On a partial differential equation involving the Jacobian determinant. (English) Zbl 0707.35041

Let \(\Omega \subset {\mathbb{R}}^ n\) be a smoothly bounded domain \((\partial \Omega \in C_{k+3,\alpha}\), say). The authors prove the existence of a diffeomorphism u: \({\bar \Omega}\to {\bar \Omega}\) with \(u\in C_{k+1,\alpha}({\bar \Omega})\), solving \[ \det \nabla u=f>0\text{ in } \Omega;\quad u(x)=x\text{ on } \partial \Omega, \] if \(f\in C_{k,\alpha}({\bar \Omega})\), \(k\geq 0\), and \(\int f=| \Omega |.\)
The proof makes use of a deformation argument and of the solution (interesting in itself) to div v\(=g\) in \(\Omega\), \(v=0\) on \(\partial \Omega\) (if \(\int g=0)\) in Hölder spaces. Let us remark the possibility of presenting the solution in closed form by Bogovskij’s formula [see M. E. Bogovskij, Sov. Math., Dokl. 20, 1094-1098 (1979); translation from Dokl. Akad. Nauk SSSR 248, 1037-1040 (1979; Zbl 0499.35022) and W. Borchers and H. Sohr, Hokkaido Math. J. 19, No.1, 67-87 (1990)] for corresponding \(L_ p\)-estimates.
The second part of the paper gives another proof, which works with the help of the implicit function theorem in \(C_ k\)-spaces and allows less boundary regularity, but lacks the expected gain of one order of differentiability for the solution.
Reviewer: M.Wiegner


35F30 Boundary value problems for nonlinear first-order PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)


Zbl 0499.35022
Full Text: DOI Numdam EuDML


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