TY - ELEC
AU - Lauri Oksanen
PY - 2019
TI - Inverse problem for a semi-linear elliptic equation
LA - eng
M3 - Moving Image
AB - We consider the Dirichlet-to-Neumann map, defined in a suitable sense, for the equation $-\Delta u + V(x,u)=0$ on a compact Riemannian
manifold with boundary. We show that, under certain geometrical assumptions, the Dirichlet-to-Neumann map determines V for a large
class of non-linearities. The proof is constructive and is based on a multiple-fold linearization of the semi-linear equation near complex
geometric optics solutions for the linearized operator, and the resulting non-linear interactions. This approach allows us to reduce
the inverse problem boundary value problem to the purely geometric problem to invert a family of weighted ray transforms, that we call
the Jacobi weighted ray transform. This is a joint work with Ali Feizmohammadi.
N2 - We consider the Dirichlet-to-Neumann map, defined in a suitable sense, for the equation $-\Delta u + V(x,u)=0$ on a compact Riemannian
manifold with boundary. We show that, under certain geometrical assumptions, the Dirichlet-to-Neumann map determines V for a large
class of non-linearities. The proof is constructive and is based on a multiple-fold linearization of the semi-linear equation near complex
geometric optics solutions for the linearized operator, and the resulting non-linear interactions. This approach allows us to reduce
the inverse problem boundary value problem to the purely geometric problem to invert a family of weighted ray transforms, that we call
the Jacobi weighted ray transform. This is a joint work with Ali Feizmohammadi.
UR - https://open.library.ubc.ca/collections/48630/items/1.0383384
ER - End of Reference