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On convexity properties of homogeneous functions of degree one. (English) Zbl 0895.49002
The authors give an explicit example of a function that is homogeneous of degree one, rank-one convex, but not convex. The function is defined by \(f(\xi)=|\xi|+\gamma\langle M\xi;\xi\rangle/| \xi| \) if \(\xi\not= 0\), \(f(0)=0\). Here \(\xi\in R^{2\times 2}\), \(| \xi|\) denotes the Euclidean norm of \(\xi\) in \(R^4\), \(\langle\cdot;\cdot\rangle\) is the scalar product in \(R^4\), \(\gamma\geq 0\) is appropriately chosen to produce a rank-one function which is not convex, and \(M\) is a symmetric matrix, which commutes with the matrix \(E\) (\(E\) is the symmetric matrix defined by \(\text{det }\xi={1\over 2}\langle E\xi;\xi\rangle\)). The following choice \(M=\left(\begin{matrix} 9&0&0&1\\ 0&6&2&0\\ 0&2&6&0\\ 1&0&0&9\end{matrix}\right)\), and \(\gamma\in ({1\over 2},1)\) gives an explicit counterexample. It is derived from a more general theorem which relies the eigenvalues of \(M\) and \(\gamma\), with the convexity and the rank-one convexity properties of \(f\).

49J10 Existence theories for free problems in two or more independent variables
26B25 Convexity of real functions of several variables, generalizations
49J45 Methods involving semicontinuity and convergence; relaxation
49K10 Optimality conditions for free problems in two or more independent variables
Full Text: DOI
[1] Sverak, Proc. Roy. Soc. Edinburgh Sect. A 120 pp 185– (1992) · Zbl 0777.49015
[2] Dacorogna, Direct Methods in the Calculus of Variations (1989) · Zbl 0703.49001
[3] Dacorogna, Progress in partial differential equations, Pont-à-Mousson, 1994 2 (1995)
[4] DOI: 10.1512/iumj.1992.41.41017 · Zbl 0736.26006
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