Limit of solutions for semilinear Hamilton–Jacobi equations with degenerate viscosity

Jianlu Zhang

doi:10.1515/acv-2022-0108

Article

Limit of solutions for semilinear Hamilton–Jacobi equations with degenerate viscosity

Jianlu Zhang

Published/Copyright: April 27, 2023

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From the journal Advances in Calculus of Variations Volume 17 Issue 4

Abstract

In the paper we prove the convergence of viscosity solutions u λ as λ → 0 + for the parametrized degenerate viscous Hamilton–Jacobi equation

H ⁢ ( x , d x ⁢ u , λ ⁢ u ) = α ⁢ ( x ) ⁢ Δ ⁢ u , α ⁢ ( x ) ≥ 0 , x ∈ 𝕋 n

under suitable convex and monotonic conditions on H : T * ⁢ M × ℝ → ℝ . Such a limit can be characterized in terms of stochastic Mather measures associated with the critical equation

H ⁢ ( x , d x ⁢ u , 0 ) = α ⁢ ( x ) ⁢ Δ ⁢ u .

Keywords: Degenerate viscous Hamilton–Jacobi equations; stochastic Mather measures; ergodic constant; nonlinear adjoint methods

MSC 2020: 35B40; 37J50; 49L25

Communicated by Hitoshi Ishii

Funding source: National Key Research and Development Program of China

Award Identifier / Grant number: 2022YFA1007500

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12231010

Award Identifier / Grant number: 11901560

Funding statement: This work is supported by the National Key R&D Program of China (No. 2022YFA1007500) and the National Natural Science Foundation of China (No. 12231010, No. 11901560).

A Adjoint equation

For λ > 0 , suppose u : 𝕋 n → ℝ is the viscosity solution of

λ ⁢ β ⁢ ( x ) ⁢ u + 〈 V ⁢ ( x ) , d x ⁢ u 〉 = ( α ⁢ ( x ) + η 2 ) ⁢ Δ ⁢ u + λ ⁢ f ⁢ ( x ) , x ∈ 𝕋 n .

Then u ≥ 0 as long as C ⁢ ( 𝕋 n , ℝ ) ∋ f , β ≥ 0 (due to the comparison principle). As its adjoint equation, there holds

λ ⁢ β ⁢ θ - div ⁡ ( V ⁢ ( x ) ⁢ θ ) = Δ ⁢ ( ( α + η 2 ) ⁢ θ ) + λ ⁢ δ x 0

for some x 0 ∈ 𝕋 n . As we can see,

∫ 𝕋 n λ ⁢ f ⁢ θ ⁢ 𝑑 x = ∫ 𝕋 n ( λ ⁢ β ⁢ ( x ) ⁢ u + 〈 V ⁢ ( x ) , d x ⁢ u 〉 - ( α ⁢ ( x ) + η 2 ) ⁢ Δ ⁢ u ) ⁢ θ ⁢ ( x ) ⁢ 𝑑 x

= ∫ 𝕋 n ( λ ⁢ β ⁢ θ - div ⁡ ( V ⁢ ( x ) ⁢ θ ) - Δ ⁢ ( ( α + η 2 ) ⁢ θ ) ) ⁢ u ⁢ 𝑑 x

= ∫ 𝕋 n λ ⁢ β ⁢ δ x 0 ⁢ u ⁢ 𝑑 x = λ ⁢ β ⁢ ( x 0 ) ⁢ u ⁢ ( x 0 ) ≥ 0

for any f , β ≥ 0 . Consequently, θ ≥ 0 on 𝕋 n and

∫ 𝕋 n β ⁢ θ ⁢ 𝑑 x = ∫ 𝕋 n δ x 0 ⁢ 𝑑 x = 1 .

Furthermore, if β > 0 , then

1 max 𝕋 n ⁡ β ≤ ∫ 𝕋 n θ ⁢ 𝑑 x ≤ 1 min 𝕋 n ⁡ β .

