Haim Brezis
Paris VI and Rutgers

New types of Sobolev-Nirenberg imbeddings, isoperimetric inequalities and elliptic estimates in

We present new estimates obtained jointly with J. Bourgain, and partially with P. Mironescu. Each one has a different flavour, but, in fact, they are closely related.

The first one asserts that

where is a closed rectifiable curve, denotes the length of , is a universal constant and is the tangent to .

The second estimate concerns the classical system, in ,

div    curl

Our new estimate asserts that

A third new estimate concerns the system

where is a divergence-free vector-field. Our new estimate asserts that

Such inequality is unusual because it is well-known that standard elliptic estimates fail in .