Linking special semiconvex functions and three-term-recurrent structures (Q2747956)

After a short review on the various notions of convexity, such as polyconvexity, quasiconvexity, rank-one convexity, that occur in many problems of the calculus of variations, the author considers in Section 2 functions \(f_\gamma: \mathbb{R}^n\to\mathbb{R}\) of the form NEWLINE\[NEWLINEf_\gamma(z)= |z|+\gamma {Az\cdot z\over|z|}NEWLINE\]NEWLINE with \(A\) a symmetric \(n\times n\) matrix. It is shown that NEWLINE\[NEWLINEf_\gamma\text{ convex }\Longleftrightarrow \gamma\leq{1\over(\mu_n- 2\mu_1)^+},NEWLINE\]NEWLINE where \(\mu_k\) are the eigenvalues of \(A\).NEWLINENEWLINENEWLINEIn Section 3 some stability properties of the functions \(f_\gamma\) above are studied with respect to small perturbations of the matrix \(A\).NEWLINENEWLINENEWLINEFinally, in Section 4 the particular case of NEWLINE\[NEWLINEf(z)= g(\lambda_1(z), \lambda_2(z))NEWLINE\]NEWLINE is studied, where \(z\) varies among \(2\times 2\) matrices and \(\lambda_1\), \(\lambda_2\) denote the singular values of \(z\), i.e., the eigenvalues of the matrix \((A^t A)^{1/2}\).