existence and compactness for weak solutions to bellman...

Necas Center for Mathematical Modeling

Existence and compactness for weaksolutions to Bellman systems with

critical growth

A. Bensoussan, M. Bulıcek and J. Frehse

Preprint no. 2011-012

Research Team 1Mathematical Institute of the Charles University

Sokolovska 83, 186 75 Praha 8http://ncmm.karlin.mff.cuni.cz/

EXISTENCE AND COMPACTNESS FOR WEAKSOLUTIONS TO BELLMAN SYSTEMS WITH

CRITICAL GROWTH

ALAIN BENSOUSSAN, MIROSLAV BULICEK, AND JENS FREHSE

Abstract. We deal with nonlinear elliptic and parabolic systemsthat are the Bellman systems associated to stochastic differentialgames as a main motivation. We establish the existence of weaksolutions in any dimension for an arbitrary number of equations(“players”). The method is based on using a renormalized sub-and super-solution technique. The main novelty consists in thenew structure conditions on the critical growth terms with allowus to show weak solvability for Bellman systems to certain classesof stochastic differential games.

1. Introduction

We consider diagonal elliptic and parabolic systems arising from theBellman systems of stochastic differential games. It means that for theelliptic case we want to solve

(1) Luν + λνuν = Hν(·,u,∇u) for ν = 1, . . . , N

in an open bounded domain Ω ⊂ Rn, while in the parabolic case, weare interested in solving

(2) Dtuν + Luν + λνuν = Hν(·,u,∇u) for ν = 1, . . . , N

in a parabolic cylinder Q := (0, T )× Ω completed by the initial data

(3) uν(0) = u0ν in Ω for all ν = 1, . . . N.

2000 Mathematics Subject Classification. 35J60, 35K55, 35J55, 35B65.Key words and phrases. Stochastic games; Bellman equation; nonlinear elliptic

equations; weak solution; Hamiltonians; weak lower- and upper- stability; renor-malized sub- and super-solutions.

A. Bensoussan and J. Frehse thank DAAD and the Hong Kong Science Foun-dation RGC for the support obtained under the Germany/Hong Kong ResearchScheme. Miroslav Bulıcek thanks to Collaborative Research Center (SFB) 611 andto Jindrich Necas Center for Mathematical Modeling, the project LC06052 financedby MSMT for their support.

1

2 ALAIN BENSOUSSAN, MIROSLAV BULICEK, AND JENS FREHSE

Here, u := (u1, . . . , uN) is the unknown function (and N ∈ N+) and theelliptic operator L is given as (here we are using the notation Di := ∂

∂xi

for all i = 1, . . . , n and Dt := ∂∂t

)

(4) L := −n∑

i,j=1

Di(aij(t, x)Dj).

In order to simplify the presentation of the assumptions consideredfor Hν , we denote U := Ω for the elliptic case and U := Q for theparabolic one. Similarly, we use the variable z := x for elliptic caseand z := (t, x) for parabolic one and in addition we reserve the symbol∇ as the gradient with respect to the spatial variable x, i.e.,

(∇u)iν :=∂uν∂xi

for all i = 1, . . . , n and all ν = 1, . . . , N.

Using this notation we introduce the assumptions on the data of (1)and/or (2). First, for the elliptic operator L, we assume that

aij ∈ L∞(U),(5)

c1|y|2 ≤n∑

i,j=1

aik(z)yiyj for all y ∈ Rn and a.a. z ∈ U(6)

with some c1 > 0. Next, in order to guarantee the L∞ bound on u, werequire that for all ν = 1, . . . , N

(7)λν ≥ 0 if U = Q,

λν > 0 if U = Ω.

The terms on the right hand sides of (1) and (2), usually called Hamil-tonians, are assumed to satisfy for all ν = 1, . . . , N and some K > 0

Hν : (U,RN ,Rn×N)→ R is Caratheodory mapping,(8)

|Hν(z,u, p)| ≤ K|p|2 +K for all u ∈ RN and all p ∈ Rn×N .(9)

First, we explain the difficulties connected to the presence of possi-bly quadratic terms on the right hand sides of (1) and (2): It is wellknown that for elliptic and/or parabolic systems of the form (1), (2)respectively with the critical growth (9) in the lower order terms Hν

one gets specific analytical difficulties concerning the regularity theoryand even the existence of a weak solution. The main reason is, that,in general, solutions to systems of type (1) and (2) fail to be Holdercontinuous and fail to satisfy corresponding a priori estimates. Evenscalar equations (i.e., N = 1) may have an unbounded solution, e.g.the equation

−4u = |∇u|2 in Be−1(0) ⊂ R2

WEAK SOLUTIONS TO BELLMAN SYSTEMS 3

subject to homogeneous Dirichlet data has a weak solution

u = ln | ln |x|| ∈ H10 (Be−1(0)).

However, in the scalar case, the theory of Ladyzhenskaya and Uralt-seva [21] (see also [18]) yields Cα regularity for bounded solutions inthe elliptic case, see the similar result for the parabolic case [20]. So,motivated by [21] we focus here only on bounded solutions to (1) or(2). However, for diagonal systems one can construct simple exampleswith a bounded, but discontinuous solution. We refer the reader to[12], where it is shown that

u1 = sin ln | ln |x||, u2 = cos ln | ln |x||solve

−4u1 = b1(x)|∇u|2,−4u2 = b2(x)|∇u|2

for certain L∞ functions b1 and b2. Another example of non-regularsolution occurs in the theory of harmonic mappings, where for say n =N = 3, u := x

|x| solves the problem −4u = b|∇u|2 with b := u = x|x| ,

see [19]. On the other hand, there are many cases where uniform Cαestimates are available. The typical example of such a system is theBellman system in game theory, where for the Hamiltonians satisfying

|Hν(z,u, p)| ≤ K|pν ||p|+K for all ν = 1, . . . , N,

or for even more general case

|Hν(z,u, p)| ≤ K|pν ||p|+ν∑

µ=1

|pµ|2 +K for all ν = 1, . . . , N.

