Min-max formulas and other properties of certain classes of nonconvex effective Hamiltonians
aa r X i v : . [ m a t h . A P ] J a n MIN-MAX FORMULAS AND OTHER PROPERTIES OFCERTAIN CLASSES OF NONCONVEX EFFECTIVEHAMILTONIANS
JIANLIANG QIAN, HUNG V. TRAN, AND YIFENG YU
Abstract.
This paper is the first attempt to systematically study properties ofthe effective Hamiltonian H arising in the periodic homogenization of some coer-cive but nonconvex Hamilton-Jacobi equations. Firstly, we introduce a new androbust decomposition method to obtain min-max formulas for a class of nonconvex H . Secondly, we analytically and numerically investigate other related interestingphenomena, such as “quasi-convexification” and breakdown of symmetry, of H from other typical nonconvex Hamiltonians. Finally, in the appendix, we showthat our new method and those a priori formulas from the periodic setting can beused to obtain stochastic homogenization for same class of nonconvex Hamilton-Jacobi equations. Some conjectures and problems are also proposed. Introduction
Overview.
Let us describe the periodic homogenization theory of Hamilton-Jacobi equations. For each ε >
0, let u ε ∈ C ( R n × [0 , ∞ )) be the viscosity solutionto ( u εt + H ( Du ε ) − V (cid:0) xε (cid:1) = 0 in R n × (0 , ∞ ) ,u ε ( x,
0) = g ( x ) on R n . (1.1)Here, the Hamiltonian H ( p ) − V ( x ) is of separable form with H ∈ C ( R n ), whichis coercive (i.e., lim | p |→∞ H ( p ) = + ∞ ), and V ∈ C ( R n ), which is Z n -periodic. Theinitial data g ∈ BUC ( R n ), the set of bounded, uniformly continuous functions on R n .It was proven by Lions, Papanicolaou and Varadhan [29] that u ε converges to u locally uniformly on R n × [0 , ∞ ) as ε →
0, and u solves the effective equation ( u t + H ( Du ) = 0 in R n × (0 , ∞ ) ,u ( x,
0) = g ( x ) on R n . (1.2) Mathematics Subject Classification.
Key words and phrases.
Cell problems; nonconvex Hamilton-Jacobi equations; effective Hamil-tonians; evenness; min-max formulas; quasi-convexification; periodic homogenization; stochastichomogenization; viscosity solutions.The work of JQ is partially supported by NSF grants 1522249 and 1614566, the work of HT ispartially supported by NSF grant DMS-1615944, the work of YY is partially supported by NSFCAREER award
The effective Hamiltonian H ∈ C ( R n ) is determined in a nonlinear way by H and V through the cell problems as following. For each p ∈ R n , it was shown in [29]that there exists a unique constant H ( p ) ∈ R such that the following cell problemhas a continuous viscosity solution H ( p + Dv ) − V ( x ) = H ( p ) in T n , (1.3)where T n is the n -dimensional flat torus R n / Z n .Although there is a vast literature on homogenization of Hamilton-Jacobi equa-tions in different settings after [29], characterizing the shape of H remains largelyopen even in basic situations. Let us summarize quickly what is known in the litera-ture about H . It is not hard to see that H is coercive thanks to the coercivity of H .If one assumes furthermore that H is convex, then H is also convex and the graphof H can contain some flat parts (i.e., (cid:8) H = min H (cid:9) has interior points). See [29]and the works of Concordel [12, 13]. Furthermore, in this convex setting, we havethe following representation formula, thanks to the results of Contreras, Iturriaga,Paternain and Paternain [14], and Gomes [23], H ( p ) = inf φ ∈ C ( T n ) max x ∈ T n ( H ( p + Dφ ( x )) − V ( x )) . (1.4)Note that the above representative formula still holds if H is quasiconvex (level-setconvex), in which case H is also quasiconvex. More interestingly, in case n = 2, H ( p ) = | p | and V ∈ C ∞ ( T ), a deep result of Bangert [9] says that the level curve (cid:8) H = c (cid:9) for every c > − min V must contain line segments (i.e., not strictly convex)unless V is a constant function. Bangert’s result relies on detailed information aboutthe structure of Aubry-Mather sets in two dimension ([8]). See also Jing, Tran, Yu[26] for discussion regarding locations of line segments of the level curves for Ma˜n´etype Hamiltonians.The first numerical computation of effective Hamiltonians is due to Qian [38]based on the so called big-T method , that is, H ( p ) = − lim t →∞ w ( x,t ) t , where w ( x, t )is the unique viscosity solution to ( w t + H ( Dw ) − V ( x ) = 0 in R n × (0 , ∞ ), w ( x,
0) = p · x on R n . For other numerical schemes, we refer to Gomes, Oberman [25], Falcone, Rorro [19],Achdou, Camilli, Capuzzo-Dolcetta [1], Oberman, Takei, Vladimirsky [37], Luo, Yu,Zhao [33] and the references therein.It is worth mentioning that cell problem (1.3) and representation formula (1.4)appear also in weak KAM theory (see E [17], Evans, Gomes [18], Fathi [20] for theconvex case, and Cagnetti, Gomes, Tran [11] for the nonconvex case). In fact, acentral goal of the weak KAM theory is to find information of underlying dynamicalsystem encoded in the effective Hamiltonian.In the case where H is nonconvex, to identify the shape of H is highly non-trivial even in the one dimensional space. This was settled only very recently by IN-MAX FORMULAS AND OTHER PROPERTIES OF H Armstrong, Tran, Yu [7], and Gao [22]. One fundamental feature obtained is the“ quasi-convexification ” phenomenon, that is, the effective Hamiltonian H becomesquasiconvex (level-set convex) when the oscillation of V is large enough. See Section3 for more precise statements. In multi-dimensional spaces, Armstrong, Tran, Yu [6]obtained a qualitative shape of H for a representative case where H ( p ) = ( | p | − .Other than [6], very little is known about finer properties of nonconvex H in multi-dimensional spaces, partly due to the extreme complexity of dynamics associatedwith nonconvex Hamiltonians. Furthermore, as far as the authors know, there is nonumerical study of H in this case.Let us also call attention to an extrinsic way to study H via inverse problems.See Luo, Tran, Yu [32].1.2. Main results.
Reducing a complex quantity to relatively simpler objects is avery natural and common idea in mathematics. For a class of nonconvex Hamil-tonians H , we introduce a new decomposition method to obtain min-max typerepresentation formulas for H . These formulas consist of effective Hamiltonians ofquasiconvex Hamiltonians which are presumably less challenging to analyze. Themost general statement is given by inductive formulas (Theorem 2.6). Two specific(but important) cases of Theorem 2.6 are provided in Theorem 2.1 and Lemma2.5. One immediate corollary is the evenness of H associated with a certain classof radially symmetric Hamiltonians, which is otherwise not obvious at all. Giventhe vast variety of nonconvex functions, our surgical approach is only a preliminarystep toward understanding the shape of nonconvex H . In Section 2.4, we presentsome natural obstacles to decomposing a nonconvex H . In particular, there is aconnection between “non-decomposability” and loss of evenness.As another interesting application, the method and the representation formulasare robust enough that we are also able to prove stochastic homogenization for thesame class of nonconvex H in the appendix. For instance, Theorem 4.1 includesthe result in [6] as a special case with a much shorter proof. The detailed discus-sion on this (including a brief overview of stochastic homogenization) is left to theappendix. We would like to point out that a priori identification of shape of H iscurrently the only available way to tackle homogenization of nonconvex Hamilton-Jacobi equations in general stationary ergodic setting.In Section 3, we provide various numerical computations of H in multi-dimensionalspaces for general radially symmetric Hamiltonians and a double-well type Hamil-tonian. These provide insights on how the changes of potential energy V affect thechanges in shape of effective Hamiltonian H . The important “quasi-convexification”phenomenon is observed in multi-dimensional cases as well. Nevertheless, verifyingit rigorously seems to be quite challenging. Interesting connections between de-composition, loss of evenness and quasi-convexification are demonstrated in Section2.4 and Remark 4. Several open problems are provided based on the numericalevidences we have in this section. JIANLIANG QIAN, HUNG V. TRAN, AND YIFENG YU Min-max formulas
Basic case.
