Global estimates for the Jacobians of orientation preserving mappings
In accordance with the notation of the authors, let \(\Omega\) be an open subset of \({\mathbb R}^n\), \(n \geq 2\), and let \(f = (f^1, \dots, f^n) : \Omega \to {\mathbb R}^n\) be a mapping of Sobolev class \(W^{1,p}_{\text{loc}} (\Omega, {\mathbb R}^n ), 1 \leq p < \infty ,\) whose distributional differential \(Df = [\frac{\partial f^i}{\partial x_j}] : \Omega \to \text{GL}(n) \) is a locally integrable function on \(\Omega\) with values in the space \(\text{GL}(n)\) of all \(n \times n\)-matrices. In addition, we say that \(f\) is orientation preserving (respectively, reversing) if its Jacobian determinant \(J(x, f) = \det Df(x)\) is nonnegative (respectively, nonpositive) almost everywhere. Then the authors presented a result of \textit{S. Müller} [J. Reine Angew. Math. 412, 20--34 (1990; Zbl 0713.49004)] that can be restated as the following Theorem. Let \(\Omega \subset {\mathbb R}^n\) be a bounded domain and \(f\) be an orientation-preserving map of Sobolev class \( W^{1, n}_{\text{loc}}(\Omega, {\mathbb R}^n)\). Then \[ \int_E J(x, f) \log \left(e + \frac{J(x, f)}{\int_{\Omega}J(y,f) \,dy} \right) dx \leq C \int_{\Omega} {| Df(x) | }^n\, dx, \tag{1} \] for each compact \(E \subset \Omega\), where \(C\) is a constant and \(| Df(x)| \) denotes the norm \(\max\{ | Df(x)h|\): \(| h| =1\} \). Shortly after, \textit{T. Iwaniec} and \textit{C. Sbordone} proved in [Arch. Ration. Mech. Anal. 119, 129--143 (1992; Zbl 0766.46016)] the following inequality. Let \(f\) be an orientation-preserving map of Orlicz-Sobolev class \(D^n \log^{-1} D(\Omega , {\mathbb R}^n).\) Then the Jacobian of \(f\) is locally integrable and for each compact \(E \subset \Omega\), the following estimate holds \[ \int_E J(x, f) \,dx \leq C \left( \int_{\Omega} \frac{{| Df(x)| }^n}{\log (e + | Df(x)| / | Df| _{\Omega})} \,dx \right), \tag{2} \] where \({| Df| }_{\Omega} \) denotes the integral mean of \(| Df| \) over \(\Omega\) and \(C\) is a constant. In this paper, the authors answer the following interesting question: does there exist a domain \(\Omega\) such that the estimates (1) and (2) become global weighted estimates on \(\Omega\)? Or equivalently, do the inequalities (1) and (2) still hold if \(E\) is replaced by \(\Omega\) in the two preceding inequalities? The authors provide an affirmative answer to these questions if \(\Omega\) is a bounded John domain or an \(L^{\varphi}(\mu)\)-domain. The proofs depend on various estimates and also, in the case of John domains, on a modified Whitney cover of cubes. Editorial remark: The paper has been retracted [cf. ibid. 56, No. 10, 2779 (2008; Zbl 1166.46307)].
scientific article; zbMATH DE number 5006960