It is then possible to obtain uniform Cα bounds (see [5]). As one cansee from the above examples, problems concerning Cα-estimates arise,if the full gradient appears in all Hamiltonians Hν with the critical(quadratic) power. Progress concerning this question has recently beendone in [1, 8, 7] for elliptic case and two players, in [2] for two playersand parabolic case and finally for N players in the elliptic case in[10]. A classical case concerning right hand sides uF0(z,u,∇u), F0

scalar and F0 ≤ 0 and related generalization is studied in [25], see alsofurther results in [24, 23] and for more general results in Lp setting see[22]. Finally, for the problems with the Hamiltonians above combinedwith the presence of the term uF0(z,u,∇u) on the right-hand side,which is the typical case in Belmann systems with discount control, theregularity of the solution was recently established in [5] for dimension


two in the elliptic case. The parabolic case and the case N > 2 howeverremain unsolved.

The Cα estimates are usually used for proving the existence of a weaksolution to (1), (2) respectively. In fact, if we approximate the system(1) by (we focus here only on the elliptic case, but the same schemecan be applied also onto the parabolic setting)

(10) Luν + λνuν =Hν(z,u,∇u)

1 +m−1|∇u|2for ν = 1, . . . , N,

one has to show that a subsequence of the approximate solutions con-verges weakly in H1(Ω;RN) to a solution of the primal system. Forthis, one first needs the uniform H1 estimates for um, which are notavailable in general but are usually guaranteed by an additional struc-ture hypothesis. Having such estimates, it is well known to prove, fora subsequence, a.e. convergence of ∇um and consequently the strongconvergence in any W 1,p(Ω;RN) for any p ∈ [1, 2). Concerning the a.econvergence of gradients, we recommend to use [22, Lemma in §4] Analternative approach can be found in [14], which is based on the useof local capacities. Moreover, for non-diagonal principal parts allowingregularity theory one can work using a dual equation. However, it isstill not sufficient to pass to the limit with terms on the right handside, since they have quadratic growth in gradients and a justificationof the limit procedure needs the strong convergence in H1(Ω;RN), see[15] for an example of a sequence of solutions to a diagonal elliptic sys-tems with the critical growth on the right hand side, which does notconverge in H1. An easy way to show such a convergence result followsfrom uniform Cα bounds (if they are available). Indeed, having suchbounds, it is easy to deduce the strong convergence of um in L∞ andconsequently by monotonicity argument one can observe∫

Ω

aijDiumν Dj(u

mν − uν) dx

=

∫Ω

λνuν(uν − umν ) +Hν(x,u

n,∇um)

1 +m−1|∇um|2(umν − uν) dx

m→∞→ 0.

Finally, due to the ellipticity of aij one can easily conclude the strongconvergence of um in H1(Ω;RN).

An alternative method for proving the existence of a weak solutionfor certain diagonal systems, not based on Cα estimates, relies on acombination of exponential domination effects with the monotonicitymethod and/or the Fatou lemma which have been first used in [13].However, the conditions in [13] are very similar to those needed for Cαestimates. In fact the conditions assumed there also do not contain


those of the present paper, and in the case treated here Cα estimateshave not been worked out yet.

From this point of view, the result of this paper is of real interest.Our conditions cover examples of Hamiltonians like (see the assump-tions (13)–(17) or (18) that replace (17))

(11) Hν(z,u,∇u) = aν |∇uν |2 − bν |∇u|2 +∇uν ·G(z,u,∇u) + fν(z)

with aν , bν > 0 satisfying∑

ν bν < minν aν . As we can see, we allowthe presence of the full gradient |∇u|2 in all Hamiltonians Hν given by(11), without smallness restriction.

Next, we formulate the assumptions imposed on the structure ofHamiltonians Hν that go beyond those presented above and that coverthe case (11). Moreover, they are motivated by Bellman systems ofstochastic differential games (see the discussion below) and they arestrongly related to those introduced in [10] as far as the structure con-ditions are concerned. First, we require a decomposition

(12) Hν(z,u, p) = H0ν (z,u, p) + pν ·G(z,u, B(z, p)) + fν(z).

Second, for such a decomposition we assume that for all ν = 1, . . . , Nthe mappings H0

ν , G and B are Caratheodory and satisfy for all ν =1, . . . , N , all u ∈ Rn, all p ∈ Rn×N and almost all z ∈ U the followinginequalities

|B(z, p)| ≤ K|p|+K,(13)

|G(z,u, y)| ≤ K|y|+K,(14)

|fν(z)| ≤ K,(15)

N∑ν=1

H0ν (z,u, p) ≥ c0|B(z, p)|2 −K,(16)

H0ν (z,u, p) ≤ K|pν |2 +K,(17)

with some constant c0 > 0. For B(z, p) one may think of B(z, p) =(p1, . . . , pN) however in applications one usually has a degeneration likeB(z, p) = (A1p1, . . . , ANpN) with non-necessarily quadratic matrices Aicausing degeneration.

Our main goal in the paper is to establish the existence of a weaksolution u ∈ L∞(U ;RN) with ∇u ∈ L2(U ;Rn×N) to (1)–(4) providedthat the Hamiltonians satisfy (9)–(15). Moreover, we want to presentan alternative proof that does not rely on proving Cα-regularity andit should be understood as one of the main novelties of the paper.Indeed, the same result was proved in [10] but with a very complicatedproof. Here we present a relatively simple proof. Moreover, we are


able to cover also the parabolic theory and therefore extend the theorypresented in [10]. Finally, due to the simplicity of the method we cango really beyond the result of [10] and extend the theory in a significantway. To be more precise, we are able to replace (17) by the followingweaker assumption

(18) H0ν (z,u, p) ≤ K|pν |2 +K|pν ||p|+K.

