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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 \(f\).

MSC:
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
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References:
[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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