An inverse problem with data on the part of the boundary

Abstract

Let u t = ∇ 2 u − q( x) u ≔ Lu in D × [0, ∞), where D ⊂ R 3 is a bounded domain with a smooth connected boundary S, and q( x) ∈ L 2( S) is a real-valued function with compact support in D. Assume that u( x, 0) = 0, u = 0 on S 1 ⊂ S, u = a( s, t) on S 2 = S⧹ S 1, where a( s, t) = 0 for t > T, a( s, t) ≢ 0, a ∈ C 1([0, T]; H 3/2( S 2)) is arbitrary. Given the extra data u N ∣ S 2 = b ( s , t ) , for each a ∈ C 1([0, T]; H 3/2( S 2)), where N is the outer normal to S, one can find q( x) uniquely. A similar result is obtained for the heat equation u t = L u ≔ ∇ · ( a ∇ u ) . These results are based on new versions of Property C.