# American Institute of Mathematical Sciences

December  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 and Applied Analysis, 2002, 1 (4) : 565-573. doi: 10.3934/cpaa.2002.1.565
 [1] Yong Xia. Convex hull of the orthogonal similarity set with applications in quadratic assignment problems. Journal of Industrial and Management Optimization, 2013, 9 (3) : 689-701. doi: 10.3934/jimo.2013.9.689 [2] Alessandro Ferriero, Nicola Fusco. A note on the convex hull of sets of finite perimeter in the plane. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 103-108. doi: 10.3934/dcdsb.2009.11.103 [3] Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems and Imaging, 2021, 15 (2) : 315-338. doi: 10.3934/ipi.2020070 [4] Xianchao Xiu, Lingchen Kong. Rank-one and sparse matrix decomposition for dynamic MRI. Numerical Algebra, Control and Optimization, 2015, 5 (2) : 127-134. doi: 10.3934/naco.2015.5.127 [5] Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83 [6] Masayuki Asaoka. Local rigidity of homogeneous actions of parabolic subgroups of rank-one Lie groups. Journal of Modern Dynamics, 2015, 9: 191-201. doi: 10.3934/jmd.2015.9.191 [7] Mao Okada. Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces. Journal of Modern Dynamics, 2021, 17: 111-143. doi: 10.3934/jmd.2021004 [8] Hong Seng Sim, Chuei Yee Chen, Wah June Leong, Jiao Li. Nonmonotone spectral gradient method based on memoryless symmetric rank-one update for large-scale unconstrained optimization. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021143 [9] Ling Zhang, Xiaoqi Sun. Stability analysis of time-varying delay neural network for convex quadratic programming with equality constraints and inequality constraints. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022035 [10] Leandro M. Del Pezzo, Nicolás Frevenza, Julio D. Rossi. Convex and quasiconvex functions in metric graphs. Networks and Heterogeneous Media, 2021, 16 (4) : 591-607. doi: 10.3934/nhm.2021019 [11] Somphong Jitman, Ekkasit Sangwisut. The average dimension of the Hermitian hull of constacyclic codes over finite fields of square order. Advances in Mathematics of Communications, 2018, 12 (3) : 451-463. doi: 10.3934/amc.2018027 [12] David L. Russell. Coefficient identification and fault detection in linear elastic systems; one dimensional problems. Mathematical Control and Related Fields, 2011, 1 (3) : 391-411. doi: 10.3934/mcrf.2011.1.391 [13] Ye Wang, Ran Tao. Constructions of linear codes with small hulls from association schemes. Advances in Mathematics of Communications, 2022, 16 (2) : 349-364. doi: 10.3934/amc.2020114 [14] Songqiang Qiu, Zhongwen Chen. An adaptively regularized sequential quadratic programming method for equality constrained optimization. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2675-2701. doi: 10.3934/jimo.2019075 [15] Frank Blume. Minimal rates of entropy convergence for rank one systems. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 773-796. doi: 10.3934/dcds.2000.6.773 [16] Raz Kupferman, Asaf Shachar. On strain measures and the geodesic distance to $SO_n$ in the general linear group. Journal of Geometric Mechanics, 2016, 8 (4) : 437-460. doi: 10.3934/jgm.2016015 [17] Sébastien Gautier, Lubomir Gavrilov, Iliya D. Iliev. Perturbations of quadratic centers of genus one. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 511-535. doi: 10.3934/dcds.2009.25.511 [18] John Sheekey. A new family of linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 475-488. doi: 10.3934/amc.2016019 [19] Tim Gutjahr, Karsten Keller. Equality of Kolmogorov-Sinai and permutation entropy for one-dimensional maps consisting of countably many monotone parts. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4207-4224. doi: 10.3934/dcds.2019170 [20] Jiyoung Han. Quantitative oppenheim conjecture for $S$-arithmetic quadratic forms of rank $3$ and $4$. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2205-2225. doi: 10.3934/dcds.2020359

2020 Impact Factor: 1.916