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 Algebra i Analiz: Year: Volume: Issue: Page: Find

 Algebra i Analiz, 1994, Volume 6, Issue 6, Pages 128–153 (Mi aa485)

Research Papers

Partial regularity of the deformation gradient for some model problems in nonlinear twodimensional elasticity

M. Fuchsa, G. A. Sereginb

a Saarland University
b St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: We consider the model problem of minimizing the functional $\int_{\Omega}\frac{1}{2}|\nabla u|^2+h(\operatorname{det}\nabla u)dx$ where $u:\mathbb R^2\supset\Omega\to\mathbb R^2$ and $h:\mathbb R\to[0,\infty]$ denotes a function which is convex and smooth on $(0,\infty)$, $\operatorname{lim}_{t\downarrow 0}h(t)=+\infty$ and $h\equiv+\infty$ on $(-\infty,0]$. In particular, we show that it is possible to introduce an approximation $\int_{\Omega}\frac{1}{2}|\nabla u|^2+h_{\delta}(\operatorname{det}\nabla u)dx$ for the energy whose minimizers $u_{\delta}$ are of class $C^1$ on some open subset $\Omega_{\delta}$ of $\Omega$ and converge strongly in $H^{1,2}(\Omega,\mathbb R^2)$ to a minimizer è of the original problem. Moreover, we have control on the measure of the exceptional set in the sense that $|\Omega-\Omega_{\delta}|\to 0$ as $\delta\to 0$.

Keywords: Nonlinear elasticity, partial regularity, approximation.

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English version:
St. Petersburg Mathematical Journal, 1995, 6:6, 1229–1248

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Citation: M. Fuchs, G. A. Seregin, “Partial regularity of the deformation gradient for some model problems in nonlinear twodimensional elasticity”, Algebra i Analiz, 6:6 (1994), 128–153; St. Petersburg Math. J., 6:6 (1995), 1229–1248

Citation in format AMSBIB
\Bibitem{FucSer94} \by M.~Fuchs, G.~A.~Seregin \paper Partial regularity of the deformation gradient for some model problems in nonlinear twodimensional elasticity \jour Algebra i Analiz \yr 1994 \vol 6 \issue 6 \pages 128--153 \mathnet{http://mi.mathnet.ru/aa485} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1322123} \zmath{https://zbmath.org/?q=an:0839.73009|0827.73010} \transl \jour St. Petersburg Math. J. \yr 1995 \vol 6 \issue 6 \pages 1229--1248