Analysis and Control of Nonlinear Infinite Dimensional Systems

a.e. in Q a.e. t E a.e. t G a.e. x E Q accretive approximating arbitrary but fixed assume assumptions Ay(t Banach space Barbu boundary value problem bounded subset Brézis C0-semigroup Cauchy problem Chapter compact completes the proof condition convex function Corollary defined definition denoted differential equations duality mapping E D(A elliptic equivalently estimate exists fill finite follows free boundary Hence Hilbert space implies infinitesimal Lebesgue measure Lemma lim sup linear locally Lipschitz lower semicontinuous m-accretive maximum principle mild solution monotone operators Moreover multiply Eq nonlinear norm obstacle problem optimal control problem optimal pair parabolic problem 1.1 Proof Let proof of Theorem Proposition 1.1 readily seen reflexive satisfies scalar product Section semigroup solution to Eq Stefan problem strong solution subdifferential tend to zero Theorem 1.1 uniformly convex unique solution variational inequalities viscosity solution weak star weakly X X X yields