Maximal Monotone Operators on Hilbert Spaces and Applications

Abstract

Let H be a real Hilbert space and A : D(A) ⊂ H → H be an unbounded, linear, self-adjoint, and maximal monotone operator. The aim of this thesis is to solve u 0 (t) + Au(t) = 0, when A is linear but not bounded. The classical theory of differential linear systems cannot be applied here because the exponential formula exp(tA) does not make sense, since A is not continuous. Here we assume A is maximal monotone on a real Hilbert space, then we use the Yosida approximation to solve. Also, we provide many results on regularity of solutions. To illustrate the basic theory of the thesis, we propose to solve the heat equation in L 2 (Ω). In order to do that, we use many important properties from Sobolev spaces, Green’s formula and Lax-Milgram’s theorem.