Lipschitz Stability In Determining Coefficients Of An Inverse Hyperbolic Problem

Partial differential equations are widely used in natural sciences and engineering technology sciences.The direct problem is to solve partial differential equations based on definite solution conditions,thereby obtaining a mathematical description of the process or the state of the object;while the inverse problem is based on known or measured condition determines the unknown condition of the equation.The inverse problem of hyperbolic equation plays an important part in the fields of pattern recognition,atmospheric measurement,non-destructive testing,image processing,especially geophysical prospecting.The inverse problem of hyperbolic equations with unknown coefficients on Riemannian manifolds is studied.For an initial boundary value problem of hyperbolic equation,the unknown potential and damping coefficients are discussed separately.For inverse problems with damping coefficients,the unknown damping coefficient is determined according to its value at the boundary.First the inverse problem is extended,then by introducing a cut off function,the Carleman estimate with the second-order hyperbolic operator on the Riemannian manifold is given.The energy estimate of unknown damping coefficient is established,and the Lipschitz stability of inverse problem is discussed.