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189 personnes travaillent au LJLL

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Chiffres mars 2019



Lundi 6 novembre 2017

Jonas Hirsch (SISSA Trieste)

Non-existence of a Wente’s L estimate for the Neumann problem

Wente’s L-estimate is a fundamental example of a ’gain’ of regularity due to the
special structure of Jacobian determinants. It concerns the following Dirichlet
problem : let V ∈ H 1 (D, R 2 )

−∆u = det(∇V ) in D
u =0 on ∂D.

Wente’s theorem states that the solution u ∈ W0 1,1 (D, R) to the above Dirichlet
problem is in the space L(D) ∩ H 0 1 (D). This estimate found many applications in
geometric analysis, for instance in the existence of immersed surfaces with constant
mean curvature.
It is natural to ask whether a similar estimate holds true for the Neumann problem
The aim of this talk will be to present a counterexample. We will present at first a
possible motivation for studying the Neumann problem. Thereafter we will try to
sketch the ideas of the proof.