1. CJM Online first
 Karpukhin, Mikhail A.

Steklov problem on differential forms
In this paper we study spectral properties of the DirichlettoNeumann
map on differential forms obtained by a slight modification of
the definition due to Belishev and Sharafutdinov.
The
resulting operator $\Lambda$ is shown to be selfadjoint on the
subspace of coclosed forms and to have purely discrete spectrum
there.
We investigate properties of eigenvalues of $\Lambda$ and prove
a HerschPayneSchiffer type inequality relating products of
those eigenvalues to eigenvalues of the Hodge Laplacian on the
boundary. Moreover, nontrivial eigenvalues of $\Lambda$ are
always at least as large as eigenvalues of the DirichlettoNeumann
map defined by Raulot and Savo
Finally, we remark
that a particular case of $p$forms on the boundary of a $2p+2$dimensional
manifold shares a lot of important properties with the classical
Steklov eigenvalue problem on surfaces.
Keywords:DirichlettoNeumann map, differential form, Steklov eigenvalue, shape optimization Categories:58J50, 58J32, 35P15 

2. CJM 2011 (vol 64 pp. 1058)
 Plakhov, Alexander

Optimal Roughening of Convex Bodies
A body moves in a rarefied medium composed of point particles at
rest. The particles make elastic reflections when colliding with the
body surface, and do not interact with each other. We consider a
generalization of Newton's minimal resistance problem: given two
bounded convex bodies $C_1$ and $C_2$ such that $C_1 \subset C_2
\subset \mathbb{R}^3$ and $\partial C_1 \cap \partial C_2 = \emptyset$, minimize the
resistance in the class of connected bodies $B$ such that $C_1 \subset
B \subset C_2$. We prove that the infimum of resistance is zero; that
is, there exist "almost perfectly streamlined" bodies.
Keywords:billiards, shape optimization, problems of minimal resistance, Newtonian aerodynamics, rough surface Categories:37D50, 49Q10 
