Startseite Mathematik Convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function
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Convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function

  • Qinbo Chen ORCID logo EMAIL logo
Veröffentlicht/Copyright: 20. Januar 2021
Advances in Calculus of Variations
Aus der Zeitschrift Advances in Calculus of Variations Band 16 Heft 1

Abstract

Motivated by the vanishing contact problem, we study in the present paper the convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function. Let H ⁢ ( x , p , u ) be a continuous Hamiltonian which is strictly increasing in u, and is convex and coercive in p. For each parameter λ > 0 , we denote by u λ the unique viscosity solution of the Hamilton–Jacobi equation

H ⁢ ( x , D ⁢ u ⁢ ( x ) , λ ⁢ u ⁢ ( x ) ) = c .

Under quite general assumptions, we prove that u λ converges uniformly, as λ tends to zero, to a specific solution of the critical Hamilton–Jacobi equation H ⁢ ( x , D ⁢ u ⁢ ( x ) , 0 ) = c . We also characterize the limit solution in terms of Peierls barrier and Mather measures.

Keywords: Nonlinear discounted systems; Hamilton–Jacobi equations; asymptotic behavior of solutions; weak KAM theory; Aubry–Mather theory
MSC 2010: 35B40; 49L25; 37J50

Communicated by Hitoshi Ishii

Funding source: H2020 European Research Council

Award Identifier / Grant number: 677793

Funding statement: This research is funded by the ERC Project 677793 StableChaoticPlanetM.

A Appendix

Here, we provide an equivalent description for assumption (H2) provided that (H1) and (H3) hold.

Proposition A.1.

If H ∈ C ⁢ ( T * ⁢ M × R ) satisfies the convexity assumption (H1) and the monotonicity assumption (H3), then the following properties are equivalent:

  1. There is a constant 𝐫 𝟎 > 0 such that lim | p | x → ∞ ⁡ H ⁢ ( x , p , - 𝐫 𝟎 ) = + ∞ , uniformly in x ∈ M .

  2. lim | p | x → ∞ ⁡ H ⁢ ( x , p , 0 ) = + ∞ , uniformly in x ∈ M .

Proof.

By the monotonicity assumption (H3), the implication (1)  ⇒  (2) is obvious.

Now, we show that (2) implies (1). To this end, only assumption (H1) will be used. Indeed, by the coercivity of H ⁢ ( x , p , 0 ) and the compactness of M, there is a constant r > 0 such that

H ⁢ ( x , p , 0 ) > max x ∈ M ⁡ H ⁢ ( x , 0 , 0 ) + 2   for all  ⁢ ( x , p ) ∈ ∂ ⁡ B r ,

where ∂ ⁡ B r denotes the compact set { ( x , p ) ∈ T * ⁢ M : x ∈ M , | p | x = r } . By continuity and compactness, there exists a small constant 𝐫 𝟎 > 0 such that

(A.1) H ⁢ ( x , p , - 𝐫 𝟎 ) > max x ∈ M ⁡ H ⁢ ( x , 0 , - 𝐫 𝟎 ) + 1   for all  ⁢ ( x , p ) ∈ ∂ ⁡ B r .

Next, for each ( x , p ) with | p | x > r , we pick a point z x , p := r | p | x ⁢ p on ∂ ⁡ B r . Then the convexity assumption (H1) implies

(A.2) H ⁢ ( x , z x , p , - 𝐫 𝟎 ) ⩽ ( 1 - r | p | x ) ⁢ H ⁢ ( x , 0 , - 𝐫 𝟎 ) + r | p | x ⁢ H ⁢ ( x , p , - 𝐫 𝟎 )   for all  ⁢ | p | x > r .

Since z x , p ∈ ∂ ⁡ B r , we infer from (A.1)–(A.2) that

H ⁢ ( x , p , - 𝐫 𝟎 ) ⩾ H ⁢ ( x , z x , p , - 𝐫 𝟎 ) - H ⁢ ( x , 0 , - 𝐫 𝟎 ) r | p | x + H ⁢ ( x , 0 , - 𝐫 𝟎 ) ⩾ 1 r ⁢ | p | x + H ⁢ ( x , 0 , - 𝐫 𝟎 )

for all | p | x > r . This implies

lim | p | x → ∞ ⁡ H ⁢ ( x , p , - 𝐫 𝟎 ) = + ∞ , uniformly in  x ∈ M . ∎

Acknowledgements

This work was initiated while I was visiting Georgia Tech (December 2018–March 2019), whose warm hospitality is gratefully acknowledged. It was completed while I was working at the University of Padova. I am particularly grateful to Albert Fathi for many stimulating discussions during my stay at Georgia Tech, and for his careful reading of the manuscript and valuable comments. I wish to thank Wei Cheng, Jun Yan and Jianlu Zhang for some helpful discussions. I also would like to thank Gabriella Pinzari for her support.

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Received: 2020-08-28
Revised: 2020-11-25
Accepted: 2020-12-21
Published Online: 2021-01-20
Published in Print: 2023-01-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. The existence of minimizers for an isoperimetric problem with Wasserstein penalty term in unbounded domains
  3. Compactness of 𝑀-uniform domains and optimal thermal insulation problems
  4. Convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function
  5. Pansu–Wulff shapes in ℍ1
  6. Integrability of the sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group
  7. p-harmonic functions by way of intrinsic mean value properties
  8. The sharp quantitative isocapacitary inequality (the case of p-capacity)
  9. A finer singular limit of a single-well Modica–Mortola functional and its applications to the Kobayashi–Warren–Carter energy
  10. Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects
  11. Remarks on the vanishing discount problem for infinite systems of Hamilton–Jacobi–Bellman equations
  12. Bilateral estimates of solutions to quasilinear elliptic equations with sub-natural growth terms
  13. Rectifiability of entropy defect measures in a micromagnetics model
https://doi.org/10.1515/acv-2020-0089
Schlagwörter für diesen Artikel
Nonlinear discounted systems; Hamilton–Jacobi equations; asymptotic behavior of solutions; weak KAM theory; Aubry–Mather theory

Artikel in diesem Heft

  1. Frontmatter
  2. The existence of minimizers for an isoperimetric problem with Wasserstein penalty term in unbounded domains
  3. Compactness of 𝑀-uniform domains and optimal thermal insulation problems
  4. Convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function
  5. Pansu–Wulff shapes in ℍ1
  6. Integrability of the sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group
  7. p-harmonic functions by way of intrinsic mean value properties
  8. The sharp quantitative isocapacitary inequality (the case of p-capacity)
  9. A finer singular limit of a single-well Modica–Mortola functional and its applications to the Kobayashi–Warren–Carter energy
  10. Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects
  11. Remarks on the vanishing discount problem for infinite systems of Hamilton–Jacobi–Bellman equations
  12. Bilateral estimates of solutions to quasilinear elliptic equations with sub-natural growth terms
  13. Rectifiability of entropy defect measures in a micromagnetics model
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