Applying previous procedure to ((AJ${{}_{e}}$)) by taking

β ⁢ ( x ) = ∂ u ⁡ H ⁢ ( x , d x ⁢ u λ η , 0 ) , V ⁢ ( x ) = ∂ p ⁡ H ⁢ ( x , d x ⁢ u λ η , 0 ) ,

we instantly get (2.12).

Proof of Lemma 2.3.

Differentiating both sides of ((HJ${{}_{e}^{\lambda,\eta}}$)) by x, we get

∂ x ⁡ H ⁢ ( x , d x ⁢ u λ η , λ ⁢ u λ η ) + ∂ p ⁡ H ⁢ ( x , d x ⁢ u λ η , λ ⁢ u λ η ) ⋅ D 2 ⁢ u λ η ⁢ ( x ) + λ ⁢ ∂ u ⁡ H ⁢ ( x , d x ⁢ u λ η , λ ⁢ u λ η ) ⁢ d x ⁢ u λ η ⁢ ( x )

= ( α + η 2 ) ⁢ Δ ⁢ ( d x ⁢ u λ η ) + d x ⁢ α ⁢ ( x ) ⁢ Δ ⁢ u λ η .

Multiplying previous equality by d x ⁢ u λ η , then we get

(A.1) 〈 ∂ x ⁡ H ⁢ ( x , d x ⁢ u λ η , λ ⁢ u λ η ) , d x ⁢ u λ η 〉 + 〈 ∂ p ⁡ H , d x ⁢ ψ ⁢ ( x ) 〉 + 2 ⁢ λ ⁢ ∂ u ⁡ H ⁢ ( x , d x ⁢ u λ η , λ ⁢ u λ η ) ⁢ ψ ⁢ ( x ) = ( α + η 2 ) ⁢ ( Δ ⁢ ψ - | D 2 ⁢ u λ η | 2 ) + 〈 d x ⁢ α , d x ⁢ u λ η 〉 ⁢ Δ ⁢ u λ η ,

where ψ ⁢ ( x ) := | d x ⁢ u λ η ⁢ ( x ) | 2 2 . Since

(A.2) | u λ η | + | d x ⁢ u λ η | L ∞ ≤ C 4 for all ⁢ λ , η ∈ ( 0 , 1 ]

for some constant C 4 > 0 , there exists a constant C 5 > 0 such that

| 〈 ∂ x ⁡ H ⁢ ( x , d x ⁢ u λ η , λ ⁢ u λ η ) , d x ⁢ u λ η 〉 | ≤ C 5 .

On the other side,

| 〈 d x ⁢ α , d x ⁢ u λ η 〉 ⁢ Δ ⁢ u λ η | ≤ | d x ⁢ α | ⋅ | d x ⁢ u λ η | ⋅ | Δ ⁢ u λ η |

≤ C 4 ⁢ | d x ⁢ α | ⋅ | Δ ⁢ u λ η | = C 4 δ ⋅ δ ⁢ | d x ⁢ α | ⋅ | Δ ⁢ u λ η |

≤ 1 2 ⁢ ( C 4 2 δ 2 + δ 2 ⁢ | d x ⁢ α | 2 ⋅ | Δ ⁢ u λ η | 2 )

≤ 1 2 ⁢ ( C 4 2 δ 2 + δ 2 ⁢ C 6 ⁢ α ⁢ ( x ) ⁢ | D 2 ⁢ u λ η | 2 )

for some constant C 6 > 0 , since α ≥ 0 , we have α ∈ Lip ⁢ ( 𝕋 n , ℝ ) in view of [26, Theorem 5.2.3]. Furthermore, the previous inequality leads to

| 〈 d x ⁢ α , d x ⁢ u λ η 〉 ⁢ Δ ⁢ u λ η | ≤ C 4 2 ⁢ C 6 2 + 1 2 ⁢ α ⁢ ( x ) ⁢ | D 2 ⁢ u λ η | 2

by taking δ 2 = 1 C 6 . Accordingly, (A.1) implies

(A.3) 〈 ∂ p ⁡ H , d x ⁢ ψ ⁢ ( x ) 〉 + 2 ⁢ λ ⁢ ∂ u ⁡ H ⁢ ( x , d x ⁢ u λ η , λ ⁢ u λ η ) ⁢ ψ ⁢ ( x ) - ( α + η 2 ) ⁢ Δ ⁢ ψ + α + η 2 2 ⁢ | D 2 ⁢ u λ η | 2 ≤ C 7 := C 5 + C 4 2 ⁢ C 6 2 .

