Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations

By Matti Lassas, T. Liimatainen, Yi-Hsuan Lin, Mikko Salo. Revista matemática iberoamericana, 2019

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We study various partial data inverse boundary value problems for the semilinear elliptic equation $\Delta u+ a(x,u)=0$ in a domain in $\mathbb R^n$ by using the higher order linearization technique introduced in [LLS 19, FO19]. We show that the Dirichlet-to-Neumann map of the above equation determines the Taylor series of $a(x,z)$ at $z=0$ under general assumptions on $a(x,z)$. The determination of the Taylor series can be done in parallel with the detection of an unknown cavity inside the domain or an unknown part of the boundary of the domain. The method relies on the solution of the linearized partial data Calderon problem [FKSU09], and implies the solution of partial data problems for certain semilinear equations $\Delta u+ a(x,u) = 0$ also proved in [KU19]. The results that we prove are in contrast to the analogous inverse problems for the linear Schrodinger equation. There recovering an unknown cavity (or part of the boundary) and the potential simultaneously are long-standing open problems, and the solution to the Calderon problem with partial data is known only in special cases when $n \geq 3$.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/979d66611b1a71998c858780242cd5e06be08fdd

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Citations in the corpus

Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities (2019, methodology): “Following [30], we shall next discuss inverse boundary problems for semilinear elliptic equations with quadratic gradient terms, in the presence of an unknown obstacle.”