The setting is this. Let H = H ( p ) : R n → R be a continuous,coercive Hamiltonian such that(H1) min R n H = 0 and there exists a bounded domain U ⊂ R n such that { H = 0 } = ∂U. (H2) H ( p ) = H ( − p ) for all p ∈ R n .(H3) There exist H , H : R n → R such that H , H are continuous and H = max { H , H } . Here, H is coercive, quasiconvex, even ( H ( p ) = H ( − p ) for all p ∈ R n ), H = H in R n \ U and H < U . The function H is quasiconcave, H = H in U , H < R n \ U and lim | p |→∞ H ( p ) = −∞ .It is easy to see that any H satisfying (H1)–(H3) can be written as H ( p ) = | F ( p ) | for some even, coercive quasiconvex function F such that min R n F <
0. Below isthe first decomposition result.
Theorem 2.1.
Let H ∈ C ( R n ) be a Hamiltonian satisfying (H1)–(H3) . Let V ∈ C ( T n ) be a potential energy with min T n V = 0 .Assume that H is the effective Hamiltonian corresponding to H ( p ) − V ( x ) . As-sume also that H i is the effective Hamiltonian corresponding to H i ( p ) − V ( x ) for i = 1 , . Then H = max { H , H , } . In particular, H is even. We would like to point out that the evenness of H will be used later and is notobvious at all although H is even. See the discussion in Subsection 2.4 for thissubtle issue. Proof.
We proceed in few steps.
Step 1.
It is straightforward that 0 ≤ H ( p ) ≤ H ( p ) for all p ∈ R n . In particular, H ( p ) = 0 for all p ∈ ∂U. (2.1)Besides, as H i ≤ H , we get H i ≤ H . Therefore, H ≥ max (cid:8) H , H , (cid:9) . (2.2)It remains to prove the reverse inequality of (2.2) in order to get the conclusion. Step 2.
Fix p ∈ R n . Assume now that H ( p ) ≥ max { H ( p ) , } . We will show that H ( p ) ≥ H ( p ). IN-MAX FORMULAS AND OTHER PROPERTIES OF H Since H is quasiconvex and even, we use the inf-max representation formula for H (see [4, 16, 36]) to get that H ( p ) = inf φ ∈ C ( T n ) max x ∈ T n ( H ( p + Dφ ( x )) − V ( x ))= inf φ ∈ C ( T n ) max x ∈ T n ( H ( − p − Dφ ( x )) − V ( x ))= inf ψ ∈ C ( T n ) max x ∈ T n ( H ( − p + Dψ ( x )) − V ( x )) = H ( − p ) . Thus, H is even. Let v ( x, − p ) be a solution to the cell problem H ( − p + Dv ( x, − p )) − V ( x ) = H ( − p ) = H ( p ) in T n . (2.3)Let w ( x ) = − v ( x, − p ). For any x ∈ T n and q ∈ D + w ( x ), we have − q ∈ D − v ( x, − p )and hence, in light of (2.3) and the quasiconvexity of H (see [10]), H ( p ) = H ( − p − q ) − V ( x ) = H ( p + q ) − V ( x ) . We thus get H ( p + q ) = H ( p ) + V ( x ) ≥
0, and therefore, H ( p + q ) = H ( p + q ).This yields that w is a viscosity subsolution to H ( p + Dw ) − V ( x ) = H ( p ) in T n . Hence, H ( p ) ≤ H ( p ). Step 3.
Assume now that H ( p ) ≥ max { H ( p ) , } . By using similar argumentsas those in the previous step (except that we use v ( x, p ) instead of v ( x, − p ) due tothe quasiconcavity of H ), we can show that H ( p ) ≥ H ( p ). Step 4.
Assume that max (cid:8) H ( p ) , H ( p ) (cid:9) <
0. We now show that H ( p ) = 0 inthis case. Thanks to (2.1) in Step 1, we may assume that p / ∈ ∂U .For σ ∈ [0 ,
1] and i = 1 ,
2, let H σ , H σi be the effective Hamiltonians correspondingto H ( p ) − σV ( x ), H i ( p ) − σV ( x ), respectively. It is clear that0 ≤ H = H ≤ H σ for all σ ∈ [0 , . (2.4)By repeating Steps 2 and 3 above, we getFor p ∈ R n and σ ∈ [0 , (cid:8) H σ ( p ) , H σ ( p ) (cid:9) = 0, then H σ ( p ) = 0. (2.5)We only consider the case p / ∈ U here. The case p ∈ U is analogous. Notice that H ( p ) = H ( p ) = H ( p ) > H ( p ) = H ( p ) < . By the continuity of σ H σ ( p ), there exists s ∈ (0 ,
1) such that H s ( p ) = 0. Notefurthermore that, as p / ∈ U , H s ( p ) ≤ H ( p ) <
0. These, together with (2.4) and(2.5), yield the desired result. (cid:3)
Remark 1.
We emphasize that Step 4 in the above proof is important. It playsthe role of a “patching” step, which helps glue H and H together.It is worth noting that the representation formula in Theorem 2.1 still holds incase H is not even in U . In fact, we do not use this point at all in the proof. Weonly need it to deduce that H is even. JIANLIANG QIAN, HUNG V. TRAN, AND YIFENG YU
Assumptions (H1)–(H3) are general and a bit complicated. A simple situationwhere (H1)–(H3) hold is a radially symmetric case where H ( p ) = ψ ( | p | ), and ψ ∈ C ([0 , ∞ ) , R ) satisfying ( ψ (0) > , ψ (1) = 0 , lim r →∞ ψ ( r ) = + ∞ ,ψ is strictly decreasing in (0 ,
1) and is strictly increasing in (1 , ∞ ) . (2.6)Let ψ , ψ ∈ C ([0 , ∞ ) , R ) be such that ( ψ = ψ on [1 , ∞ ) , and ψ is strictly increasing on [0 , ,ψ = ψ on [0 , , ψ is strictly decreasing on [1 , ∞ ) , and lim r →∞ ψ ( r ) = −∞ . (2.7)See Figure 1 below. Set H i ( p ) = ψ i ( | p | ) for p ∈ R n , and for i = 1 ,
2. It is clearthat (H1)–(H3) hold provided that (2.6)–(2.7) hold. An immediate consequence of r ψ ψ ψ Figure 1.
Graphs of ψ, ψ , ψ Theorem 2.1 is
Corollary 2.2.
Let H ( p ) = ψ ( | p | ) , H i ( p ) = ψ i ( | p | ) for i = 1 , and p ∈ R n , where ψ, ψ , ψ satisfy (2.6) – (2.7) . Let V ∈ C ( T n ) be a potential energy with min T n V = 0 .Assume that H is the effective Hamiltonian corresponding to H ( p ) − V ( x ) . As-sume also that H i is the effective Hamiltonian corresponding to H i ( p ) − V ( x ) for i = 1 , . Then H = max (cid:8) H , H , (cid:9) . Remark 2.
A special case of Corollary 2.2 is when H ( p ) = ψ ( | p | ) = (cid:0) | p | − (cid:1) for p ∈ R n , which was studied first by Armstrong, Tran and Yu [6]. The method here is muchsimpler and more robust than that in [6]. IN-MAX FORMULAS AND OTHER PROPERTIES OF H By using Corollary 2.2 and approximation, we get another representation formulafor H which will be used later. Corollary 2.3.
Assume that (2.6) – (2.7) hold. Set ˜ ψ ( r ) = max { ψ , } = ( for ≤ r ≤ ,ψ ( r ) for r > . Let H ( p ) = ψ ( | p | ) , ˜ H ( p ) = ˜ ψ ( p | ) and H ( p ) = ψ ( | p | ) for p ∈ R n . Let V ∈ C ( T n ) be a potential energy with min T n V = 0 .Assume that H, ˜ H , H are the effective Hamiltonian corresponding to H ( p ) − V ( x ) , ˜ H ( p ) − V ( x ) , H ( p ) − V ( x ) , respectively. Then H = max n ˜ H , H o . See Figure 2 for the graphs of ψ, ˜ ψ , ψ . r ψ ψ ψ Figure 2.
Graphs of ψ, ˜ ψ , ψ When the oscillation of V is large enough, we have furthermore the followingresult. Corollary 2.4.
