2D Complex-Valued Eikonal Equation2D Complex-Valued Eikonal Equation

Peijia LiuDepartment of MathematicsThe University of Texas at Austin

Eikonal Equation

Assume

Eikonal Equation

Substituting into the Helmholtz equation,

Helmholtz equation

2D Complex-Valued Eikonal Equation

Replace with

Let

Set real and imaginary parts to zero

Consider as vectors

2D Complex-Valued Eikonal Equation

If , we get the

If , using ,

If and its gradient is always non-zero

Solvability of equation in divergence form

Solvability of equation in divergence form

Is (D) the equation of certain functional?

Variational Problem

Where

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

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

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]

Solvability of equation in non-divergence form

satisfies the equation

where

By De Giorgi theorem, is , thus is .

2D Complex-Valued Eikonal Equation

RemarkThe 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

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

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!

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