We study existence and regularity of solutions to the Dirichlet problem for the prescribed Jacobian equation, \(\det D u =f\), where f is integrable and bounded away from zero. In particular, we take \(f\in L^p\), where \(p>1\), or in \(L\log L\). We prove that for a Baire-generic f in either space there are no solutions with the expected regularity.

中文翻译：

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临界和超临界 Sobolev 空间中雅可比方程的狄利克雷问题

我们研究规定的雅可比方程(\det D u =f\)的狄利克雷问题解的存在性和规律性，其中f是可积的并且有界远离零。特别是，我们取\(f\in L^p\)，其中\(p>1\)或\(L\log L\)。我们证明，对于任一空间中的 Baire 泛型f，都没有具有预期规律性的解。