Let H ∈ C ( R n ) be a coercive Hamiltonian satisfying (H1)–(H3) ,except that we do not require H to be quasiconcave. Assume that osc T n V = max T n V − min T n V ≥ max U H = max R n H . Then H = max n H , − min T n V o . In particular, H is quasiconvex in this situation. It is worth noting that the result of Corollary 2.4 is interesting in the sense thatwe do not require any structure of H in U except that H >
JIANLIANG QIAN, HUNG V. TRAN, AND YIFENG YU
Proof.
Without loss of generality, we assume that min T n V = 0. Choose a quasicon-cave function H +2 ∈ C ( R n ) such that { H = 0 } = { H +2 = 0 } = ∂U,H ≤ H +2 in U, and max U H = max R n H +2 , lim | p |→∞ H +2 ( p ) = −∞ . Denote H + ∈ C ( R n ) as H + ( p ) = max { H, H +2 } = ( H ( p ) for p ∈ R n \ U , H +2 ( p ) for p ∈ U .
Also denote by H + and H +2 the effective Hamiltonians associated with H + ( p ) − V ( x )and H +2 ( p ) − V ( x ), respectively. Apparently,max (cid:8) H , (cid:9) ≤ H ≤ H + . (2.8)On the other hand, by Theorem 2.1, the representation formula for H + is H + = max n H , H +2 , o = max { H , } , (2.9)where the second equality is due to H +2 ≤ max R n H +2 − max T n V = max ¯ U H − max R n V ≤ . We combine (2.8) and (2.9) to get the conclusion. (cid:3)
A more general case.
We first extend Theorem 2.1 as following. To avoidunnecessary technicalities, we only consider radially symmetric cases from now on.The results still hold true for general Hamiltonians (without the radially symmetricassumption) under corresponding appropriate conditions.Let H : R n → R be such that(H4) H ( p ) = ϕ ( | p | ) for p ∈ R n , where ϕ ∈ C ([0 , ∞ ) , R ) such that ( ϕ (0) > , ϕ (2) = 0 , lim r →∞ ϕ ( r ) = + ∞ ,ϕ is strictly increasing on [0 ,
1] and [2 , ∞ ), and is strictly decreasing on [1 , . (H5) H i ( p ) = ϕ i ( | p | ) for p ∈ R n and 1 ≤ i ≤
3, where ϕ i ∈ C ([0 , ∞ ) , R ) such that ϕ = ϕ on [2 , ∞ ) , ϕ is strictly increasing on [0 , ,ϕ = ϕ on [1 , , ϕ is strictly decreasing on [0 ,
1] and [2 , ∞ ) , lim r →∞ ϕ ( r ) = −∞ ,ϕ = ϕ on [0 , , ϕ is strictly increasing on [1 , ∞ ) , and ϕ > ϕ in (1 , ∞ ) . Lemma 2.5.
Let H ( p ) = ϕ ( | p | ) , H i ( p ) = ϕ i ( | p | ) for ≤ i ≤ and p ∈ R n ,where ϕ, ϕ , ϕ , ϕ satisfy (H4)–(H5) . Let V ∈ C ( T n ) be a potential energy with min T n V = 0 . IN-MAX FORMULAS AND OTHER PROPERTIES OF H r ϕ ϕ ϕ ϕ Figure 3.
Graphs of ϕ, ϕ , ϕ , ϕ Assume that H is the effective Hamiltonian corresponding to H ( p ) − V ( x ) . As-sume also that H i is the effective Hamiltonian corresponding to H i ( p ) − V ( x ) for ≤ i ≤ . Then H = max (cid:8) , H , K (cid:9) = max (cid:8) , H , min (cid:8) H , H , ϕ (1) − max T n V (cid:9)(cid:9) . Here K is the effective Hamiltonian corresponding to K ( p ) − V ( x ) for K : R n → R defined as K ( p ) = min { ϕ ( | p | ) , ϕ ( | p | ) } = ( ϕ ( | p | ) for | p | ≤ ,ϕ ( | p | ) for | p | ≥ . In particular, both H and K are even.Proof. Considering − K ( − p ), thanks to the representation formula and evennessfrom Theorem 2.1, K = min n H , H , ϕ (1) − max T n V o . Define ˜ ϕ = min { ϕ , ϕ (1) } . Let ˜ H ( p ) = ˜ ϕ ( | p | ) and ˜ H be the effective Hamil-tonian corresponding to ˜ H ( p ) − V ( x ). Then, thanks to Corollary 2.3, we also havethat K = min n ˜ H , H o . (2.10)Our goal is then to show that H = max (cid:8) , H , K (cid:9) . To do this, we again dividethe proof into few steps for clarity. Readers should notice that the proof below doesnot depend on the quasiconvexity of H . It only uses the fact that H ≥ H . This isessential to prove the most general result, Theorem 2.6. Step 1.
Clearly 0 ≤ H ≤ H . This implies further that H ( p ) = 0 for all | p | = 2 . (2.11) We furthermore have that
K, H ≤ H as K, H ≤ H . Thus, H ≥ max (cid:8) , H , K (cid:9) (2.12)We now show the reverse inequality of (2.12) to finish the proof. Step 2.
Fix p ∈ R n . Assume that H ( p ) ≥ max (cid:8) , K ( p ) (cid:9) . Since H is quasiconvex,we follow exactly the same lines of Step 2 in the proof of Theorem 2.1 to deducethat H ( p ) ≥ H ( p ). Step 3.
Assume that K ( p ) ≥ max (cid:8) , H ( p ) (cid:9) . Since K is not quasiconvex orquasiconcave, we cannot directly copy Step 2 or Step 3 in the proof of Theorem 2.1.Instead, there are two cases that need to be considered.Firstly, we consider the case that K ( p ) = ˜ H ( p ) ≤ H ( p ). Let v ( x, p ) be asolution to the cell problem˜ H ( p + Dv ( x, p )) − V ( x ) = ˜ H ( p ) ≥ T n . (2.13)Since ˜ H is quasiconcave, for any x ∈ T n and q ∈ D + v ( x, p ), we have˜ H ( p + q ) − V ( x ) = ˜ H ( p ) ≥ , which gives that ˜ H ( p + q ) ≥ H ( p + q ) ≥ H ( p + q ). Therefore, v ( x, p )is a viscosity subsolution to H ( p + Dv ( x, p )) − V ( x ) = ˜ H ( p ) in T n . We conclude that K ( p ) = ˜ H ( p ) ≥ H ( p ).Secondly, assume that K ( p ) = H ( p ) ≤ ˜ H ( p ). Since ϕ ≥ ϕ , H ( p ) ≥ H ( p ).Combining with H ( p ) ≥ K ( p ) in (2.12), we obtain K ( p ) = H ( p ) in this step. Step 4.
Assume that 0 > max (cid:8) H ( p ) , K ( p ) (cid:9) . Our goal now is to show H ( p ) = 0.Thanks to (2.11) in Step 1, we may assume that | p | 6 = 2.For σ ∈ [0 ,
1] and i = 1 ,
2, let H σ , H σ , K σ be the effective Hamiltonians corre-sponding to H ( p ) − σV ( x ) , H ( p ) − σV ( x ), K ( p ) − σV ( x ), respectively. It is clearthat 0 ≤ H = H ≤ H σ for all σ ∈ [0 , . (2.14)By repeating Steps 2 and 3 above, we getFor p ∈ R n and σ ∈ [0 , (cid:8) H σ ( p ) , K σ ( p ) (cid:9) = 0, then H σ ( p ) = 0. (2.15)We only consider the case | p | < | p | > H ( p ) = K ( p ) = K ( p ) > K ( p ) = K ( p ) < . By the continuity of σ K σ ( p ), there exists s ∈ (0 ,
1) such that K s ( p ) = 0. Notefurthermore that, as | p | < H s ( p ) ≤ H ( p ) <
0. These, together with (2.14) and(2.15), yield the desired result. (cid:3)
IN-MAX FORMULAS AND OTHER PROPERTIES OF H rϕs s s m Figure 4.
Graph of ϕ in first general case2.3. General cases.
