RECORD
Stochastic homogenization and effective Hamiltonians of HJ equations in one space dimension: The double-well case
Analysis of PDEs · Submitted 15 Jul 2020
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Under the extra condition that $G$ is a double-well function (i.e., it has precisely two local minima), we give a new and fully constructive proof of homogenization which yields a formula for the effective Hamiltonian $\overline H$.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
We consider Hamilton-Jacobi equations in one space dimension with Hamiltonians of the form $H(p,x,\omega) = G(p) + \beta V(x,\omega)$, where $V(\cdot,\omega)$ is a stationary and ergodic potential of unit amplitude. The homogenization of such equations is established in a 2016 paper of Armstrong, Tran and Yu for all continuous and coercive $G$. Under the extra condition that $G$ is a double-well function (i.e., it has precisely two local minima), we give a new and fully constructive proof of homogenization which yields a formula for the effective Hamiltonian $\overline H$. We use this formula to provide a complete list of the heights at which the graph of $\overline H$ has a flat piece. We illustrate our results by analyzing basic classes of examples, highlight some corollaries that clarify the dependence of $\overline H$ on $G$, $\beta$ and the law of $V(\cdot,\omega)$, and discuss a generalization to even-symmetric triple-well Hamiltonians.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider Hamilton-Jacobi equations in one space dimension with Hamiltonians of the form $H(p,x,\omega) = G(p) + \beta V(x,\omega)$, where $V(\cdot,\omega)$ is a stationary and ergodic potential of unit amplitude. | background 0.95 | 0.30 | 0.03 | asserted 0.89 |
| s1 | The homogenization of such equations is established in a 2016 paper of Armstrong, Tran and Yu for all continuous and coercive $G$. | background 1.00 | 0.03 | 0.05 | asserted 1.00 |
| s2 | Under the extra condition that $G$ is a double-well function (i.e., it has precisely two local minima), we give a new and fully constructive proof of homogenization which yields a formula for the effective Hamiltonian $\overline H$. | contribution 0.99 | 0.96 | 0.76 | asserted 1.00 |
| s3 | We use this formula to provide a complete list of the heights at which the graph of $\overline H$ has a flat piece. | result 0.86 | 0.95 | 0.22 | asserted 1.00 |
| s4 | We illustrate our results by analyzing basic classes of examples, highlight some corollaries that clarify the dependence of $\overline H$ on $G$, $\beta$ and the law of $V(\cdot,\omega)$, and discuss a generalization to even-symmetric triple-well Hamiltonians. | contribution 0.92 | 0.93 | 0.09 | asserted 0.99 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
We consider Hamilton-Jacobi equations in one space dimension with Hamiltonians of the form $H(p,x,\omega) = G(p) + \beta V(x,\omega)$, where $V(\cdot,\omega)$ is a stationary and ergodic potential of unit amplitude. The homogenization of such equations is established in a 2016 paper of Armstrong, Tran and Yu for all continuous and coercive $G$. Under the extra condition that $G$ is a double-well function (i.e., it has precisely two local minima), we give a new and fully constructive proof of homogenization which yields a formula for the effective Hamiltonian $\overline H$. We use this formula to provide a complete list of the heights at which the graph of $\overline H$ has a flat piece. We illustrate our results by analyzing basic classes of examples, highlight some corollaries that clarify the dependence of $\overline H$ on $G$, $\beta$ and the law of $V(\cdot,\omega)$, and discuss a generalization to even-symmetric triple-well Hamiltonians.
Human review
How the literature uses it
1 arXiv paper cite this in 1 sentence
Papers citing it
Cites it as background 1 paper
-
When $H$ is not assumed to be convex (non-convex case), only very specific frameworks were treated: for level-set convex Hamiltonians [ref] , when the law of $H$ is invariant by rotation [ref] , in one-dimension [ref] , when the law of $H$ satisfies a finite range condition and $H$ is positively homogenous or star-shaped [ref] .
Percolation games · Stochastic homogenization · 0.91
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.