Carleman estimates with no lower-order terms for general Riemann wave equations. Global uniqueness and observability in one shot
This long paper deals with a new and interesting study of a fully general Riemann wave equation \[ w_{tt}= \Delta w+ F(w)+ f\quad\text{in } (0, T)\times\Omega\tag{1} \] on a finite-dimensional Riemannian manifold with energy level terms on an open bounded, connected, compact set of \(M\) with smooth boundary, where \(\Delta\) is the Laplace operator on the manifold \(M\), \(F(w)\) is the energy-level differential term \[ F(w)= \langle P(t, x), Dw\rangle+ p(t, x)w_t+ p_0(t, x)w, \] \(p_0\), \(p_1\) are functions on \([0, T]\times\Omega\) and \(P(t)\) is a vector field on \(M\) for \(t> 0\), and \(f\) is a forcing term in (1). The terms \(F(w)\) and \(f\) satisfy the following conditions: (i) there exists a constant \(C_T> 0\) such that \[ |F(w)|^2\leq C_t\{|Dw|^2+ w^2_t+ w^2\},\quad x,t\in Q, \] where \(Dw = \nabla_gw\) for the scalar function \(w\), \(g\) is the metric on \(M\) and \(D\) is the Levi-Cività connection on \(M\) and \(F\in L(0,T; \Omega)= L_2(Q)\), \[ \int_Q f^2 dQ< \infty, \] where \(dQ= d\Omega dt\) and \(d\Omega\) is the volume element of the manifold \(M\) in its Riemann metric \(g\). The authors prove Carleman-type estimates which contain no lower-order term. They provide Carleman-type inequalities: first pointwise, for \(C^2\)-solutions and then in integral form for \(H^{1,1}(Q)\)-solutions. Their other main results include global uniqueness for overdetermined problems and continuous observability/uniform stabilization inequalities. Indeed, the authors generalize results in the Riemannian setting in the more different case of purely Neumann boundary conditions, and in the case of Dirichlet boundary conditions.
Dirichlet and Neumann boundary conditions
scientific article; zbMATH DE number 1966429