The assumption (18) allows us to cover more general situations. Tosupport this we briefly explain what kind of Hamiltonians usually ap-pear in the Bellman systems of differential games.

We refer the reader to the usual derivation of the Bellman systemsof stochastic differential games in [17, 11, 9] (see also [3, 7]). In thesepapers the Hamiltonians Hν are derived from the Lagrange functions

Lν(z, v, p) = lν(z, v) + pν ·G(z, v) + fν(z),

where G is associated to the dynamics of the stochastic differentialequations. The ν-th player chooses the control function vν accordingto an optimal strategy which leads to an equilibrium control v∗ =v∗(z, p). Depending on the rules of the game this might lead to aNash equilibrium, Stackelberg equilibrium, or also other choices arepermitted. The Hamiltonians Hν are then defined by

(19) Hν(z, p) := L(z, v∗(z, p), p).

When applying our main theorem, the quantity B(z, p) corresponds toB(z, p) = v∗(z, p) and so one sees that the condition (16) follows from a(reasonable) condition for the quantities lν(z, v) in the cost functional,namely the condition

(20)N∑ν=1

lν(z, v) ≥ c0|v|2 −K.

The upper condition (17) follows when the optimal strategies v∗ satisfycondition

(21) Lν(z, v∗, p) ≤ Lν(z, v

∗1, . . . , v

∗ν−1, 0, v

∗ν+1, . . . , v

∗ν),

which is very reasonable from the game theoretic point of view in thecase of Nash equilibrium, and if

(22) lν(z, v∗1, . . . , v

∗ν−1, 0, v

∗ν+1, . . . , v

∗ν) ≤ K,

which is, e.g., satisfied for quadratic forms

lν = vTν

N∑j=1

Qjvj


and related generalizations. In [7] the games with the conditions (21)–(22) were treated for dimension n = 2. Therefore, we are able now toget the existence result in arbitrary dimension n.

We close the introduction by stating the main results. First, weconsider the elliptic problem (1) subject to the homogeneous Neumannboundary condition.

Theorem 1.1 (Neumann-elliptic). Let Ω be an open bounded set withLipschitz boundary. Let L be given by (4) and satisfy (5)–(6), and letthe coefficient λν satisfy (7). In addition, assume that HamiltoniansHν satisfy (8)–(9), (12)–(16) and (18). Then there exists u ∈ L∞ ∩H1(Ω;RN) solving (1) in the following sense∫

Ω

n∑i,j=1

aij(·)DjuνDjψ dx+ λν

∫Ω

uνψ dx =

∫Ω

Hν(·,u,∇u)ψ dx

for all ψ ∈ L∞ ∩H1(Ω) and all ν = 1, . . . , N.

(23)

Second, we state the existence result for the parabolic problem (2)–(3), again subject to the homogeneous Neumann boundary data.

Theorem 1.2 (Neumann-parabolic). Let Ω be an open bounded setwith Lipschitz boundary and T > 0 be given. Let L be given by (4)and satisfy (5)–(6). Assume that the coefficient λν satisfy (7) andHamiltonians Hν satisfy (8)–(9), (12)–(16) and (18). Finally, let theinitial data u0 : Ω→ RN fulfill

(24) ‖u0‖L∞(Ω;RN ) ≤ K.

Then there exists u ∈ L∞(0, T ;L∞(Ω;RN))∩L2(0, T ;H1(Ω;RN)) solv-ing (2)–(3) in the following sense

−∫Q

uνDtψ dx dt+

∫Q

n∑i,j=1

aij(·)DjuνDjψ dx dt+ λν

∫Q

uνψ dx dt

=

∫Q

Hν(·,u,∇u)ψ dx dt+

∫Ω

u0νψ(0) dx

for all ψ ∈ D(−∞, T ;L∞ ∩H1(Ω)) and all ν = 1, . . . , N.

(25)

The next theorem establishes the existence of a weak solution for theDirichlet stationary problem.

Theorem 1.3 (Dirichlet-elliptic). Let Ω be an open bounded set. LetL be given by (4) and satisfy (5)–(6), the coefficient λν satisfy (7)and Hamiltonians Hν satisfy (8)–(16) and (18). Then there exist u ∈


L∞ ∩H1loc(Ω;RN) solving (1) in the following sense∫

Ω

n∑i,j=1

aij(·)DjuνDjψ dx+ λν

∫Ω

uνψ dx =

∫Ω

Hν(·,u,∇u)ψ dx

for all ψ ∈ D(Ω) and all ν = 1, . . . , N.

(26)

Moreover, the boundary data is attained in the following sense1

(27)

(N∑ν=1

σνuν

)+

∈ H10 (Ω),

(N∑ν=1

uν

)−

∈ H10 (Ω)

for all σν 1 with ν = 1, . . . , N .

Note that (27) is a generalized notion of attainment of the homo-geneous Dirichlet data introduced in [10]. Indeed, once having thatu ∈ H1(Ω;RN) then (27) directly implies that u ∈ H1

0 (Ω;RN). Fi-nally, for unsteady systems of the form (1), the main results are thefollowing.

Theorem 1.4 (Dirichlet-parabolic). Let Ω be an open bounded set andT > 0 be given. Let L be given by (4) and satisfy (5)–(6). Assume thatthe coefficient λν satisfy (7) and Hamiltonians Hν satisfy (8)–(16) and(18). Finally, let the initial data u0 : Ω→ RN fulfill (24). Then thereexists u ∈ L∞(0, T ;L∞(Ω;RN)) ∩ L2(0, T ;H1

loc(Ω;RN)) solving (1) inthe following sense

−∫Q

uνDtψ dx dt+

∫Q

n∑i,j=1

aij(·)DjuνDjψ dx dt+ λν

∫Q

uνψ dx dt

=

∫Q

Hν(·,u,∇u)ψ dx dt+

∫Ω

u0νψ(0) dx

for all ψ ∈ D(−∞, T ;D(Ω)) and all ν = 1, . . . , N.

