Existence and \({\mathcal L}^ {2,\mu}\) regularity for a class of nonlinear systems with monotonicity hypothesis (Q1825367)

The operator \(A: H\to H,\) where H is a Hilbert space, is said to be monotone if \(A(0)=0\) and there exists a constant \(M\geq 0\), such that \(x,y\in H\) implies \(\| A(x)-A(y)\|_ H\leq M\| x-y\|_ H,\) \((A(x)-A(y),x-y)_ H\geq 0.\) If the second inequality is replaced by \((A(x)-A(y),x-y)_ H\geq \nu \| x-y\|^ 2,\) \(M\geq \nu >0\), the notion of strictly monotone operator is obtained. It is proved, that if A: \(H\to H\) is a strictly monotone operator, then for arbitrary \(\lambda >0\), \(x,y\in H\), the following inequality holds \[ \| \lambda (x-y)-A(x)+A(y)\|_ H\leq \lambda K(\lambda)\| x- y\|_ H, \] where \[ K(\lambda)=(1+(M^ 2/\lambda^ 2)-(2\nu /\lambda))^{1/2}\quad and\quad K(\lambda)<1\quad for\quad \lambda >(M^ 2/2\nu). \] As a consequence if \(A: H\to H\) is strictly monotone, then for each \(f\in H\), there exists a unique \(x\in H\) such that \(A(x)=f\) and \(\| x\|_ H\leq (1/\nu)\| f\|.\) This statement applied to the system \[ \sum_{i}D_ ia^ i(x,Du+Dg)+b(x,u+h)=-\sum_{i}D_ if^ i+f^ 0,\quad u\in H^ 1(\Omega), \] allows to assert that if the vector a(x,p) is uniformly continuous in x, strictly monotone in p and the vector b(x,p) is measurable in x, monotone in u, then this problem has a unique solution. The \({\mathcal L}^{2,\mu}\) regularity of this solution is shown, under the assumptions that Dg, \(f^ 1,...,f^ n\) belongs to \({\mathcal L}^{2,\mu}(\Omega)\) and h, \(f^ 0\) belong to \({\mathcal L}^{2,(\mu - 2)\nu_ 0}(\Omega)\). Here \(\mu\in (0,{\bar \alpha})\) is arbitrary and \({\bar \alpha}\in (2,n]\) is a fixed number. In particular if \(n-2<\mu <{\bar \alpha}\) Hölder regularity of the solution u in \(\Omega\) is obtained.