Esistenza di soluzioni per una disequazione ellittica quasilineare derivante da un problema di frontiera libera

Abstract

In this work we prove the existence of (at least) one solution of theinequality:                a(u, v − u) + l(u, v − u) ≥ 0    for any v ∈ M(u^◦ )                u ∈ M(u^◦ ) ∩ L^∞ (Omega)where M(u^◦) = {v ∈ H^{1,2}(Omega) : v = u^◦ on Gamma^+ and v ≤ u^◦ on Gamma^◦}, a and l are non linear forms, whose coefficients satisfy Caratheodory's conditions and suitable growth's assumptions, Gamma^+ and Gamma^◦ are parts of ∂Omega.The above introduced inequality represents a mathematical generalization of a free boundary problem studied in [1], where, in the same space M(u^◦), the author looks for solutions of :    Integral_Omega  k(u)∇(v − u)a(·)(∇u + e(·, u)) dx ≥ 0 for any v ∈ M(u^◦),where e is bounded and satisfying Caratheodory's conditions, k piecewise continuous, bounded and not negative, a bounded and uniformly elliptic.