# Convergence & rates for Hamilton–Jacobi equations with Kirchoff junction conditions

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## Abstract

We investigate rates of convergence for two approximation schemes of time-independent and time-dependent Hamilton–Jacobi equations with Kirchoff junction conditions. We analyze the vanishing viscosity limit and monotone finite-difference schemes. Following recent work of Lions and Souganidis, we impose no convexity assumptions on the Hamiltonians. For stationary Hamilton–Jacobi equations, we obtain the classical \(\epsilon ^{\frac{1}{2}}\) rate, while we obtain an \(\epsilon ^{\frac{1}{7}}\) rate for approximations of the Cauchy problem. In addition, we present a number of new techniques of independent interest, including a quantified comparison proof for the Cauchy problem and an equivalent definition of the Kirchoff junction condition.

## Keywords

Hamilton–Jacobi equations Junction problems Stratification problems Vanishing viscosity limit Monotone finite difference schemes## Mathematics Subject Classification

35F20 65N12 65M12## Notes

### Acknowledgements

It is a pleasure to acknowledge P.E. Souganidis for suggesting this problem and for enlightening discussions. Credit is due as well to the anonymous reviewers for their sage advice and for pointing out a number of typos, and to M. Sardarli for helpful comments. The author was partially supported by the National Science Foundation Research Training Group Grant DMS-1246999.

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