By using induction, we can obtain min-max (max-min) formu-las for H in case H ( p ) = ϕ ( | p | ) where ϕ satisfies some certain conditions describedbelow. We consider two cases corresponding to Figures 4 and 5.In this first general case corresponding to Figure 4, we assume that(H6) ϕ ∈ C ([0 , ∞ ) , R ) satisfying there exist m ∈ N and 0 = s < s < . . . s m < ∞ = s m +1 such that ϕ is strictly increasing in ( s i , s i +1 ), and is strictly decreasing in ( s i +1 , s i +2 ), ϕ ( s ) > ϕ ( s ) > . . . > ϕ ( s m ), and ϕ ( s ) < ϕ ( s ) < . . . < ϕ ( s m +1 ) = ∞ .For 0 ≤ i ≤ m , • let ϕ i : [0 , ∞ ) → R be a continuous, strictly increasing function such that ϕ i = ϕ on [ s i , s i +1 ] and lim s →∞ ϕ i ( s ) = ∞ . Also ϕ i ≥ ϕ i +2 . • let ϕ i +1 : [0 , ∞ ) → R be a continuous, strictly decreasing function such that ϕ i +1 = ϕ on [ s i +1 , s i +2 ] and lim s →∞ ϕ i +1 ( s ) = −∞ . Also ϕ i +1 ≤ ϕ i +3 .Define H m − ( p ) = max { ϕ ( | p | ) , ϕ m − ( | p | ) } = ( ϕ ( | p | ) for | p | ≤ s m − , ϕ m − ( | p | ) for | p | > s m − and k m − ( s ) = min { ϕ ( s ) , ϕ m − ( s ) } = ( ϕ ( s ) for s ≤ s m , ϕ m − ( s ) for s > s m .Denote H m − , H m , K m − , Φ j as the effective Hamiltonians associated with theHamiltonians H m − ( p ) − V ( x ), ϕ ( | p | ) − V ( x ), k m − ( | p | ) − V ( x ) and ϕ j ( | p | ) − V ( x )for 0 ≤ j ≤ m , respectively.The following is our main decomposition theorem in this paper. Theorem 2.6.
Assume that (H6) holds for some m ∈ N . Then H m = max n K m − , Φ m , ϕ ( s m ) − min T n V o , (2.16) and K m − = min n H m − , Φ m − , ϕ ( s m − ) − max T n V o . (2.17) In particular, H m and K m − are both even. Again, we would like to point out that the evenness of H m and K m − is far frombeing obvious although H m and K m are both even. See the discussion in Subsection2.4 for this subtle issue. Proof.
We prove by induction. When m = 1, the two formulas (2.16) and (2.17)follow from Lemma 2.5 and Theorem 2.1.Assume that (2.16) and (2.17) hold for m ∈ N . We need to verify these equalitiesfor m + 1. Using similar arguments as those in the proof Lemma 2.5, noting thestatement in italic right above Step 1, we first derive that K m = min n H m , Φ m +1 , ϕ ( s m +1 ) − max T n V o . Then again, by basically repeating the proof of Lemma 2.5, we obtain H m +1 = max n K m , Φ m +2 , ϕ ( s m +2 ) − min T n V o . (cid:3) Remark 3. (i) By approximation, we see that representation formulas (2.16) and(2.17) still hold true if we relax (H6) a bit, that is, we only require that ϕ satisfies ( ϕ is increasing in ( s i , s i +1 ), and is decreasing in ( s i +1 , s i +2 ), ϕ ( s ) ≥ ϕ ( s ) ≥ . . . ≥ ϕ ( s m ), and ϕ ( s ) ≤ ϕ ( s ) ≤ . . . < ϕ ( s m +1 ) = ∞ .(ii) According to Corollary 2.4, if osc T n V = max T n V − min T n V ≥ ϕ ( s m − ) − ϕ ( s m ), then H is quasiconvex and H m = max n Φ m , ϕ ( s m ) − min T n V o . The second general case corresponds to the case where H ( p ) = − k ( | p | ) for all p ∈ R n as described in Figure 5. After changing the notations appropriately, weobtain similar representation formulas as in Theorem 2.6. We omit the details here.2.4. “Non-decomposability” and Breakdown of symmetry. A natural ques-tion is whether we can extend Theorem 2.6 to other nonconvex H , i.e., there existquasiconvex/concave H i (1 ≤ i ≤ m ) such that H is given by a “decomposition”formula (e.g., min-max type) involving H i , min V and max VH = G ( H , ..., H m , min V, max V ) , (2.18) IN-MAX FORMULAS AND OTHER PROPERTIES OF H rs s s m − − k Figure 5.
Graph of − k in the second general casefor any V ∈ C ( T n ). Here H and H i are effective Hamiltonians associated with H − V and H i − V . Note that for quasiconvex/concave function F , using the inf-max formula (1.4), it is easy to see that the effective Hamiltonians associated with F ( p ) − V ( x ) and F ( p ) − V ( − x ) are the same. Hence if such a “decomposition”formula indeed exists for a specific nonconvex H , effective Hamiltonians associatedwith H ( p ) − V ( x ) and H ( p ) − V ( − x ) have to be identical as well. In particular, if H is an even function, this is equivalent to saying that H is even too, which leadsto the following question. Question 1.
Let H ∈ C ( R n ) be a coercive and even Hamiltonian, and V ∈ C ( T n ) be a given potential. Let H be the effective Hamiltonian associated with H ( p ) − V ( x ) .Is it true that H is also even? In general, we may ask what properties of the originalHamiltonian will be preserved under homogenization. Even though that this is a simple and natural question, it has not been studiedmuch in the literature as far as the authors know. We give below some answers anddiscussions to this: • If H is quasiconvex, the answer is “yes” due to the inf-max formula H ( p ) = inf φ ∈ C ( T n ) max x ∈ R n ( H ( p + Dφ ( x )) − V ( x ))as shown in the proof of Lemma 2.1. • For genuinely nonconvex H , if H can be written as a min-max formula involvingeffective Hamiltonians of even quasiconvex (or quasiconcave) Hamiltonians, then H is still even (e.g., see Corollary 2.2, Lemma 2.5, and Theorem 2.6). • However, in general, the evenness is lost as presented in Remark 1.2 in [32].Let us quickly recall the setting there. We consider the case n = 1, and choose H ( p ) = ϕ ( | p | ) for p ∈ R , where ϕ satisfies (H8) (see Figure 7 below) with m = and M = . Fix s ∈ (0 , V s ( x ) = min (cid:8) xs , − x − s (cid:9) for x ∈ [0 , V to R in a periodic way. Then H is not even unless s = . In particular, this impliesthat a decomposition formula for H does not exist. Also, see Figure 12 below forloss of evenness when the Hamiltonian is of double-well type. • It is extremely interesting if we can point out some further general requirementson H and V in the genuinely nonconvex setting, under which H is even. Theinterplay between H and V plays a crucial role here (see Remark 4 for intriguingobservations).Some related discussions and interesting applications of evenness can also befound in [40].3. Quasi-convexification phenomenon in multi-dimensional spaces
Intuitively, homogenization, a nonlinear averaging procedure, makes the effectiveHamiltonian less nonconvex. The question is how to describe this in a rigorous andsystematic way. Some special cases have been handled in Remark 3. In this sec-tion, we look at more generic and important situations: general radially symmetricHamiltonians and a typical double-well Hamiltonian. These two types of Hamil-tonians more or less capture essential features of nonconvexity. In some sense,quasi-convexification represents a scenario where there is no genuine decompositionof H . Due to the difficulty in rigorous analysis, we focus more on numerical com-putations. The Lax-Friedrichs based big-T method is used to compute the effectiveHamiltonian.3.1. Radially symmetric Hamiltonians.
Assume that H ( p ) = ϕ ( | p | ) for all p ∈ R n , where ϕ : [0 , ∞ ) → R is a given function. The following is quite a generalcondition on ϕ .(H7) ϕ ∈ C ([0 , ∞ ) , R ) satisfying that lim s →∞ ϕ ( s ) = + ∞ and ( there exist m ∈ N and 0 = s < s < . . . s m < ∞ = s m +1 such that ϕ is strictly increasing in ( s i , s i +1 ), and is strictly decreasing in ( s i +1 , s i +2 ).It is clear that (H7) is more general than (H6). In fact, any coercive function ψ ∈ C ([0 , ∞ ) , R ) can be approximated by ϕ satisfying (H7).Denote by ( M i = ϕ ( s i − ) for 1 ≤ i ≤ m,m j = ϕ ( s j ) for 1 ≤ j ≤ m. We propose the following conjecture.
