Partial data inverse problems for the Hodge Laplacian

Abstract

Abstract. We prove uniqueness results for a Calderón type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or absolute-to-relative boundary value maps uniquely determine a zeroth order potential. The method is based on Carleman estimates for the Hodge Laplacian with relative or absolute boundary conditions, and on the construction of complex geometrical optics solutions which reduce the Calderón type problem to a tomography problem for 2-tensors. The arguments in this paper allow to establish partial data results for elliptic systems that generalize the scalar results due to Kenig-Sjöstrand-Uhlmann.

Partial data inverse problems for Maxwell equations via Carleman estimates

“…If u is a graded form we write u ⊥ = ν ∧ i ν u and u || = u − u ⊥ . The proof of the following estimate was first given in [CST13]. Here · = · M and · ∂M are the relevant L 2 norms in M and ∂M.…”

“…Note that our definition is made to guarantee that v| ∂M = 0, t∇ ν v = h, and i ν v = 0 in a neighbhorhood of the boundary of M. Now by Lemma 3.4 of [CST13],…”

“…We can rewrite the boundary terms in terms of the relative and absolute boundary values of u, by using the following formulas valid for any 1-form ξ and any k-form η (see for example [CST13]): Thus for any u ∈ H ∆ , we may use the above formulas to define (tu, tδu) as an element of H −1/2 (∂M, Λ(∂M)) × H −3/2 (∂M, Λ(∂M)). Applying this argument to * u shows that (t * u, tδ * u) are well defined for u ∈ H ∆ in a similar way.…”

“…We will then apply the recent work [CST13] where the authors studied Carleman estimates and complex geometrical optics solutions for the Hodge Laplace operator with relative and absolute boundary conditions. The crucial point is that the Carleman estimate with boundary terms proved in [CST13] for the Hodge Laplacian is sufficiently powerful to yield information also in the Maxwell case.…”

“…The crucial point is that the Carleman estimate with boundary terms proved in [CST13] for the Hodge Laplacian is sufficiently powerful to yield information also in the Maxwell case. However, we stress that the present situation for Maxwell equations does not reduce to that of [CST13], since the boundary measurements for Maxwell determine in a sense only part of the boundary measurements for the Hodge system. It is essential to use the special structure of the Maxwell system to show that the coefficients are uniquely determined.…”