Abstract
We study two inverse problems on a globally hyperbolic Lorentzian manifold (M, g). The problems are:
Passive observations in spacetime: consider observations in an open set . The light observation set corresponding to a point source at is the intersection of V and the light-cone emanating from the point q.
Let be an unknown open, relatively compact set. We show that under natural causality conditions, the family of light observation sets corresponding to point sources at points determine uniquely the conformal type of W.
Active measurements in spacetime: we develop a new method for inverse problems for non-linear hyperbolic equations that utilizes the non-linearity as a tool. This enables us to solve inverse problems for non-linear equations for which the corresponding problems for linear equations are still unsolved. To illustrate this method, we solve an inverse problem for semilinear wave equations with quadratic non-linearities. We assume that we are given the neighborhood V of the time-like path and the source-to-solution operator that maps the source supported on V to the restriction of the solution of the wave equation to V. When M is 4-dimensional, we show that these data determine the topological, differentiable, and conformal structures of the spacetime in the maximal set where waves can propagate from and return back to mu.
| Original language | English |
|---|---|
| Journal | Inventiones Mathematicae |
| Volume | 212 |
| Issue number | 3 |
| Pages (from-to) | 781-857 |
| Number of pages | 77 |
| ISSN | 0020-9910 |
| DOIs | |
| Publication status | Published - Jun 2018 |
| MoE publication type | A1 Journal article-refereed |
Fields of Science
- CUSP
- ELASTOGRAPHY
- OPERATORS
- PROGRESSING WAVES
- SCATTERING
- SEMILINEAR WAVE-EQUATIONS
- SINGULAR SYMBOLS
- SPACETIMES
- TIME
- UNIQUE CONTINUATION
- 111 Mathematics
- 112 Statistics and probability
Cite this
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Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations. / Kurylev, Yaroslav; Lassas, Matti; Uhlmann, Gunther.
In: Inventiones Mathematicae, Vol. 212, No. 3, 06.2018, p. 781-857.Research output: Contribution to journal › Article › Scientific › peer-review
TY - JOUR
T1 - Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations
AU - Kurylev, Yaroslav
AU - Lassas, Matti
AU - Uhlmann, Gunther
PY - 2018/6
Y1 - 2018/6
N2 - We study two inverse problems on a globally hyperbolic Lorentzian manifold (M, g). The problems are:Passive observations in spacetime: consider observations in an open set . The light observation set corresponding to a point source at is the intersection of V and the light-cone emanating from the point q.Let be an unknown open, relatively compact set. We show that under natural causality conditions, the family of light observation sets corresponding to point sources at points determine uniquely the conformal type of W.Active measurements in spacetime: we develop a new method for inverse problems for non-linear hyperbolic equations that utilizes the non-linearity as a tool. This enables us to solve inverse problems for non-linear equations for which the corresponding problems for linear equations are still unsolved. To illustrate this method, we solve an inverse problem for semilinear wave equations with quadratic non-linearities. We assume that we are given the neighborhood V of the time-like path and the source-to-solution operator that maps the source supported on V to the restriction of the solution of the wave equation to V. When M is 4-dimensional, we show that these data determine the topological, differentiable, and conformal structures of the spacetime in the maximal set where waves can propagate from and return back to mu.
AB - We study two inverse problems on a globally hyperbolic Lorentzian manifold (M, g). The problems are:Passive observations in spacetime: consider observations in an open set . The light observation set corresponding to a point source at is the intersection of V and the light-cone emanating from the point q.Let be an unknown open, relatively compact set. We show that under natural causality conditions, the family of light observation sets corresponding to point sources at points determine uniquely the conformal type of W.Active measurements in spacetime: we develop a new method for inverse problems for non-linear hyperbolic equations that utilizes the non-linearity as a tool. This enables us to solve inverse problems for non-linear equations for which the corresponding problems for linear equations are still unsolved. To illustrate this method, we solve an inverse problem for semilinear wave equations with quadratic non-linearities. We assume that we are given the neighborhood V of the time-like path and the source-to-solution operator that maps the source supported on V to the restriction of the solution of the wave equation to V. When M is 4-dimensional, we show that these data determine the topological, differentiable, and conformal structures of the spacetime in the maximal set where waves can propagate from and return back to mu.
KW - CUSP
KW - ELASTOGRAPHY
KW - OPERATORS
KW - PROGRESSING WAVES
KW - SCATTERING
KW - SEMILINEAR WAVE-EQUATIONS
KW - SINGULAR SYMBOLS
KW - SPACETIMES
KW - TIME
KW - UNIQUE CONTINUATION
KW - 111 Mathematics
KW - 112 Statistics and probability
U2 - 10.1007/s00222-017-0780-y
DO - 10.1007/s00222-017-0780-y
M3 - Article
VL - 212
SP - 781
EP - 857
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
SN - 0020-9910
IS - 3
ER -