Conjecture 1.
Assume that (H7) holds. Assume further that ϕ (0) = min ϕ = 0 .Let H ( p ) = ϕ ( | p | ) for all p ∈ R n , and V ∈ C ( T n ) be a given potential function. Let H be the effective Hamiltonian corresponding to H ( p ) − V ( x ) . If osc T n V = max T n V − min T n V ≥ max i,j ( M i − m j ) , IN-MAX FORMULAS AND OTHER PROPERTIES OF H then the effective Hamiltonian H is quasiconvex. When n = 1, the above conjecture was proven in [7] based on some essentially onedimensional approaches. In multi-dimensional spaces, this conjecture seems quitechallenging in general (Remark 3 is a special case). Let us now consider a basicsituation, which we believe is an important step toward proving Conjecture 1.(H8) ϕ ∈ C ([0 , ∞ ) , R ) such that there exist 0 < s < s < ∞ satisfying ( ϕ (0) = 0 < ϕ ( s ) = m < ϕ ( s ) = M < lim s →∞ ϕ ( s ) = + ∞ , ϕ is strictly increasing on [0 , s ] and [ s , ∞ ) and ϕ is strictly decreasing on [ s , s ].See Figure 7. For this particular case, the conjecture says that H is quasiconvex ifosc T n V ≥ M − m . This is clear in terms of numerical results (Figure 6).
Numerical example 1.
Let n = 2. We consider the following setting H ( p ) = min (cid:26) q p + p , (cid:12)(cid:12)(cid:12)(cid:12)q p + p − (cid:12)(cid:12)(cid:12)(cid:12) + 1 (cid:27) for p = ( p , p ) ∈ R ,V ( x ) = S ∗ (1 + sin(2 πx ))(1 + sin(2 πx )) for x = ( x , x ) ∈ T . The constant S serves as the scaling parameter to increase or decrease the effect ofthe potential energy V . For this specific case, M − m = 2 − T V = 4 S .So the threshold value is S = 0 . x domain [0 , is discretized by 401 ×
401 mesh pointsand the p domain [ − , is sampled by 21 ×
21 mesh points. The initial conditionfor the big-T method is taken to be cos(2 πx ) sin(2 πx ). See Figure 6.(a) −1 −0.5 0 0.5 1−1−0.500.5100.511.52 pxpy (b) −1 −0.5 0 0.5 1−1−0.500.5100.511.52 pxpy (c) −1 −0.5 0 0.5 1−1−0.500.5100.511.52 pxpy (d) −1 −0.5 0 0.5 1−1−0.500.5100.511.52 pxpy (e) −1 −0.5 0 0.5 1−1−0.500.5100.511.52 pxpy Figure 6. (a) The original Hamiltonian: S = 0. (b) S = 0 . S = 0 .
25. (d) S = 0 .
30. (e) S = 0 . However, we are only able to rigorously verify this for level sets above m . Thispartially demonstrate the “quasi-convexification” since the nonconvexity of the orig-inal H appears on level sets between m and M . Denote H = max { H, m } and H ( p ) = ( H ( p ) for | p | ≤ s , max { M , H ( p ) } for | p | ≥ s . (3.1)Note that H is a “decomposable” nonconvex function from Remark 3 and H isquasiconvex. Precisely speaking, Theorem 3.1.
Assume that (H8) holds. Let H ( p ) = ϕ ( | p | ) for all p ∈ R n , and V ∈ C ( T n ) be a given potential energy such that osc T n V ≥ M − m . Let H be the effective Hamiltonian corresponding to H ( p ) − V ( x ) . Then for any µ ≥ m , the level set { H ≤ µ } is convex. rs M s m ϕ Figure 7.
Graphs of ϕ Proof.
Without loss of generality, we assume that min T n V = 0. Let H be theeffective Hamiltonian of H − V . Clearly H is quasiconvex by Remark 3. So itsuffices to show that for every µ > m , H ( p ) = µ if and only if H ( p ) = µ. Since H ≤ H , we only need to show that for fixed p ∈ R n , if H ( p ) > m , then H ( p ) = H ( p ). In fact, let v ( x, p ) be a solution to H ( p + Dv ) − V = H ( p ) > m in T n . Note that H ( p ) + V > m . It is straightforward that v is also a solution to H ( p + Dv ) − V = H ( p ) in T n . Thus, H ( p ) = H ( p ). The proof is complete. (cid:3) IN-MAX FORMULAS AND OTHER PROPERTIES OF H Remark 4.
Here is an interesting transition between min-max decomposition, even-ness and quasi-convexification when n = 1. Assume min T V = 0. • If c = max T V < M − m , it is not hard to obtain a representation formulafor H (“conditional decomposition”) H = min (cid:8) H , H (cid:9) . (3.2)Here, H falls into the category of item (i) in Remark 3. And H is the effectiveHamiltonian associated with the quasiconvex H in (3.1). In particular, H is evenbut not quasiconvex . The shape of H is qualitatively similar to that of H . It isnot clear to us whether this decomposition formula holds when n ≥
2. The key isto answer Question 3 in the appendix first. • If c = max T V = M − m , H is both even and quasiconvex . • If c = max T V > M − m and V s ( x ) = c min (cid:8) xs , − x − s (cid:9) for x ∈ [0 ,
1] (extend V to R periodically), then H is quasiconvex but loses evenness . More precisely,by adapting Step 1 in the proof of Theorem 1.4 in [32], we can show that the levelset (cid:8) H ≤ µ (cid:9) is not even for any µ ∈ ( M − c , m ) if s = . Hence the aboveformula or decomposition (3.2) no longer holds.See also Figure 8 below for numerical computations of a specific example. Numerical example 2.
Let n = 1. We consider the following setting H ( p ) = min { | p | , | | p | − | + 1 } , for p ∈ R ,V ( x ) = S ∗ min (cid:26) x,
32 (1 − x ) (cid:27) , for x ∈ [0 , , and extend V to R in a periodic way. The constant S serves as the scaling parameterto increase or decrease the effect of the potential energy V . For this case, c = 1and M − m = 1.The Lax-Friedrichs based big-T method is used to compute this effective Hamil-tonian. The p domain [ − ,
1] is sampled by 41 mesh points. The computational x domain [0 ,
1] is discretized by 401 mesh points. The initial condition for the big-Tmethod is taken to be cos(2 πx ). The results are shown in Figure 8.(a) −1 −0.5 0 0.5 100.20.40.60.811.21.41.61.82 p H b a r (b) −1 −0.5 0 0.5 100.20.40.60.811.21.41.61.82 p H b a r (c) −1 −0.5 0 0.5 100.20.40.60.811.21.41.61.82 p H b a r Figure 8. S = 0 .
5. (b): S = 1 .
0. (c): S = 1 . A double-well type Hamiltonian.
Let n = 2. We consider a prototypicalexample H ( p ) = min {| p − e | , | p + e |} for p = ( p , p ) ∈ R , where e = (1 , H is more sensitive to the structure of the potential V instead of just the oscillation.3.2.1. A unstable potential.
We consider the following situation ( H ( p ) = min {| p − e | , | p + e |} for p = ( p , p ) ∈ R ,V ( x ) = S ∗ (1 + sin 2 πx )(1 + sin 2 πx ) for x = ( x , x ) ∈ T . (3.3)The constant S > V . Note that V attains its minimum along lines x = − / Z and x = − / Z , which is clearly not a stable situation.For this kind of nonconvex Hamiltonian H and V , complete quasi-convexificationdoes not occur. However, we still see that H eventually becomes a “less nonconvex”function. Let H ( p, S ) be the effective Hamiltonian corresponding to H ( p ) − V ( x ).We have that Theorem 3.2.
Assume that (3.3) holds. Then lim S →∞ H ( p, S ) = max {| p | , min {| p − | , | p + 1 |}} for p = ( p , p ) ∈ R . Proof.