(28)

Moreover, the boundary data is attained in the following sense

(29)

(N∑ν=1

σνuν

)+

∈ L2(0, T ;H10 (Ω)),

(N∑ν=1

uν

)−

∈ L2(0, T ;H10 (Ω))

for all σν 1 with ν = 1, . . . , N .

The paper is structured as follows. In Section 2 we show that forany sequence um of solutions to (23) and (25), respectively, thatis bounded in the corresponding spaces we can find a weak limit uthat satisfies (23) and (25), respectively, but with the inequality sign“ ≤ ”. We provide two proofs. The first one is related to the case when

1Here a+ := max(0, a) and a− := min(0, a).


Hamiltonians Hν satisfy the restriction (17) assumed in [10]. Then wegeneralize the proof so that we are able to replace (17) by the weakerone (18). The main reason for such a splitting is that we want to presentan alternative simpler proof of Theorem1.1 that was also proved in [10],provided that (17) holds. Next, in Section 3 we show that for the samesequence of um we can sum (23) and (25), respectively with respectto ν and we show that in the limit we can replace the equality signin the sum by the inequality sign “ ≥ ”. Finally, in Section 4 weconclude that for any sequence um the weak limit satisfies (23) and(25), respectively, and finish the paper with the proof of Theorem 1.1and Theorem 1.2. The proof for the homogeneous Dirichlet data, i.e.localized version (Theorem 1.3 and Theorem 1.1) is not outlined here,since the difference consists only in replacing the test function by adifferent one having compact support. The only essential difference isthe attainment of the Dirichlet data (properties (27), (29) respectively),however for such estimates we refer the interested reader to [10].

2. Weak lower-stability

This section is devoted to the proof of the weak lower-stability of (1),(2) respectively. Thus, for the elliptic problem subject to Neumannboundary conditions, the main statement is the following.

Lemma 2.1 (Neumann-elliptic). Let Ω ⊂ Rn be an open bounded Lips-chitz domain. Assume that there is a sequence umm∈N and a sequenceHm

ν such that um u weakly in H1(Ω;RN), ‖um‖∞ ≤ K for allm ∈ N, Hamiltonians Hm

ν satisfy (8)–(9), (12)–(16), (18) and

(30) Hmν → Hν strongly in L∞(Ω; C(RN × Rn×N)).

In addition assume that for all m ∈ N the unknown um and the Hamil-tonians Hm

ν solve

(31)

∫Ω

n∑i,j=1

aij(·)Diumν Djψ + λνu

mν ψ dx =

∫Ω

Hmν (·,um,∇um)ψ dx

for all ψ ∈ L∞ ∩H1(Ω) and all ν = 1, . . . , N. Then u and Hν satisfy

(32)

∫Ω

n∑i,j=1

aij(·)DiuνDjψ + λνuνψ dx ≤∫

Ω

Hν(·,u,∇u)ψ dx

for all ν = 1, . . . , N and all nonnegative ψ ∈ L∞ ∩H1(Ω).

Since Lemma 2.1 deals with the Neumann boundary data, we alsoformulate the localized version that is suitable for proving the existence


result for the Dirichlet problem. Here, we only state the result withoutproof because it is basically the same as the proof of Lemma 2.1.

Lemma 2.2 (local stability-elliptic). Let Ω ⊂ Rn be an open boundeddomain. Assume that there is a sequence umm∈N and a sequenceHm

ν such that um u weakly in H1(Ω;RN), ‖um‖∞ ≤ K for allm ∈ N, Hamiltonians Hm

ν satisfy (8)–(9), (12)–(16), (18) and (30). Inaddition assume that for all m ∈ N the unknown um and the Hamilto-nians Hm

ν solve

(33)

∫Ω′

n∑i,j=1


mν ψ dx =

∫Ω′Hmν (·,um,∇um)ψ dx

for all ψ ∈ L∞ ∩ H10 (Ω′), all Ω′ ⊂⊂ Ω and all ν = 1, . . . , N. Then u

and Hν satisfy

(34)

∫Ω′

n∑i,j=1

aij(·)DiuνDjψ + λνuνψ dx ≤∫

Ω′Hν(·,u,∇u)ψ dx

for all for all ν = 1, . . . , N , all nonnegative ψ ∈ L∞ ∩H10 (Ω′) and all

Ω′ ⊂⊂ Ω.

Finally, we formulate the weak lower-stability result for the parabolicsystem (2)–(3).

Lemma 2.3 (Neumann-parabolic). Let Ω ⊂ Rn be an open boundedLipschitz domain and T > 0 be given. Assume that there is a se-quence umm∈N and a sequence Hm

ν such that um u weakly inL2(0, T ;H1(Ω;RN)), ‖um‖∞ ≤ K for all m ∈ N, Hamiltonians Hm

ν

satisfy (8)–(16), (18) and

(35) Hmν → Hν strongly in L∞(Q; C(RN × Rn×N)).

In addition assume that for all m ∈ N the unknown um and the Hamil-tonians Hm

ν solve

−∫Q

umν Dtψ dx dt+

∫Q

n∑i,j=1


mν ψ dx dt

=

∫Q

Hmν (·,um,∇um)ψ dx dt

(36)


for all ψ ∈ W 1,10 (0, T ;L∞ ∩H1(Ω)) and all ν = 1, . . . , N. Then u and

Hν satisfy

−∫Q

uνDtψ dx dt+

∫Q

n∑i,j=1

aij(·)DiuνDjψ + λνuνψ dx dt

≤∫Q

Hν(·,u,∇u)ψ dx dt

(37)

for all nonnegative ψ ∈ W 1,10 (0, T ;L∞ ∩H1(Ω)) and all ν = 1, . . . , N.

