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Rigidity and flatness of the image of certain classes of mappings having tangential laplacian

Abugirda, H., Ayanbayev, B. and Katzourakis, N. (2020) Rigidity and flatness of the image of certain classes of mappings having tangential laplacian. Rocky Mountain Journal of Mathematics, 50 (2). pp. 383-396. ISSN 0035-7596

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To link to this item DOI: 10.1216/rmj.2020.50.383

Abstract/Summary

In this paper we consider the PDE system of vanishing normal projection of the Laplacian for C2 maps u:Rn⊇Ω→RN: [[Du]]⊥Δu=0 in Ω. 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∈[2,∞]. For p=∞, the ∞-Laplace system is the archetypal equation describing extrema of supremal functionals in vectorial calculus of variations in L∞. 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∈[2,∞].

Item Type:Article
Refereed:Yes
Divisions:Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:95576
Publisher:Rocky Mountain Mathematics Consortium

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