JournalsaihpcVol. 7, No. 1pp. 1–26

On a partial differential equation involving the Jacobian determinant

  • Bernard Dacorogna

    École Polytechnique, Lausanne, Suisse

  • Jürgen Moser

    Eidg. Techn. Hochschule, Zürich, Suisse
On a partial differential equation involving the Jacobian determinant cover
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Abstract

Let Ω ⊂ ℝn a bounded open set and f > 0 in \mathrm{\Omega }\limits^{¯} satisfying ∫Ωf(x) dx=meas Ω\int _{\mathrm{\Omega }}f(x)\:dx = \mathrm{meas}\:\mathrm{\Omega }. We study existence and regularity of diffeomorphisms u:\mathrm{\Omega }\limits^{¯}\rightarrow \mathrm{\Omega }\limits^{¯} such that

{det⁡∇ u  (x)=f(x), x∈Ωu (x)=x, x∈∂Ω.\left\{\begin{matrix} \det \nabla \:u\:\:(x) = f(x),\:x \in \mathrm{\Omega } \\ u\:(x) = x,\:x \in ∂\mathrm{\Omega }. \\ \end{matrix}\right.

Résumé

Soit Ω ⊂ ℝn un ouvert borné et soit ƒ > 0 dans \mathrm{\Omega }\limits^{¯} satisfaisant ∫Ωf(x) dx=mes Ω\int _{\mathrm{\Omega }}f(x)\:dx = \mathrm{mes}\:\mathrm{\Omega }. On étudie l’existence et la régularité de difféomorphismes u:\mathrm{\Omega }\limits^{¯}\rightarrow \mathrm{\Omega }\limits^{¯} tels que

{det⁡∇ u  (x)=f(x), x∈Ωu (x)=x, x∈∂Ω.\left\{\begin{matrix} \det \nabla \:u\:\:(x) = f(x),\:x \in \mathrm{\Omega } \\ u\:(x) = x,\:x \in ∂\mathrm{\Omega }. \\ \end{matrix}\right.

Cite this article

Bernard Dacorogna, Jürgen Moser, On a partial differential equation involving the Jacobian determinant. Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), no. 1, pp. 1–26

DOI 10.1016/S0294-1449(16)30307-9