From Jacobian to Hessian: distributional form and relaxation. (English) Zbl 1101.49011

The authors study various definitions of determinant of the second derivative of functions on Euclidean domains, focusing mainly on three of them, which apply to suitable Sobolev classes: the pointwise Hessian determinant, the weak Jacobian of the gradient of \(\nabla u\) and a yet weaker formulation which exploit the second order structure of the Hessian determinant. The paper contains several stability statements under weak convergence, comparison results between the different notions and applications to relaxation problems. The authors give also examples showing that some second order problems considered in this setting cannot be reduced to first order ones.


49J45 Methods involving semicontinuity and convergence; relaxation
49K20 Optimality conditions for problems involving partial differential equations