2020/05/14 by Francesca Prinari, Prinari, Francesca, Michela Eleuteri +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analytic and geometric function theory #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2005.06774
openalex publication_date 2020/05/14 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
We study the Γ-convergence of the functionals Fn(u):= || f(⋅,u(⋅),Du(⋅))||pn(⋅) and Fn(u):= ∫Ω (1)/(pn(x)) fpn(x)(x,u(x),Du(x))dx defined on X∈ \L1(Ω,ℝd), L^∞(Ω,ℝd), C(Ω,ℝd)\ (endowed with their usual norms) with effective domain the Sobolev space W1,pn(⋅)(Ω, ℝd ). Here Ω⊆ ℝN is a bounded open set, N,d ≥ 1 and the measurable functions pn: Ω → (1, + ∞) satisfy the conditions \mathop\rm ess sup Ω pn ≤ β \mathop\rm ess inf Ω pn for a fixed constant β> 1 and \mathop\rm ess inf Ω pn → + ∞ as n → + ∞. We show that when f(x,u,⋅) is level convex and lower semicontinuous and it satisfies a uniform growth condition from below, then, as n→ ∞, the sequences (Fn)n Γ-converges in X to the functional F represented as F(u)= || f(⋅,u(⋅),Du(⋅))||∞ on the effective domain W1,∞(Ω, ℝd ). Moreover we show that the Γ-limn \mathcal Fn is given by the functional F(u):=\\begin arraylll & 0 & \hboxif || f(⋅,u(⋅),Du(⋅)) ||∞≤ 1, & +∞ & \hboxotherwise in X. \endarray.