1. 2D Complex-Valued Eikonal Equation Peijia Liu Department of Mathematics The University of Texas at Austin

2. Eikonal Equation Assume Eikonal Equation Substituting into the Helmholtz equation, Helmholtz equation

3. 2D Complex-Valued Eikonal Equation Replace with Let Set real and imaginary parts to zero Consider as vectors

4. 2D Complex-Valued Eikonal Equation If, we get the If, using, If and its gradient is always non-zero

5. Solvability of equation in divergence form Is (D) the equation of certain functional? Variational Problem Where

6. Solvability of equation in divergence form Equivalent norm Then satisfies the following conditions has a unique minimizer and it satisfies The minimizer satisfies (D) only if the set of critical points has zero measure

7. Solvability of equation in non-divergence form It’s a degenerate-elliptic partial differential equation Add a viscosity term to make it uniformly elliptic Stability Result in Viscosity Theory

8. is the unique minimizer of, a “weak solution” of (F) and does not depend on the particular subsequence of Solvability of equation in non-divergence form The solution of is the unique minimizer of the functional Where and Moreover, are uniformly bounded in for any locally uniformly The functional uniformly converge to, then [Magnanini,Talenti; 2003]

9. Solvability of equation in non-divergence form satisfies the equation where By De Giorgi theorem, is, thus is.

10. 2D Complex-Valued Eikonal Equation Remark The problem (F) does not always have a unique solution. However, we can always find a particular viscosity solution uniquely by using this viscosity method above is, thus all the coefficients are is estimate

11. Counter Example g is a solution of the equation in non-divergence form (F) g is not a minimizer of J and thus differs from the solution we get using the viscosity method above

12. Summary (F) is degenerate elliptic There exists a continuous solution to (F) by using viscosity method The solution to (F) may not be unique Elliptic problems require boundary conditions extra boundary conditions needed!

13. Complex Eikonal Equation in 2D proposed boundary conditions use to match the known central ray of GB Initial-boundary value problem of the 2D complex Eikonal equation