Problemas de fronteira livre na forma divergente decorrentes de perturbações singulares.

Abstract

This master thesis intents to study the Free Boudary Problem arising as a limit of the singular approximated problems { Lu = div(A(x)∇u) = Γ(x)β ε (u) em Ω u = ϕ em ∂Ω } where Ω and ϕ are smooth enough and the matrix coefficient A = A(x) is H¨older continuous, Γ is strictly positive, boundary and continuous; and β ε converges to the Dirac delta δ 0 , as ε goes to 0 + . Naturally, we pursue establish uniform properties in ε for the solutions u ε of the ε-pertubated problem above . Regularity properties will be transported to the solution of the limit problem. it will be obtained the convergence, in the Hausdorff distance, of some level sets {u ε ≥ ε}. Thus, the limit function and its free boundary will enjoy the same geometric properties associated to the minimizers u ε , of the variational problems above. Moreover, we establish a free boundary condition in the integral sense.