it was cited in this paper and this paper as a personal communication.

This note originated from a brief conversation we had with Moritz in 2012. We were interested in the problem of the concavity of the first eigenfunction of the fractional Laplacian in a bounded interval. We came up with a simple proof in the case that the power of the Laplacian is of the form $1/m$ for some positive integer $m$. That proof is written in this note. We tried to extend the result to all exponents of the Laplacian in $(0,1)$ but we were never able to make progress in that direction. We left the problem behind and this short note was never sent for publication. An alternative proof of the same result is given in this paper.

This article gives a non optimal upper bound for multiphase composites which improves the Hashin-Shtrikman bounds in some range of parameters. I abandomed the project when I found out about these other bounds which are similar (although not exactly the same).

After initially submiting this paper for publication, we learned that the result follows from the work of Giusti and Giaquinta. In particular from the paper Sharp estimates for the derivatives of local minima of variational integrals. (Bollettino della unione matematica Italiana 3.2 (1984): 239-248) which in turn applies the techniques developped in here, here and here. Our article has the modest advantage of providing a self contained proof.