Existence and nonexistence of solutions to a variational problem on a square (Q2705795)

The authors consider the problem of minimizing the functional NEWLINE\[NEWLINE\int_\Omega [h(\|\nabla v\|)+ v] dx,NEWLINE\]NEWLINE where \(h: [0,\infty)\to [0,\infty]\) takes finite values only at \(t= 1\) and \(t=2\), namely \(h(1)= 0\) and \(h(2)= 1\). Therefore, any minimizer must have 1 or 2 a.e. in \(\Omega\) as the modulus of its gradient. For \(\Omega\) a circular disk, one can solve this problem explicitly and obtains a cone or intersection of two cones as the shape of the optimal solution. Here, the case that \(\Omega\) is a square is investigated. A relaxed formulation of the problem leads to an Euler-Lagrange equation involving the 1-Laplacian operator. While there exists always a solution to the relaxed problem (in which \(h\) is replaced by its convex envelope \(h^{**}\) on \([0,\infty)\)), a solution to the original problem exists if and only if the length of the square does not exeed 2.NEWLINENEWLINENEWLINEThe reviewer wonders why his name is again misspelled.