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The failure of rank-one connections. (English) Zbl 1007.35014

The authors study the interface problems for Lipschitz mappings \(f_{+}: \mathbb{R}^{n}_{+} \rightarrow \mathbb{R}^{m}\) and \(f_{-}: \mathbb{R}^{n}_{-} \rightarrow \mathbb{R}^{m}\) in the half spaces, which agree on the common boundary \(\mathbb{R}^{n-1}=\partial \mathbb{R}^{n}_{+}= \partial \mathbb{R}^{n}_{-}\). The main task is to determine the relationship between the sets of values of the differentials \(Df_{+}\) and \(Df_{-}\). Some examples show that neither the polyconvex hulls \([Df_{+}]^{pc}\) and \([Df_{-}]^{pc}\) satisfy Hadamard’s jump condition nor are rank-one connected. Moreover, the authors obtained some estimates of the Jacobian in the course of solving Monge-Ampère inequalities and they construct uniformly elliptic systems of first-order partial differential equations in the same homotopy class as the familiar Cauchy-Riemann equations, for which the unique continuation property fails.

MSC:

35F05 Linear first-order PDEs
26B25 Convexity of real functions of several variables, generalizations
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