Mathematical Research Letters

Volume 27 (2020)

Number 6

Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities

Pages: 1801 – 1824



Katya Krupchyk (Department of Mathematics, University of California, Irvine, Calif., U.S.A.)

Gunther Uhlmann (Department of Mathematics, University of Washington, Seattle, Wa., U.S.A.; and Institute for Advanced Study, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong)


We show that the linear span of the set of scalar products of gradients of harmonic functions on a bounded smooth domain $\Omega \subset \mathbb{R}^n$ which vanish on a closed proper subset of the boundary is dense in $L^1 (\Omega)$. We apply this density result to solve some partial data inverse boundary problems for a class of semilinear elliptic PDE with quadratic gradient terms.

The research of K.K. is partially supported by the National Science Foundation (DMS 1815922). The research of G.U. is partially supported by NSF and a Si-Yuan Professorship of HKUST. Part of the work was supported by the NSF grant DMS-1440140 while both authors were in residence at MSRI in Berkeley, California, during Fall 2019 semester.

Received 8 October 2019

Accepted 13 March 2020

Published 17 February 2021