# American Institute of Mathematical Sciences

2002, 1(4): 565-573. doi: 10.3934/cpaa.2002.1.565

## On equality of relaxations for linear elastic strains

 1 School of Mathematical Sciences, University of Sussex, Falmer, Brighton, BN1 9QH, United Kingdom

Received  October 2001 Revised  July 2002 Published  September 2002

We use the quadratic rank-one convex envelope $qr_e(f)$ for $f:M_s^{n} \to \mathbb R$ defined on the space of linear elastic strains with $n\geq 2$ to study conditions for equality of semiconvex envelopes. We also use the corresponding quadratic rank-one convex hull $qr_e(K)$ for compact sets $K\subset M_s^{n}$ to give a condition for equality of semiconvex hulls. We show that $L^e_c(K)=C(K)$ if and only if $qr_e(K)=C(K)$, where $L^e_c(K)$ is the closed lamination convex hull on linear strains. We also establish that for functions satisfying $f(A)\geq c|A|^2-C_1$ for $A\in M_s^{n}$, $R_e(f)=C(f)$ if and only if $qr_e(f)=C(f)$.
Citation: Kewei Zhang. On equality of relaxations for linear elastic strains. Communications on Pure & Applied Analysis, 2002, 1 (4) : 565-573. doi: 10.3934/cpaa.2002.1.565
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