6533b86ffe1ef96bd12ce785
RESEARCH PRODUCT
Inverse problems for elliptic equations with power type nonlinearities
Yi-hsuan LinTony LiimatainenMatti LassasMikko Salosubject
Mathematics - Differential GeometryGLOBAL UNIQUENESSGeneral MathematicsConformal mapCALDERON PROBLEMTransversally anisotropic01 natural sciencesinversio-ongelmatMathematics - Analysis of PDEsSimple (abstract algebra)Euclidean geometryFOS: Mathematics111 MathematicsApplied mathematics0101 mathematicsMathematicsInverse boundary value problemosittaisdifferentiaaliyhtälötCalderón problemGeometrical opticsSemilinear equationApplied Mathematics010102 general mathematicstransversally anisotropicInverse problemManifold010101 applied mathematicssemilinear equationNonlinear systemDifferential Geometry (math.DG)inverse boundary value problemLinear equationAnalysis of PDEs (math.AP)description
We introduce a method for solving Calder\'on type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. Assuming the knowledge of a nonlinear Dirichlet-to-Neumann map, we determine both a potential and a conformal manifold simultaneously in dimension $2$, and a potential on transversally anisotropic manifolds in dimensions $n \geq 3$. In the Euclidean case, we show that one can solve the Calder\'on problem for certain semilinear equations in a surprisingly simple way without using complex geometrical optics solutions.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2021-01-01 | Journal de Mathématiques Pures et Appliquées |