2018/02/19 by Maria Stella Gelli, Gelli, Maria Stella, Francesca Prinari +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1802.06687
openalex publication_date 2018/02/19 · openalex created_date 2022/09/02 · openalex updated_date 2026/07/28
We consider a supremal functional of the form F(u)=
mathop
rm ess
: sup\nx
in
Omega f(x,Du(x)) where \Ω\⊆ mathbf RN is a\nregular bounded open set, u\∈ W1,\∞(\Ω) and f is a Borel\nfunction. Assuming that the intrinsic distances d\λF(x,y):= \sup\n \ u(x) - u(y): , F(u)\≤ \λ \ are locally equivalent to the\neuclidean one for every \λ>\infW1,\∞(\Ω) F, we give a\ndescription of the sublevel sets of the weak^*-lower semicontinuous envelope\nof F in terms of the sub-level sets of the difference quotient functionals\nRd^\λF(u):=\supx not =y \(u(x)-u(y))/(d^\λF(x,y)). As a\nconsequence we prove that the relaxed functional of positive 1-homogeneous\nsupremal functionals coincides with Rd1F. Moreover, for a more general\nsupremal functional F (a priori non coercive), we prove that the sublevel\nsets of its relaxed functionals with respect to the weak^* topology, the\nweak^* convergence and the uniform convergence are convex. The proof of these\nresults relies both on a deep analysis of the intrinsic distances associated to\nF and on a careful use of variational tools such as \Γ-convergence.\n