The same statement holds also locally (in Ω) but since we describedit in a detail for the elliptic case, we omit it here for the parabolic case.

Next, in order to simplify the presentation, we introduce severalabbreviations used later in this section. For any real θνNν=1, λ and σand any vector u ∈ RN we define

w(θ1, . . . , θN , λ,u) :=N∑ν=1

θνeλuν ,

exp(θ1, . . . , θN , λ,u, σ) := eσw(θ1,...,θN ,λ,u).

(38)

Moreover, if it is clear from the context we use the notation exp :=exp(θ1, . . . , θN , λ,u, σ) and w := w(θ1, . . . , θN , λ,u).

Finally, we focus on tools/inequalities that are sufficient to proveabove weak lower-stability results. The main ingredient is that theweak limit u satisfies a kind of a renormalized inequality. First, weformulate it under the assumption (17) (that was supposed in [10]and for which we would like to provide a simpler proof) and then wegeneralize the result in order to cover also the structure assumption(18). Moreover, we do not formulate explicitly the local stability resultsince it can be proved in the same way and focus only on the Neumannboundary data.

Lemma 2.4 (Neumann-elliptic-renormalized I). Let all assumptionsof Lemma 2.2 be satisfied. Moreover, assume2 that Hamiltonians Hm

ν

satisfy (17). Assume that θνNν=1 and λ > 0 are arbitrary such that

(39) 0 < θν ≤ 1 and λ ≥ 2K

c1

.

Define Θ := minν=1,...,N θνe−Kλ and assume that σ > 0 be arbitrary

such that

(40) σ ≥ NK2

Θc21λ

2.

2In fact we do need to assume (16) here.


Then u, w and exp (defined in (38)) satisfy∫Ω

n∑i,j=1

aij

N∑ν=1

DjuνDi

(ϕθνe

λuν exp)

+N∑ν=1

ϕλνuνθνeλuν exp dx

≤∫

Ω

ϕN∑ν=1

Hνθνeλuν exp dx

(41)

for all nonnegative ϕ ∈ L∞ ∩H1(Ω).

Finally, the next result covers the more general situation when (17)is not needed and is replaced by (18).

Lemma 2.5 (Neumann-elliptic-renormalized II). Let all assumptionsof Lemma 2.2 be satisfied and let N = 2. Assume that θν2

ν=1 > 0 andλ > 0 are arbitrary such that

λ ≥ 32K

c1

,

min

θ1

θ2

,θ2

θ1

≤ c1λ

8Ke2λK.

(42)

Define Θ := minν=1,...,2 θνe−Kλ and assume that σ > 0 is arbitrary such

that

(43) σ ≥ 16K2

Θc21λ

2.

Then u, w and exp (defined in (38)) satisfy∫Ω

n∑i,j=1

aij

2∑ν=1

DjuνDi

(ϕθνe

λuν exp)

+2∑

ν=1

ϕλνuνθνeλuν exp dx

≤∫

Ω

ϕ2∑

ν=1

Hνθνeλuν exp dx

(44)


Finally, we end up this part of the paper with the renormalizedinequalities for the parabolic case. For simplicity we restrict ourselvesto the more difficult case when (18) is assumed and provide the prooffor only two players. (Note that we could also provide a “parabolic”version of Lemma 2.4 but since the proof is the same we again donot present it here.) The proof for N players is however similar andrequires use of the iterative exponentials and therefore we omit it here.Since the iterative exponentials scheme can be found in previous worksof Bensoussan and Frehse (in a context of the regularity of solution),


we give here only a brief description where the interested reader canfind the corresponding procedure. The proof for the case of for twoplayers is based on [1], see also [3]. The case N = 3, together with anoutlook to more than three players was first presented in [16], whereone uses a triple iteration of exponential as a test function. A simplediscussion of the case N = 3 was repeated in [6]. The general case ofN players was worked out in the parabolic case, but the argument isthe same in the elliptic case, this was done in [4]. We recommend thistext for the reader concerning the general case, where N -times iteratedexponentials of the solution was used as a test function.

Lemma 2.6 (Neumann-parabolic-renormalized). Let all assumptionsof Lemma 2.3. Assume that θν2

ν=1 > 0 and λ > 0 be arbitrary suchthat (42) holds. Define Θ := minν=1,2 θνe

−Kλ and assume that σ > 0be arbitrary satisfying (43). Then u satisfies∫

Q

− 1

λσexpDtϕ+

n∑i,j=1

aij

2∑ν=1

DjuνDi

(ϕθνe

λuν exp)dx dt

+

∫Q

2∑ν=1

ϕλνuνθνeλuν exp dx dt ≤

∫Q

ϕ2∑

ν=1

Hνθνeλuν exp dx dt

(45)

for all nonnegative ψ ∈ W 1,10 (0, T ;L∞ ∩H1(Ω)).

2.1. Proof of Lemma 2.1 - easier case. In this subsection we givethe proof of Lemma 2.1 that will be based on using of Lemma 2.4.

Proof of Lemma 2.1 - easier version. To show (32) we use (41). Thus,

for arbitrary positive θνNν=1, we define λ := 2KC1

and σ := NK2

ΘC21λ

2 . With

this setting we see that all assumptions of Lemma 2.4 are satisfied andtherefore we know that u satisfies (41). Then we set ϕ := η(exp)−1 in(41) with arbitrary nonnegative η ∈ L∞ ∩ H1(Ω) to get (such settingis possible since u ∈ L∞ ∩H1(Ω;RN))

N∑ν=1

∫Ω

n∑i,j=1

aijDjuνDi

(ηθνe

λuν)

+ ηλνuνθνeλuν dx

≤N∑ν=1

∫Ω

ηHνθνeλuν dx.

