2D Complex-Valued Eikonal Equation
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2D Complex-Valued Eikonal Equation2D Complex-Valued Eikonal Equation
Peijia LiuDepartment of MathematicsThe University of Texas at Austin
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Eikonal Equation
Assume
Eikonal Equation
Substituting into the Helmholtz equation,
Helmholtz equation
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2D Complex-Valued Eikonal Equation
Replace with
Let
Set real and imaginary parts to zero
Consider as vectors
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2D Complex-Valued Eikonal Equation
If , we get the
If , using ,
If and its gradient is always non-zero
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Solvability of equation in divergence form
Solvability of equation in divergence form
Is (D) the equation of certain functional?
Variational Problem
Where
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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
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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
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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]
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Solvability of equation in non-divergence form
satisfies the equation
where
By De Giorgi theorem, is , thus is .
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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
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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
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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!
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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