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 the
inequality:
                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.