(46)

The most important property of (46) is that there is no dependence onexp and consequently on σ. Moreover, since λ does not depend on thechoice of θν we set θν0 = 1 for some fixed ν0 and for others ν 6= ν0 weset ν := k−1. Thus, letting k → ∞ we observe by using the Lebesgue


dominated convergence theorem that for any ν0 = 1, . . . , N (ν0 wasfixed but arbitrary)∫

Ω

n∑i,j=1

aijDjuν0Di

(ηeλuν0

)+ ηλν0uν0e

λuν0 dx

≤∫

Ω

ηHν0eλuν0 dx.

(47)

Finally, setting η := ψe−λuν0 in (47) (such setting is again possible) wededuce (32).

2.2. Proof of Lemma 2.1 - general case. Here, we prove Lemma 2.1in the most general setting and the proof relies on Lemma 2.5.

Proof of Lemma 2.1 - general. The proof is very similar to the one inthe preceding subsection. For simplicity, we prove (32) for ν = 1. Forother ν’s the proof is similar. Hence, we set λ := 32K

c1, θ2 := 1 and let

θ1 ≥ 8Ke2λKλ−1c−11 be arbitrary. Next, Θ is defined by the relation in

Lemma 2.5 and σ is finally given by

(48) σ :=16K2

Θc21λ

2.

Hence, we see that all assumptions of Lemma 2.5 are satisfied andtherefore (44) holds. Thus, setting ϕ := η(exp)−1 in (44) with arbitraryη ∈ L∞ ∩H1(Ω) (such setting is possible since u ∈ L∞ ∩H1(Ω;RN))we obtain∫

Ω

n∑i,j=1

aij

2∑ν=1

DjuνDi

(ηθνe

λuν)

+2∑

ν=1

ηλνuνθνeλuν dx

≤∫

Ω

η2∑

ν=1

Hνθνeλuν dx.

(49)

Next, dividing the result by θ1 and letting θ1 →∞ we get∫Ω

n∑i,j=1

aijDju1Di

(ηeλu1

)+ ηλ1u1e

λu1 dx ≤∫

Ω

ηH1eλu1 dx.(50)

Consequently, setting η := ψe−λu1 we end up with (32) for 1. Since 1was chosen arbitrarily we see that (32) holds for all ν = 1, 2.


2.3. Proof of Lemma 2.3. Here, we deal with the weak lower-stabilityin the parabolic setting. Since we already described differences betweenusing the assumption (17) and (18) we prove here the result in the fullgenerality with the only restriction considering two players (but withstraightforward generalization onto N players). Of course, one canalso follow the elliptic case and give two proofs (the easier one and thegeneral one) assuming (17) and (18) respectively.

Proof of Lemma 2.3. The proof follows the ideas from the elliptic case.Hence, we set λ := 32K

c1, θ2 := 1 and let θ1 > 8Kλ−1c−1

1 e2λK be ar-

bitrary. Then Θ is defined as in Lemma 2.5 and σ is given by (48).So all assumptions of Lemma 2.6 are satisfied and we see that (45)holds. Again as in the elliptic case we want to set ϕ := η(exp)−1 in(45). Since we do not have enough regularity of exp in time directionwe need to proceed more carefully. So, let κε be a standard regulariza-tion kernel of radius ε depending only on the time variable. Then, wedenote expε := κε ∗ exp, where convolution is taken only in the timevariable. Next, for arbitrary δ > ε, we set ϕ := κε ∗ (η(expε)−1) in(45) with η ∈ W 1,1

0 (δ, T − δ;L∞ ∩ H1(Ω;RN)). Now such a setting ispossible due to the time regularization. Next, we let ε → 0+. In allterms except the one with time derivative, we can easily pass to thelimit, so we discuss only the term with the time derivative. Hence, wehave

− 1

λσ

∫Q

expDt(κε ∗ (η(expε)−1)) dx dt

= − 1

λσ

∫Q

expεDt(η(expε)−1) dx dt

=1

λσ

∫Q

Dt expε η(expε)−1 dx dt

=1

λσ

∫Q

Dt ln(expε)η dx dt = − 1

λσ

∫Q

ln(expε)Dtη dx dt

ε→0+→ − 1

λσ

∫Q

ln(exp)Dtη dx dt = −1

λ

∫Q

wDtη dx dt.

Hence we see that (45) reduces to

−1

λ

∫Q

wDtη dx dt+

∫Q

n∑i,j=1

aij

2∑ν=1

DjuνDi

(ϕθνe

λuν)dx dt

+

∫Q

2∑ν=1

ϕλνuνθνeλuν dx dt ≤

∫Q

ϕ2∑

ν=1

Hνθνeλuν dx dt.

(51)


Next, dividing (51) by θ1 and letting θ1 →∞ we deduce

−1

λ

∫Q

eλu1Dtη dx dt+

∫Q

n∑i,j=1

aijDju1Di

(ηeλu1

)dx dt

+

∫Q

ηλ1u1eλu1 dx dt ≤

∫Q

ηH1eλu1 dx dt

(52)

for all η ∈ W 1,10 (0, T ;L∞ ∩ H1(Ω)). Thus, finally setting η := ψe−λu1

and repeating the regularization procedure once again we end up with(37). So the proof is complete.

2.4. Proof of Lemma 2.4. In this subsection we give a detailed proofof Lemma 2.4, where the restrictive assumption (17) is assumed.

Proof of Lemma 2.4. First, having uniform bound for um in H1 we canuse the assumptions on Hamiltonians (9) to get that

(53) ‖Hmν (z,um,∇um)‖1 ≤ K for all ν = 1, . . . , N.

