## 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

### MSC:

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

### Keywords:

volume preserving diffeomorphism; Jacobian determinant

Zbl 0499.35022
Full Text:

### References:

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