Some applications of the coarea formula to partial differential equations. (English) Zbl 0879.35028

Bethuel, Fabrice

Pràstaro, A. (ed.) et al., Geometry in partial differential equations. Singapore: World Scientific. 1-17 (1994).

Summary: We give two applications of the coarea formula of Federer and Fleming to problems in PDEs. The first question we address concerns a functional inequality involving a Jacobian (know as Wente’s estimate), and the use of the coarea formula allows us to show that the constant appearing in this inequality can be chosen independently of the domain. In the second part we address a totally different question which is related to the stationary Euler equations for inviscid incompressible fluids. We propose a new proof of the fact that, in some circumstances, weak limits of approximate solutions are still weak solutions. This result is then applied to an existence result for weak solutions to a damped Euler equation.
For the entire collection see [Zbl 0867.00017].

MSC:

35B45	A priori estimates in context of PDEs
49Q15	Geometric measure and integration theory, integral and normal currents in optimization
28A75	Length, area, volume, other geometric measure theory
58C35	Integration on manifolds; measures on manifolds

Keywords:

Wente’s estimate; stationary Euler equations