The conditional stability for unique continuation on the sphere for Helmholtz equations

Authors：Chen, Yu; Cheng, Jin^*; Jiang, Yu

Source：MATHEMATICAL METHODS IN THE APPLIED SCIENCES, 2022.

DOI：10.1002/mma.8469

• conditional stability
• convergence rate
• Helmholtz equations
• Tihkonov regularization
• unique continuation on a sphere

Unique continuation is one of the most important properties for the solution of partial differential equation, which means the local information of the solution can determine the global one. In this paper, we discuss a special unique continuation for Helmholtz equations on a sphere in Double-struck capital R-3, which is different with the classical unique continuation. The unique continuation holds only on the sphere and may not be extended to the domain in Double-struck capital R-3. A Holder type conditional stability of unique continuation is proved by complex extension method. Then a Tikhonov regularized scheme is proposed and the convergence rate is obtained by using the conditional stability estimate. Numerical examples are presented to show the performance of the scheme. It should be remarked here that our results may be applied to the problem of recovering a far field pattern with its local information.

• Institution
  复旦大学