Suppose θ λ η ⁢ ( x ) is now the solution of the following adjoint equation:

2 ⁢ λ ⁢ ∂ u ⁡ H ⁢ ( x , d x ⁢ u λ η , λ ⁢ u λ η ) ⁢ θ λ η - div ⁡ ( ∂ p ⁡ H ⁢ ( x , d x ⁢ u λ η , λ ⁢ u λ η ) ⁢ θ λ η ) = Δ ⁢ ( ( α + η 2 ) ⁢ θ λ η ) + 2 ⁢ λ ⁢ δ x 0 .

Integrating both sides of (A.3) by θ λ η ⁢ ( x ) ⁢ d ⁢ x , we get

∫ 𝕋 n ( α + η 2 ) ⁢ | D 2 ⁢ u λ η | 2 ⁢ θ λ η ⁢ ( x ) ⁢ 𝑑 x ≤ 2 ⁢ λ ⁢ | ψ ⁢ ( x 0 ) | + 2 ⁢ C 4 ⁢ ∫ 𝕋 n θ λ η ⁢ 𝑑 x

≤ λ ⁢ C 1 2 + 2 ⁢ C 7 min x ∈ 𝕋 n ⁢ ∂ u ⁡ H ⁢ ( x , d x ⁢ u λ η , λ ⁢ u λ η ) ,

which further indicates

∫ 𝕋 n | D 2 ⁢ u λ η | 2 ⁢ θ λ η ⁢ ( x ) ⁢ 𝑑 x ≤ C 8 η 2 for all ⁢ η ∈ ( 0 , 1 ]

for some constant C 8 > 0 due to (A.2).

Secondly, since u λ η ⁢ ( x ) is smooth of η ∈ ( 0 , 1 ] , we can take the derivative of ((HJ${{}_{e}^{\lambda,\eta}}$)) with respect to η such that

〈 ∂ p ⁡ H ⁢ ( x , d x ⁢ u λ η , λ ⁢ u λ η ) , ∂ x ⁢ η 2 ⁡ u λ η 〉 + λ ⁢ ∂ u ⁡ H ⋅ ∂ η ⁡ u λ η = 2 ⁢ η ⁢ Δ ⁢ u λ η + ( α + η 2 ) ⁢ Δ ⁢ ( ∂ η ⁡ u λ η ) .

Consequently,

∫ 𝕋 n 2 ⁢ λ ⁢ ∂ u ⁡ H ⁢ ( x , d x ⁢ u λ η , λ ⁢ u λ η ) ⁢ ∂ η ⁡ u λ η ⁢ θ λ η ⁢ d ⁢ x

= ∫ 𝕋 n λ ⁢ ∂ u ⁡ H ⁢ ( x , d x ⁢ u λ η , λ ⁢ u λ η ) ⁢ ∂ η ⁡ u λ η ⁢ θ λ η ⁢ d ⁢ x + ∫ 𝕋 n ( 2 ⁢ η ⁢ Δ ⁢ u λ η + ( α + η 2 ) ⁢ Δ ⁢ ( ∂ η ⁡ u λ η ) - 〈 ∂ p ⁡ H , ∂ x ⁢ η 2 ⁡ u λ η 〉 ) ⁢ θ λ η ⁢ 𝑑 x

which can be further transferred into

2 ⁢ η ⁢ ∫ 𝕋 n Δ ⁢ u λ η ⁢ θ λ η ⁢ 𝑑 x + λ ⁢ ∫ 𝕋 n ∂ u ⁡ H ⁢ ( x , d x ⁢ u λ η , λ ⁢ u λ η ) ⁢ ∂ η ⁡ u λ η ⁢ θ λ η ⁢ d ⁢ x = 2 ⁢ λ ⁢ ∫ 𝕋 n δ x 0 ⁢ ∂ η ⁡ u λ η ⁢ d ⁢ x = 2 ⁢ λ ⁢ ∂ η ⁡ u λ η ⁢ ( x 0 ) .