Then, using the standard theory of diagonal elliptic systems with L1

right hand side it is not hard to deduce that (see [22])

(54) um → u strongly in W 1,p(Ω;RN) for all p ∈ [1, 2).

Consequently, using (30) we observe (for a not relabeled subsequence)that for all ν = 1, . . . , N

(55) Hmν (·,um,∇um)→ Hν(·,u,∇u) a.e. in Ω.

Note that (54)–(55) is not enough to pass to the limit on the right handside of (31) due to the quadratic growth of Hm

ν and for such a limitprocedure we would need to guarantee (54) with p = 2 which does nottake place in general. Therefore in what follows we use the structureassumptions on Hamiltonians to get the inequality (41).

Thus, let θν , λ and σ be given parameters satisfying (39)–(40) and letϕ ∈ L∞∩H1(Ω) be arbitrary nonnegative function. First, we define ab-breviations wm := w(θ1, . . . , θN , λ,u

m, σ) and expm := exp(θ1, . . . , θN , λ,um, σ)

and set

(56) ψ := ϕθνeλumν expm


in the ν-th equation of (31). Next, we take the sum over all ν =1, . . . , N and get the following identity

0 =N∑ν=1

∫Ω

n∑i,j=1

aijDiϕDjumν θνe

λumν expm dx

+

∫Ω

ϕλνumν θνe

λumν expm dx

+

∫Ω

ϕn∑

i,j=1

aijDjumν Di

(θνe

λumν expm)dx

−∫

Ω

ϕHmν θνe

λumν expm dx

=:

4∑`=1

∫Ω

T`(um) dx.

(57)

Our main goal is to show that

(58) lim infm→∞

4∑`=1

∫Ω

T`(um) dx ≥

4∑`=1

∫Ω

T`(u) dx.

Indeed, having (58) we can let m → ∞ in (57) to observe (41). First,using the uniform L∞ estimates on un and the weak convergence ofun in H1(Ω;RN), combined with the compact embedding, it is easy todeduce that

(59) limm→∞

∫Ω

T1(um) + T2(um) dx =

∫Ω

T1(u) + T2(u) dx.

Moreover, using (54)–(55) we get (for a subsequence) that for ` = 3, 4

(60) T`(um)→ T`(u) a.e. in Ω.

Therefore, once we prove that there exists some K > 0 such that forall m ∈ N and almost all x ∈ Ω

(61) T3(um) + T4(um) ≥ −K,

we can use Fatou’s lemma to observe that

(62) lim infm→∞

∫Ω

T3(um) + T4(um) dx ≥∫

Ω

T3(u) + T4(u) dx,

which combined with (59) gives (58). Thus, it remains to prove (61).First, we start with T3. Using (6) we can deduce that for any v ∈


H1(Ω;RN) such that ‖v‖∞ ≤ K we have a.e. in Ω

T3(v) = ϕN∑ν=1

n∑i,j=1

aijDjvνDi

(θνe

λvν exp(v))

= ϕλN∑ν=1

θν

n∑i,j=1

aijDjvνDivνeλvν exp(v)

+ ϕσ

N∑ν=1

n∑i,j=1

aijDjvνθνeλvνDiw(v) exp(v)

= ϕλ exp(v)N∑ν=1

θνeλvν

n∑i,j=1

aijDjvνDivν

+ ϕσλ−1 exp(v)n∑

i,j=1

aijDjw(v)Diw(v)

≥ c1ϕλ exp(v)N∑ν=1

θνeλvν |∇vν |2

+ c1ϕσλ−1 exp(v)|∇w(v)|2 =: Y1(v) + Y2(v).

(63)

Next, we want to show that T3 really dominates T4 provided that thecoefficients satisfy (39)–(40). First, using the decomposition (12) wesee that

−T4(v) =N∑ν=1

ϕHν(v,∇v)θνeλvν exp(v)

=N∑ν=1

ϕH0ν (v,∇v)θνe

λvν exp(v)

+N∑ν=1

ϕ∇vν ·G(v,∇v)θνeλvν exp(v)

+N∑ν=1

ϕfνθνeλvν exp(v) =: I1(v) + I2(v) + I3(v).

(64)

First, using (15) and the bound |v| ≤ K, we see that

(65) I3(v) ≤ K(K, θ1, . . . , θN , λ, σ, ϕ).


Next, to estimate I1(v) we use (17) and the bound |v| ≤ K to observethat

I1(v) ≤ ϕKN∑ν=1

(1 + |∇vν |2)θνeλvν exp(v)

≤ K(K, θ1, . . . , θN , λ, σ, ϕ) +Kϕ exp(v)N∑ν=1

θν |∇vν |2eλvν

= K(K, θ1, . . . , θN , λ, σ, ϕ) +K

c1λY1(v),

(66)

where for the last inequality we used the definition of Y1(v) given in(63). Finally, to estimate I2 we use Young’s inequality, the definition ofY2(v) (see (63)), the assumptions (13)–(14) and the fact that |v| ≤ Kto get (here again K := K(K, θ1, . . . , θN , λ, σ, ϕ))

I2(v) = ϕ exp(v)λ−1∇w(v) ·G(v,∇v)

≤ ϕ exp(v)λ−1|∇w(v)||G(v,∇v)|≤ 2−1Y2(v) + ϕ(c1σλ)−1 exp(v)|G(v,∇v)|2

≤ 2−1Y2(v) + K +K2ϕ(c1σλ)−1 exp(v)|∇v|2

≤ 2−1Y2(v) + K +NΘ−1(c1λ)−2K2σ−1Y1(v).

