Regularity for Energy-Minimizing Area-Preserving Deformations |
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Authors: | Aram L Karakhanyan |
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Institution: | 1. School of Mathematics, The University of Edinburgh and Maxwell Institute for Mathematical Sciences, Mayfield Road, James Clerk Maxwell Building, King’s Buildings, EH9 3JZ, Edinburgh, UK
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Abstract: | In this paper we establish the square integrability of the nonnegative hydrostatic pressure p, that emerges in the minimization problem $$\inf_{\mathcal{K}}\int_{\varOmega}|\nabla \textbf {v}|^2, \quad\varOmega\subset \mathbb {R}^2 $$ as the Lagrange multiplier corresponding to the incompressibility constraint det?v=1 a.e. in Ω. Our method employs the Euler-Lagrange equation for the mollified Cauchy stress C satisfied in the image domain Ω ?=u(Ω). This allows to construct a convex function ψ, defined in the image domain, such that the measure of the normal mapping of ψ controls the L 2 norm of the pressure. As a by-product we conclude that $\textbf {u}\in C^{\frac{1}{2}}_{\textrm {loc}}(\varOmega)$ if the dual pressure (introduced in Karakhanyan, Manuscr. Math. 138:463, 2012) is nonnegative. |
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