Observability inequalities for wave equations with variable coefficients

We consider some observability inequalities from boundary for wave equations with variable coefficients in space. At first, an estimate is established by the geometric multiplier method in the case that no boundary conditions are imposed under some checkable geometric conditions. Then our results yield continuous observability estimates for two kinds of boundary conditions which have a physical meaning with an explicit observability time; hence, by duality, exact controllability results. Next, a number of nontrivial examples are presented to verify the observability inequality. In particular, a counterexample is given for which the observability inequality does not hold true, where the observability portion is the entire boundary