Convex sets with obstacle and gradient constraints

• Autor(a) principal: Santos, Lisa
• Data de Publicação: 2023
• Outros Autores: Azevedo, Assis, Azevedo, Davide Manuel Santos
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: https://hdl.handle.net/1822/84696
• topic: Hamilton-Jacobi equation with obstacle; Convex sets; Variational inequalities; Mosco convergence; Ciências Naturais::Matemáticas; Science & Technology

For 1 <p<∞, given g and ψ non-negative functions belonging to L^\infty(\Omega) and W^{1,p}_0(\Omega) ∩C(\overline\Omega), respectively, we show that there exists a pseudo-metric L_gdefined in \overline\Omega such that u(x) :=miny_{y\in\overline\Omega}{ψ(y) +Lg(x, y)} is a subsolution of the Hamilton-Jacobi equation with obstacle max{|∇u| −g, u −ψ} =0. Be-sides, for K_{g,ψ}={v∈W^{1,p}_0(\Omega) :|∇v| ≤g, v≤ψ}, we have u(x) =\bigvee{v(x) :v∈Kg,ψ}. As a consequence, we prove the Mosco convergence of K_{g_n,ψ_n} to K_{g,ψ}, as long as g_n converges to g in L^∞(\Omega)and ψ_n to ψ in C(\overline\Omega). As an application, we prove a stability result for the solutions u_n of variational inequalities defined in the convex sets K_{g_n,ψ_n}.