We first show that for any
S > H ( p, S ) ≥ max {| p | , min {| p − | , | p + 1 |}} . (3.4)Let v be a viscosity solution to H ( p + Dv ) − V ( x ) = H ( p, S ) in T . Without loss of generality, suppose that v is semi-convex and differentiable at( x , − /
4) for a.e. x ∈ R . Otherwise, we may use super-convolution to get asubsolution and look at a nearby line by Fubini’s theorem. Accordingly, for a.e. x ∈ R , H ( p, S ) ≥ min {| p + v x ( x , − / − | , | p + v x ( x , − /
4) + 1 |} . Assume that x v ( x , − /
4) attains its maximum at x ∈ R . Then v x ( x , − /
4) =0 due to the semi-convexity of v . Hence H ( p, S ) ≥ min {| p − | , | p + 1 |} . Now, similarly, we can show that H ( p, S ) ≥ | p + v x ( − / , x ) | for a.e. x ∈ T . Taking the integration on both side over [0 ,
1] and using Jensen’s inequality, wederive H ( p, S ) ≥ | p | . Thus, (3.4) holds.Next we show thatlim S → + ∞ H ( p, S ) ≤ max {| p | , min {| p − | , | p + 1 |}} . (3.5) IN-MAX FORMULAS AND OTHER PROPERTIES OF H In fact, for any δ >
0, it is not hard to construct φ ∈ C ( T ) such that H ( p + Dφ ) ≤ max {| p | , min {| p − | , | p + 1 |}} + δ on ( T × {− / } ) [ ( {− / } × T ) . Clearly, H ( p, S ) ≤ max x ∈ T ( H ( p + Dφ ( x )) − S (1 + sin 2 πx )(1 + sin 2 πx )) . Sending S → ∞ and then δ →
0, we obtain (3.5). (cid:3)
Remark 5.
Write F ∞ ( p ) = max {| p | , min {| p − | , | p + 1 |}} . Simple computa-tions show that • For r ∈ [0 , { F ∞ = r } consists of two disjoint squares centered at ( ± , • For r = 1, { F ∞ = 1 } consists of two adjacent squares centered at ( ± , • For r > { F ∞ = r } is a rectangle centered at the origin.See Figure 9 below. We can say that F ∞ ( p ) looks more convex than the originalHamiltonian H ( p ) = min {| p − e | , | p + e |} . { F ∞ = 2 }{ F ∞ = 1 } F ∞ = 0 . F ∞ = 0 . Figure 9.
Level curves of F ∞ A stable potential.
We consider the following ( H ( p ) = min {| p − e | , | p + e |} for p = ( p , p ) ∈ R ,V ( x ) = S ∗ (cid:0) sin (2 πx ) + sin (2 πx ) (cid:1) for x = ( x , x ) ∈ T . (3.6)The constant S > V . Note that V attains its minimum at points ( x , x ) such that 2( x , x ) ∈ Z , which is a stable situation.For this kind of potential, it is easy to show thatlim S →∞ H ( p, S ) = 0 locally uniformly in R . More interestingly, numerical computations (Figures 10 and 11) below suggest that H becomes quasiconvex at least when S ≥ Question 2.
Assume that (3.6) holds. Does there exist L such that when S > L , H is quasiconvex? Numerical example 3.
We consider setting (3.6). The constant S serves as thescaling parameter to increase or decrease the effect of the potential.We use the Lax-Friedrichs based big-T method to compute the effective Hamil-tonian. The computation for time-dependent Hamilton-Jacobi equations is done byusing the LxF-WENO 3rd-order scheme. The initial condition for the big-T methodis taken to be cos(2 πx ) sin(2 πx ). See Figure 10 and Figure 11 for results.(a) −1 −0.5 0 0.5 1−1−0.500.5100.511.5 pxpy (b) −1 −0.5 0 0.5 1−1−0.500.5100.20.40.60.81 pxpy (c) −1 −0.5 0 0.5 1−1−0.500.5100.20.40.60.8 pxpy (d) −4 −2 0 2 4−4−202400.511.5 pxpy (e) −4 −2 0 2 4−4−202400.10.20.30.40.5 pxpy Figure 10.
Surface plots. (a) The original Hamiltonian: S = 0. (b) S = 0 .
50. (c) S = 1 .
0. (d) S = 2 .
0. (e) S = 4 . Numerical example 4.
We consider ( H ( p ) = min {| p − e | , | p + e |} for p = ( p , p ) ∈ R ,V ( x ) = S ∗ (3 + sin(2 πx ) + sin(4 πx ) + sin(2 πx )) for x = ( x , x ) ∈ T . This is the case that V is not even. The constant S serves as the scaling parameterto increase or decrease the effect of the potential. See Figure 12 below. Clearly, H is not even when S = 0 . S = 0 . S = 0 . S = 0 .
95 and S = 1 .
0. Loss ofevenness for all S implies that H can not have a decomposition formula like (2.18)regardless of the oscillation of the V .4. Appendix: Some application in Random homogenization
As a bypass product, we show that all Hamiltonians in Section 2 are actuallyregularly homogenizable in the stationary ergodic setting. Let us first give a briefoverview of stochastic homogenization.
IN-MAX FORMULAS AND OTHER PROPERTIES OF H (a) px p y −1 −0.5 0 0.5 1−1−0.8−0.6−0.4−0.200.20.40.60.81 00.20.40.60.811.21.4 (b) px p y −1 −0.5 0 0.5 1−1−0.8−0.6−0.4−0.200.20.40.60.81 0.20.30.40.50.60.70.8 (c) px p y −1 −0.5 0 0.5 1−1−0.8−0.6−0.4−0.200.20.40.60.81 0.150.20.250.30.350.40.450.50.55 (d) px p y −2 −1 0 1 2−2.5−2−1.5−1−0.500.511.522.5 0.20.40.60.811.2 (e) px p y −2 −1 0 1 2−2.5−2−1.5−1−0.500.511.522.5 0.050.10.150.20.250.30.350.4 Figure 11.
Contour plots. (a) The original Hamiltonian: S = 0.(b) S = 0 .
50. (c) S = 1 .
0. (d) S = 2 .
0. (e) S = 4 . px p y −1 −0.5 0 0.5 1−1−0.8−0.6−0.4−0.200.20.40.60.81 00.20.40.60.811.21.4 (b) px p y −1 −0.5 0 0.5 1−1−0.8−0.6−0.4−0.200.20.40.60.81 0.10.20.30.40.50.60.70.80.9 (c) px p y −1 −0.5 0 0.5 1−1−0.8−0.6−0.4−0.200.20.40.60.81 0.10.20.30.40.50.60.7 (d) px p y −1 −0.5 0 0.5 1−1−0.8−0.6−0.4−0.200.20.40.60.81 00.050.10.150.20.250.30.350.40.45 (e) px p y −1 −0.5 0 0.5 1−1−0.8−0.6−0.4−0.200.20.40.60.81 0.020.040.060.080.10.120.14 (f) px p y −1 −0.5 0 0.5 1−1−0.8−0.6−0.4−0.200.20.40.60.81 00.020.040.060.080.1 Figure 12. (a) The original Hamiltonian: S = 0. (b) S = 0 . S = 0 .
25. (d) S = 0 .
5. (e) S = 0 .
95. (f) S = 1 . Brief overview of stochastic homogenization.
