On a partial differential equation involving the Jacobian determinant

Abstract

Let Ω ⊂ ℝ^n a bounded open set and f > 0 in \bar{\mathrm{\Omega }} satisfying \int _{\mathrm{\Omega }}f(x)\:dx = \text{meas}\:\mathrm{\Omega } . We study existence and regularity of diffeomorphismsn u:\bar{\mathrm{\Omega }}\rightarrow \bar{\mathrm{\Omega }} such that \begin{cases} \det \nabla \:u\:\:(x) = f(x),&\:x \in \mathrm{\Omega } \\ u\:(x) = x,&\:x \in ∂\mathrm{\Omega }. \end{cases} Résumé Soit Ω ⊂ ℝ^n un ouvert borné et soit ƒ > 0 dans \bar{\mathrm{\Omega }} satisfaisant \int _{\mathrm{\Omega }}f(x)\:dx = \text{mes}\:\mathrm{\Omega } . On étudie l’existence et la régularité de difféomorphismes u:\bar{\mathrm{\Omega }}\rightarrow \bar{\mathrm{\Omega }} tels que \begin{cases} \det \nabla \:u\:\:(x) = f(x),&\:x \in \mathrm{\Omega } \\ u\:(x) = x,&\:x \in ∂\mathrm{\Omega }. \end{cases}