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

Dacorogna, Bernard; Moser, Jürgen

Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7, No. 1, 1-26 (1990).

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

Citations:

Zbl 0499.35022

References:

[1]	M. Abraham and R. Becker, {\it Classical Theory of Electricity and Magnetism}, Glasgow, 1950.
[2]	Alpern, S., New proofs that weak mixing is generic, Inventiones Math., Vol. 32, 263-279, (1976) · Zbl 0338.28012
[3]	Anosov, D. V.; Katok, A. B., New examples in smooth ergodic theory. ergodic diffeomorphisms, Trudy Moskov Mat. Obsc. Tom., Trans. Moscow Math. Soc., Vol. 23, 1-35, (1970) · Zbl 0255.58007
[4]	Banyaga, A., Formes volume sur LES variétés à bord, Enseignement Math., Vol. 20, 127-131, (1974) · Zbl 0281.58001
[5]	Coddington, E. A.; Levinson, N., Theory of ordinary differential equations, (1955), MacGraw-Hill New York · Zbl 0042.32602
[6]	Dacorogna, B., A relaxation theorem and its application to the equilibrium of gases, Arch. Ration. Mech. Anal., Vol. 77, 359-385, (1981) · Zbl 0492.49002
[7]	Greene, R. E.; Shiohama, Diffeomorphisms and volume preserving embeddings of non compact manifolds, Trans. Am. Math. Soc., Vol. 255, 403-414, (1979) · Zbl 0418.58002
[8]	Hörmander, L., The boundary problems of physical geodesy, Arch. Ration. Mech. Anal., Vol. 62, 1-52, (1976) · Zbl 0331.35020
[9]	Ladyzhenskaya, O. A.; Uraltseva, N. N., Linear and quasilinear elliptic equations, (1968), Academic Press New York · Zbl 0164.13002
[10]	A. E. H. Love, {\it Treatise on the Mathematical Theory of Elasticity}, New York, 1944. · Zbl 0063.03651
[11]	Meisters, G. H.; Olech, C., Locally one to one mappings and a classical theorem on schlicht functions, Duke Math. J., Vol. 30, 63-80, (1963) · Zbl 0112.37702
[12]	Moser, J., On the volume elements on a manifold, Trans. Am. Math. Soc., Vol. 120, 286-294, (1965) · Zbl 0141.19407
[13]	L. Tartar, private communication, 1979.
[14]	Zehnder, E., Note on smoothing symplectic and volume preserving diffeomorphisms, Springer, Lect. Notes Math., Vol. 597, 828-855, (1976) · Zbl 0363.58004