Let (Ω , F , P ) be a proba-bility space. Suppose that { τ y } y ∈ R n is a measure-preserving translation group actionof R n on Ω which satisfies that(1) ( Semi-group property ) τ x ◦ τ y = τ x + y for all x, y ∈ R n . (2) ( Ergodicity ) For any E ∈ F , τ x ( E ) = E for all x ∈ R n ⇒ P ( E ) = 0 or P ( E ) = 1 . The potential V ( x, ω ) : R n × Ω → R is assumed to be stationary, bounded anduniformly continuous. More precisely, V ( x + y, ω ) = V ( x, τ y ω ) for all x, y ∈ R n and ω ∈ Ω, ess sup Ω | V (0 , ω ) | < + ∞ and | V ( x, ω ) − V ( y, ω ) | ≤ c ( | x − y | ) for all x, y ∈ R n and ω ∈ Ω , for some function c : [0 , ∞ ) → [0 , ∞ ) satisfying lim r → c ( r ) = 0.For ε >
0, denote u ε ( x, t, ω ) as the unique viscosity solution to ( u εt + H ( Du ε ) − V (cid:0) xε , ω (cid:1) = 0 in R n × (0 , ∞ ) ,u ε ( x, , ω ) = g ( x ) on R n . (4.1)Here H ∈ C ( R n ) is coercive. A basic question is whether u ε , as ε →
0, convergesto the solution to an effective deterministic equation (1.2) almost surely as in theperiodic setting.The stochastic homogenization of Hamilton-Jacobi equations has received muchattention in the last seventeen years. The first results were due to Rezakhanlouand Tarver [39] and Souganidis [41], who independently proved convergence resultsfor general convex, first-order Hamilton-Jacobi equations in stationary ergodic set-ting. These results were extended to the viscous case with convex Hamiltoniansby Kosygina, Rezakhanlou and Varadhan [27] and, independently, by Lions andSouganidis [30]. New proofs of these results based on the notion of intrinsic dis-tance functions (maximal subsolutions) appeared later in Armstrong and Souganidis[4] for the first-order case and in Armstrong and Tran [5] for the viscous case. SeeDavini, Siconolfi [16], Armstrong, Souganidis [4] for homogenization of quasiconvex,first-order Hamilton-Jacobi equations.One of the prominent open problems in the field is to prove/disprove homogeniza-tion in the genuinely nonconvex setting. In [6], Armstrong, Tran and Yu showedthat, for H ( p ) = ( | p | − , (4.1) homogenizes in all space dimensions n ≥
1. In thenext paper [7], Armstrong, Tran and Yu proved that, for n = 1, (4.1) homogenizesfor general coercive H . Gao [22] generalized the result in [7] to the general non sepa-rable Hamiltonians H ( p, x, ω ) in one space dimension. A common strategy in papers [6, 7, 22] is to identify the shape of H in the periodic setting first and then recoverit in the stationary ergodic setting. In particular, in contrast to previous works,our strategy does not depend on finding some master ergodic quantities suitable forsubadditive ergodic theorems. Such kind of ergodic quantities may not exist at allfor genuinely nonconvex H .Ziliotto [42] gave a counterexample to homogenization of (4.1) in case n = 2. Seealso the paper by Feldman and Souganidis [21]. Basically, [42, 21] show that, if H has a strict saddle point, then there exists a potential energy V such that H − V isnot homogenizable.Based on min-max formulas established in Section 2, we prove that, for the Hamil-tonians H appear in Theorem 2.1, Corollary 2.4, Lemma 2.5, and Theorem 2.6, H − V is always regularly homogenizable in all space dimensions n ≥
1. See theprecise statements in Theorems 4.1, 4.3, and Corollary 4.2 in Subsection 4.2. The-orem 4.1 includes the result in [6] as a special case. Also, the result of Corollary 4.2
IN-MAX FORMULAS AND OTHER PROPERTIES OF H implies that, in some specific cases, even if H has strict saddle points, H − V is stillregularly homogenizable for every V with large oscillation. See the comments afterits statement and some comparison between this result and the counterexamples in[42, 21].The authors tend to believe that a prior identification of the shape of H mightbe necessary in order to tackle homogenization in the general stationary ergodicsetting. In certain special random environment like finite range dependence (i.i.d),the homogenization was established for a class of Hamiltonians in interesting worksof Armstrong, Cardaliaguet [2], Feldman, Souganidis [21]. Their proofs are basedon completely different philosophy and, in particular, rely on specific properties ofthe random media.In the viscous case (i.e., adding − ε ∆ u ε to equation (4.1)), the stochastic homog-enization problem for nonconvex Hamiltonians is more formidable. For example,the homogenization has not even been proved or disproved for simple cases like H ( p, x ) = ( | p | − − V ( x ) in one dimension. Min-max formulas in the inviscidcase are in general not available here due to the nonlocal effect (or regularity) fromthe viscous term. Nevertheless, see a preliminary result in one dimensional spaceby Davini and Kosygina [15].The following definition was first introduced in [7]. Definition 1.
We say that H − V is regularly homogenizable if for every p ∈ R n ,there exists a unique constant H ( p ) ∈ R such that, for every R > and for a.s. ω ∈ Ω , lim sup λ → max | x |≤ Rλ (cid:12)(cid:12) λv λ ( x, p, ω ) + H ( p ) (cid:12)(cid:12) = 0 . (4.2) Here, for λ > , v λ ( · , p, ω ) ∈ W , ∞ ( R n ) is the unique bounded viscosity solution to λv λ + H ( p + Dv λ ) − V ( x, ω ) = 0 in R n . According to Lemma 5.1 in [4], (4.2) is equivalent to saying that for a.s. ω ∈ Ω,lim λ → (cid:12)(cid:12) λv λ (0 , p, ω ) + H ( p ) (cid:12)(cid:12) = 0 . Clearly, if H − V is regularly homogenizable, then random homogenization holds,that is, solution u ε of (4.1) converges to u , the solution to (1.2) with H defined by(4.2), as ε → Stochastic homogenization results.
The main claim is that H ( p ) − V ( x, ω )is regularly homogenizable provided that H ( p ) is of a form in Theorem 2.1, Corollary2.4, Lemma 2.5, and Theorem 2.6. The proof is basically a repetition of arguments inthe proofs of the aforementioned results except that the cell problem in the periodicsetting is replaced by the discount ergodic problem in the random environment.This is because of the fact that the cell problem in the random environment mightnot have sublinear solutions at all. Here are the precise statements of the results. Theorem 4.1.
Assume that H ∈ C ( R n ) satisfies (H1)–(H3) . Assume further that ess inf Ω V (0 , ω ) = 0 . Then H − V is regularly homogenizable and H = max (cid:8) H , H , (cid:9) . As mentioned, this theorem includes the result in [6] as a special case. An im-portant corollary of this theorem is the following:
Corollary 4.2.
Let H ∈ C ( R n ) be a coercive Hamiltonian satisfying (H1)–(H3) ,except that we do not require H to be quasiconcave. Assume that ess inf Ω V (0 , ω ) =0 , ess sup Ω V (0 , ω ) = m , and m > max U H = max R n H . Then H − V is regularly homogenizable and H = max (cid:8) H , (cid:9) . In particular, H is quasiconvex in this situation. It is worth emphasizing that we do not require any structure of H in U exceptthat H > H is allowed to have strict saddle points in U .Therefore, Corollary 4.2 implies that, in some specific cases, even if H has strictsaddle points, H − V is still regularly homogenizable provided that the oscillationof V is large enough. In a way, this is a situation when the potential energy V hasmuch power to overcome the depths of all the wells created by the kinetic energy H and it “irons out” all the nonconvex pieces to make H quasiconvex. This alsoconfirms that the counterexamples in [42, 21] are only for the case that V has smalloscillation, in which case V only sees the local structure of H around its strict saddlepoints, but not its global structure.Let us now state the most general result in this stochastic homogenization contextthat we have. Theorem 4.3.
Assume that (H6) holds for some m ∈ N . Assume further that ess inf Ω V (0 , ω ) = 0 , ess sup Ω V (0 , ω ) = m . Then ϕ ( | p | ) − V ( x, ω ) and k m − ( | p | ) − V ( x, ω ) are regularly homogenizable. Moreover, (2.16) and (2.17) hold in this ran-dom setting as well H m = max (cid:8) K m − , Φ m , ϕ ( s m ) (cid:9) , and K m − = min (cid:8) H m − , Φ m − , ϕ ( s m − ) − m (cid:9) . In particular, H m and K m − are both even. Here we use same notations as inTheorem 2.6. We also have the following conjecture which was proven to be true in one dimen-sion [7].
IN-MAX FORMULAS AND OTHER PROPERTIES OF H Conjecture 2.
Assume that ϕ : [0 , ∞ ) → R is continuous and coercive. Set H ( p, x, ω ) = ϕ ( | p | ) − V ( x, ω ) for ( p, x, ω ) ∈ R n × R n × Ω . Then H is regularlyhomogenizable. We believe that Conjecture 1 should play a significant role in proving the aboveconjecture as in the one dimensional case. An initial step might be to obtain sto-chastic homogenization for the specific ϕ satisfying (H8). Below is closely relatedelementary question Question 3.
Let w be a periodic semi-concave (or semi-convex) function. Denote D as the collection of all regular gradients, that is, D = { Dw ( x ) : w is differentiable at x } . Is D a connected set? The periodic assumption is essential. Otherwise, it is obviously false, e.g., w ( x ) = − ( | x | + · · · + | x n | ) for x = ( x , . . . , x n ) ∈ R n . When n = 1, the connectedness of D follows easily from the periodicity and a simple mean value property (Lemma 2.6in [7]).As for the double-well type Hamiltonian H ( p ) = min {| p − e | , | p + e |} in thetwo dimensional space, the following question is closely related to Question 2 andcounterexamples in [42, 21]. Question 4.
