Estimation of Point Sources and the Applications to Inverse Problems

Let Ω ⊂ RN be a bounded domain. We introduce a distance ‖A− B‖ between two n-tuples of point sources A = (a1, ..., an), B = (b1, ...., bn) ∈ Ω and we establish upper and lower estimates between ‖A−B‖ and a norm of Pn j=1 δaj − Pn j=1 δbj . Next we will apply the estimates to the following two inverse problems and prove best possible conditional stability estimates: (i) inverse problem of determining point heat sources by final overdetermining data. (ii) inverse problem of determining point wave sources by lateral overdetermining data.