Shape Hessian for a Nondifferentiable Variational Free Boundary Problem
Yves Guido, Jean Paul Zolésio · 1997
Proof. Assume Hj(r) = {u eHj(TD) / u = 0 q.e in Tc] and H ( r c ) = {v = v\rc/v 6Hj ( r D ) , v = g q.e in T}. We have (y — g)\r € Hj( r ) , using the weak maximum principle (y — g)+\r € Hj( r ) and IT |Vr(» - 9)+\2 ~ g in T.