Variational inequalities for quasilinear operators of hyperbolic type (Q1117405)

In the cylinder \(Q_ T=[0,T]\times \Omega\) in which \(T>0\) and \(\Omega \subset R^ m\) is a domain with the smooth boundary \(\partial \Omega\), the author studies a mixed problem for the quasilinear equation \[ (1)\quad L(u)\equiv u_{tt}-\sum^{n}_{i=1}(a_ i(u_{x_ i}))_{x_ i}=f(t,x) \] with the boundary condition \[ (2)\quad u(t,x)=0,\quad (t,x)\in (0,T)\times \partial \Omega \] and with the initial data \[ (3)\quad u(0,x)=u_ 0(x),\quad u_ t(0,x)=u_ 1(x),\quad x\in \Omega. \] Under the assumption of the hyperbolicity of (1): \(a_ i\in C^ 2(R)\) and \(a_ i(\xi)\geq c>0\) for all \(\xi\in R\) and \(i=1,...,n\), the author proves, as a fundamental result, the existence and uniqueness of solution \(u\in L_{\infty}(0,T;\tilde W^ 2)\) of the variational inequality \[ \int^{T}_{0}(L(u),v-u_ t)(t)dt\geq \int^{T}_{0}(f,v-u)(t)dt \] for every \(v\in L_ 2(0,T;\tilde W^ p)\). (The exact definition of the space \(\tilde W^ p\) can be seen in the reviewed paper.)