Since x 0 ∈ 𝕋 n is freely chosen, we can make

| ∂ η ⁡ u λ η ⁢ ( x 0 ) | = max x ∈ 𝕋 n ⁡ | ∂ η ⁡ u λ η ⁢ ( x ) | .

If so, we get

2 ⁢ η ⁢ | ∫ 𝕋 n Δ ⁢ u λ η ⁢ θ λ η ⁢ 𝑑 x | = | 2 ⁢ λ ⁢ ∂ η ⁡ u λ η ⁢ ( x 0 ) - λ ⁢ ∫ 𝕋 n ∂ u ⁡ H ⁢ ( x , d x ⁢ u λ η , λ ⁢ u λ η ) ⁢ ∂ η ⁡ u λ η ⁢ θ λ η ⁢ d ⁢ x |

≥ 2 ⁢ λ ⁢ | ∂ η ⁡ u λ η ⁢ ( x 0 ) | - λ ⁢ ∫ 𝕋 n max x ∈ 𝕋 n ⁡ | ∂ η ⁡ u λ η ⁢ ( x ) | ⁢ ∂ u ⁡ H ⋅ θ λ η ⁢ d ⁢ x

= λ ⁢ | ∂ η ⁡ u λ η ⁢ ( x 0 ) | .

On the other side,

2 ⁢ η ⁢ | ∫ 𝕋 n Δ ⁢ u λ η ⁢ θ λ η ⁢ 𝑑 x | ≤ 2 ⁢ η ⁢ ∫ 𝕋 n | Δ ⁢ u λ η | ⁢ θ λ η ⁢ 𝑑 x

≤ 2 ⁢ η ⁢ ∫ 𝕋 n | D 2 ⁢ u λ η | 2 ⁢ θ λ η ⁢ 𝑑 x ⋅ ∫ 𝕋 n θ λ η ⁢ 𝑑 x

= 2 ⁢ η ⁢ C 8 η ⋅ 1 min x ∈ 𝕋 n ⁢ ∂ u ⁡ H ⁢ ( x , d x ⁢ u λ η , λ ⁢ u λ η )

due to the Hölder’s inequality. Combining these two conclusions, we get

| ∂ η ⁡ u λ η ⁢ ( x ) | ≤ 2 ⁢ C 8 λ ⁢ min x ∈ 𝕋 n ⁢ ∂ u ⁡ H ⁢ ( x , d x ⁢ u λ η , λ ⁢ u λ η ) .

Then integrating both sides with respect to η ∈ ( 0 , 1 ] , we get

| u λ η - u λ | L ∞ ≤ C ′ ⁢ η λ for all ⁢ λ , η ∈ ( 0 , 1 ]

for some constant C ′ > 0 . ∎

Data availability statement.

The datasets analyzed during the current study are available from the corresponding author on reasonable request.

Acknowledgements

The author would like to thank the Laboratory of Mathematics for Nonlinear Science, Fudan University (LNMS) for the hospitality, where this research was initiated during the author’s visit in April 2021. The author also would like to thank the anonymous referee for helpful suggestions to improve the presentation.

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Received: 2022-12-17

Accepted: 2023-03-22

Published Online: 2023-04-27

Published in Print: 2024-10-01

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Articles in the same Issue

https://doi.org/10.1515/acv-2022-0108

Keywords for this article

Degenerate viscous Hamilton–Jacobi equations; stochastic Mather measures; ergodic constant; nonlinear adjoint methods