Seminar Analysis: Bernard Dacorogna (EPFL)

Some recent results on the Jacobian equation

Given two functions f and g,we want to find a map φ such that

g(φ(x)) det∇φ(x)=f(x)   x∈Ω,
                   φ(x)=x       x∈∂Ω.

Local case. We first consider the (local) existence, uniqueness and optimal regularity for the problem

g_i(φ(x)) det∇φ(x)=f_i(x)  for every 1≤i≤n

where g_i·f_i>0.

Global case. A necessary condition is then

∫_Ω f =∫_Ω g.                             (1)

(i) We discuss the case where g·f>0 and give three different ideas for the existence problem with optimal regularity.

(ii) We then briefly comment on the case where g>0 but f is allowed to change sign.

A problem without the condition (1). We consider a more general problem of the form

            det∇φ(x)=f(x,φ(x),∇φ(x))   x∈Ω,
                   φ(x)=x                        x∈∂Ω.

where no constraint of the type (1) is needed.

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