Assume that n = 2 and H ( p ) = min {| p − e | , | p + e |} for all p ∈ R ,where e = (1 , . Does there exist L > such that, if osc R × Ω V = ess sup Ω V (0 , ω ) − ess inf Ω V (0 , ω ) > L, then H − V is regularly homogenizable? Proof of Theorem 4.1.
As a demonstration, we only provide the proof ofTheorem 4.1 in details here. The extension to Theorem 4.3 is clear. Compared withthe proof for the special case in [6], the following proof is much clearer and simpler.We need the following comparison result.
Lemma 4.4.
Fix λ ∈ (0 , , R > . Suppose that u, v are respectively a viscositysubsolution and a viscosity supersolution to λw + H ( p + Dw ) − V = 0 in B (0 , R/λ ) . (4.3) Assume further that there exists
C > such that ( λ ( | u | + | v | ) ≤ C on B (0 , R/λ ) , | H ( p ) − H ( q ) | ≤ C | p − q | for all p, q ∈ R n . Then λ ( u − v ) ≤ C ( | x | + 1) / R + C R in B (0 , R/λ ) . Proof.
Let ˜ v ( x ) = v ( x ) + C ( | x | + 1) / R + C Rλ for x ∈ B (0 , R/λ ).Then, ˜ v is still a viscosity supersolution to (4.3) and furthermore, ˜ v ≥ u on ∂B (0 , R/λ ). Hence, the comparison principle yields ˜ v ≥ u in B (0 , R/λ ). (cid:3) Proof of Theorem 4.1.
Fix p ∈ R n . For λ >
0, let v λ ( y, p ) be the unique boundedcontinuous viscosity solution to λv λ + H ( p + Dv λ ) − V ( y ) = 0 in R n . (4.4)In order to prove Theorem 4.1, it is enough to show that P h lim λ → (cid:12)(cid:12) λv λ (0 , p ) + H ( p ) (cid:12)(cid:12) = 0 i = 1 . (4.5)Let us note first that, as H is quasiconvex and H is quasiconcave, H − V and H − V are regularly homogenizable (see [16, 4]). It is clear thatmax (cid:8) H , H (cid:9) ≤ H. (4.6)Once again, we divide our proof into few steps. Step 1.
Assume that H ( p ) ≥ max (cid:8) H ( p ) , (cid:9) . We proceed to show that P h lim λ → (cid:12)(cid:12) λv λ (0 , p ) + H ( p ) (cid:12)(cid:12) = 0 i = 1 . (4.7)Since H ≥ H , by the usual comparison principle, it is clear that P h lim inf λ → − λv λ (0 , p ) ≥ H ( p ) i = 1 . (4.8)It suffices to show that P (cid:20) lim sup λ → − λv λ (0 , p ) ≤ H ( p ) (cid:21) = 1 . (4.9)Let v λ ( y, − p ) be the viscosity solution to λv λ + H ( − p + Dv λ ) − V = 0 in R n . (4.10)Since H − V is regularly homogenizable, we get that, for any R > P (cid:20) lim λ → max y ∈ B (0 ,R/λ ) (cid:12)(cid:12) λv λ ( y, − p ) + H ( − p ) (cid:12)(cid:12) = 0 (cid:21) = 1 . (4.11)Fix R >
0. Pick ω ∈ Ω such that (4.11) holds. For each ε > λ ( R, ω, ε ) > λ < λ ( R, ω, ε ),max y ∈ B (0 ,R/λ ) (cid:12)(cid:12) λv λ ( y, − p, ω ) + H ( − p ) (cid:12)(cid:12) ≤ ε. Note that by inf-sup representation formula and the even property of H , we alsohave that H is also even, i.e., H ( − p ) = H ( p ). In particular, − λv λ ( y, − p, ω ) ≥ H ( p ) − ε ≥ − ε for y ∈ B (0 , R/λ ) . IN-MAX FORMULAS AND OTHER PROPERTIES OF H Due to the quasiconvexity of H , this implies that, for q ∈ D − v λ ( y, − p, ω ) for some y ∈ B (0 , R/λ ), H ( − p + q ) = − λv λ ( y, − p, ω ) + V ( y ) ≥ − ε. (4.12)In particular, | H ( − p + q ) − H ( − p + q ) | ≤ δ ε , where δ ε = max { H ( p ) − H ( p ) : H ( p ) ≥ − ε } .Denote by w = − v λ ( y, − p, ω ) − H ( p ) λ . Then w is a viscosity subsolution to λw + H ( p + Dw ) − V = 2 ε + δ ε in B (0 , R/λ ) . Hence w − ε + δ ε λ is a subsolution to (4.3). By Lemma 4.4, we get λw (0) − λv λ (0 , p, ω ) ≤ CR + 2 ε + δ ε . Hence, (4.9) holds. Compare this to Step 2 in the proof of Theorem 2.1 for similarity.
Step 2.
Assume that H ( p ) ≥ max (cid:8) H ( p ) , (cid:9) . We proceed in the same way as inStep 1 (except that we use v λ ( y, p ) instead of v λ ( y, − p ) because of the quasicon-cavity of H ) to get that P h lim λ → (cid:12)(cid:12) λv λ (0 , p ) + H ( p ) (cid:12)(cid:12) = 0 i = 1 . (4.13)Compare this to Step 3 in the proof of Theorem 2.1 for similarity. Step 3.
We now consider the case max (cid:8) H ( p ) , H ( p ) (cid:9) <
0. Our goal is to show P h lim λ → | λv λ (0 , p ) | = 0 i = 1 . (4.14)This step basically shares the same philosophy as Step 4 in the proof of Theorem2.1. Let us still present a proof here.Thanks to the assumption that ess inf Ω V (0 , ω ) = 0 and the fact that H ≥ P h lim inf λ → − λv λ (0 , p ) ≥ i = 1 . (4.15)We therefore only need to show P (cid:20) lim sup λ → − λv λ (0 , p ) ≤ (cid:21) = 1 . (4.16)For σ ∈ [0 , H i − σV are still regularly homogenizable for i = 1 ,
2. Let H σi bethe effective Hamiltonian corresponding to H i − σV . By repeating Steps 1 and 2above, we get that: ( For p ∈ R n and σ ∈ [0 , (cid:8) H σ ( p ) , H σ ( p ) (cid:9) = 0,then H − σV is regularly homogenizable at p and H σ ( p ) = 0 , (4.17)where H σ ( p ) is its corresponding effective Hamiltonian. Take p ∈ R n so thatmax (cid:8) H σ ( p ) , H σ ( p ) (cid:9) = 0 . (4.18) for some σ ∈ [0 , λ >
0, let v σλ be the viscosity solution to λv σλ + H ( p + Dv σλ ) − σV = 0 in R n , then by (4.17), P h lim λ → | λv σλ (0 , p ) | = 0 i = 1 . Since V ≥
0, the usual comparison principle gives us that v σλ ≤ v λ . Hence, (4.16)holds true.It remains to show that, if max (cid:8) H ( p ) , H ( p ) (cid:9) <
0, then (4.18) holds for some σ ∈ [0 , H ( p ) = H ( p ) = 0 for p ∈ ∂U , we only need to consider the case p / ∈ ∂U . There are two cases, either p ∈ U or p ∈ R n \ U . Again, it is enough toconsider the case that p ∈ U . For this p , we have that H ( p ) = H ( p ) > H ( p ) = H ( p ) < . By the continuity of σ H σ ( p ), there exists σ ∈ [0 ,
1] such that H σ ( p ) = 0. This,together with the fact that H σ ( p ) ≤ H ( p ) <
0, leads to (4.18). (cid:3)
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Department of Mathematics and Department of ComputationalMathematics, Science and Engineering, Michigan State University, East Lansing,MI 48824 , USA
E-mail address : [email protected] (Hung V. Tran) Department of Mathematics, University of Wisconsin Madison,Van Vleck hall, 480 Lincoln drive, Madison, WI 53706, USA
E-mail address : [email protected] (Yifeng Yu) Department of Mathematics, University of California, Irvine, 410GRowland Hall, Irvine, CA 92697, USA
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