The minimum of quadratic functionals of the gradient on the set of convex functions (Q1864172)

The authors prove that given an \(N\times N\) symmetric matrix \(M\), then \[ \lambda_1(M)\leq \inf _{u\in A}\int _\Omega M Du Du \leq \inf _{x\in \partial \Omega} M\nu(x) \nu(x) =\lambda, \] where \(\Omega\subset \mathbb{R}^N\), \(A=\{ u\in H_0^1(\Omega): \int _\Omega |Du|^2\leq 1, u\) convex \(\}\), \(\nu(x)\) is the exterior normal vector to \(\Omega\) and \(\lambda_1(M)\) is the minimum eigenvalue of \(M\); they also prove that if \(\Omega\) is a \(C^1\) domain, then \(\lambda=\lambda_1(M)\), but in the case \(\lambda>\lambda_1(M)\) it is given an example showing that the inequalities are strict. The authors study also the case when \(M\) is \(x\)--dependent, giving a similar result.