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Rigidity and flatness of the image of certain classes of mappings having tangential Laplacian. (English) Zbl 1453.35096

Authors’ abstract: In this paper we consider the PDE system of vanishing normal projection of the Laplacian for \(C^2\) maps \(u:\mathbb{R}^n \supseteq \Omega \to \mathbb{R}^N:\) \[[[Du]]^\perp\Delta u=0\mbox{ in }\Omega.\] This system has discontinuous coefficients and geometrically expresses the fact that the Laplacian is a vector field tangential to the image of the mapping. It arises as a constituent component of the \(p\)-Laplace system for all \(p\in [2, \ \infty]\). For \(p=\infty\), the \(\infty\)-Laplace system is the archetypal equation describing extrema of supremal functionals in vectorial calculus of variations in \(L^\infty\). Herein we show that the image of a solution \(u\) is piecewise affine if either the rank of \(Du\) is equal to one or \(n=2\) and \(u\) has additively separated form. As a consequence we obtain corresponding flatness results for \(p\)-Harmonic maps for \(p\in [2, \ \infty]\).

MSC:

35J70 Degenerate elliptic equations
35B06 Symmetries, invariants, etc. in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35D99 Generalized solutions to partial differential equations
49N60 Regularity of solutions in optimal control
49N99 Miscellaneous topics in calculus of variations and optimal control
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References:

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