(67)

Hence, combining finally (63)–(67) we find that

T3(v) + T4(v) ≥ Y1(v) + Y2(v)− I1(v)− I2(v)− I3(v)

≥ −K + Y1(v)(1−K(c1λ)−1 −NΘ−1(c1λ)−2K2σ−1

)≥ −K,

where the last inequality follows from (39)–(40). Hence, (61) holds andthe proof is complete.

2.5. Proof of Lemma 2.5. This subsection is devoted to the mostgeneral case in the elliptic setting and is an important improvementconcerning the applicability of the paper. As we have already men-tioned in the introduction, we deal with two players here only for sim-plicity. However, the complete proof can be deduced by using thescheme of iterative exponentials as discussed after Lemma 2.5.

Proof of Lemma 2.5. The proof follows the scheme in the previous sec-tion. Therefore, we keep the notation, we use the same test functionand we end up with (57). Moreover, by the same argument we seethat (59) holds and all we need to finish the proof is to check whetherthe estimate (61) is valid for parameters satisfying (42)–(43). First,


the estimate (63) for T3(v) remains unchanged. Moreover, using theidentity (64), we see that also the estimates for I2(v) and I3(v) remainvalid. Hence, we focus on the estimate of I1(v) in what follows.

For simplicity, we consider in what follows that θ1 ≥ θ2. Thus, using(18) we observe that (we assume in what follows that |v| ≤ K)

I1(v) :=2∑

ν=1

ϕH0ν (x,v,∇v)θνe

λvν exp(v)

≤ K +K exp(v)ϕ2∑

ν=1

θνeλvν |∇vν ||∇v|

+K exp(v)ϕ2∑

ν=1

θνeλvν |∇vν |2

=: K + Z1(v) + Z2(v).

(68)

First, using (63) it is evident that

(69) Z2(v) =K

c1λY1(v).

Next, we estimate Z1(v). First, we define G(x,v,∇v) := ∇v1|∇v1| |∇v| if

∇v1 6= 0 and G(x,v,∇v) = 0 otherwise. Then we can use Young’sinequality and estimate Z1(v) as

Z1(v) = K exp(v)ϕ(

(θ1eλv1∇v1 + θ2e

λv2∇v2) · G(v,∇v)

+θ2eλv2(|∇v2||∇v| − ∇v2 · G(v,∇v)

)≤ K exp(v)ϕ

(λ−1|∇w(v)||∇v|+ 2θ2e

λv2|∇v2||∇v|)

≤ I2(v)

2+

(K2

c1σλ+ 2Kθ2e

λv2

)|∇v|2 exp(v)ϕ

≤ I2(v)

2+

(K2

c21σλ

2Θ+

2K

c1λ+

2Kθ2e2λK

θ1λc1

)Y1(v)

(70)


Thus, using above estimates and also (65) and (67) we see that

T3(v) + T4(v) ≥ Y1(v) + Y2(v)− I1(v)− I2(v)− I3(v)

≥ Y1(v)(1− 2Θ−1(c1λ)−2K2σ−1

)+ 2−1Y2(v)

− K − Z1(v)− Z2(v)

≥ −K + Y1(v)

(1− 4K2

Θσ(c1λ)2− 8K

c1λ− 2Kθ2e

2λK

θ1λc1

)≥ K,

(71)

where for the last inequality we used (42) and (43). Thus, we see that(61) is satisfied and therefore we can use Fatou’s lemma to prove (44).So the proof is complete.

2.6. Proof of Lemma 2.6. This subsection is devoted to the renor-malized property of the solution in the parabolic setting. Since wealready described the difference between assuming (16) and (18), wefocus here only on the second case. Moreover, since the proof is verysimilar to the elliptic setting, we focus here only on description of thekey difference, that is the presence of time derivative.

Proof of Lemma 2.6. Similarly as in Subsection 2.4, we can use thestandard theory for the heat equation with L1 right hand side and byusing the assumed weak convergence results on um and (35) we candeduce that for all p ∈ [1, 2)

um → u stronly in Lp(0, T ;W 1,p(Ω;RN)),

Hmν (·,um,∇um)→ Hν(·,u,∇u) a.e. in (0, T )× Ω.

Hence similarly as before, we set ψ := ϕθνeλumλ expm in (36), where

ϕ ∈ W 1,10 (0, T ;L∞ ∩ H1(Ω;RN)) is arbitrary. Here this step is only

formal due to the presence of Dtψ, but since we already discussed thisissue in Subsection 2.3 we omit rigorous derivation. So taking the sum


over ν = 1, . . . , N , we get

0 =N∑ν=1

∫Q

n∑i,j=1

aijDiϕDjumν θνe

λumν expm dx dt

+

∫Q

ϕλνumν θνe

λumν expm dx dt

+

∫Q

ϕ

n∑i,j=1

aijDjumν Di

(θνe

λumν expm)dx dt

−∫Q

ϕHmν θνe

λumν expm dx dt

+

∫Q

Dtumν θνe

λumν expm ϕ dx dt

=:

5∑`=1

∫Q

T`(um) dx dt.

(72)

Although, we have now a parabolic setting, the limiting procedure interms T`(u

m) for ` = 1, . . . , 4 is the same as in the elliptic case. So itremains to discuss

limm→∞

∫Q

T5(um) dx dt.

But, using the following identity∫Q

T5(um) dx dt =

∫Q

N∑ν=1

Dtumν θνe

λumν expm ϕ dx dt

=

∫Q

λ−1Dtwm expm ϕ dx dt

=

∫Q

(σλ)−1Dt expm ϕ dx dt

= −∫Q

(σλ)−1 expmDtϕ dx dt

it is easy to conclude (by using the bound |um| ≤ K and a. e. conver-gence of um) that

limm→∞

∫Q

T5(um) dx dt = −∫Q

(σλ)−1 expDtϕ dx dt

and (45) follows.


3. Weak upper-stability of (1)

In this section we establish a result on the weak-upper stability thatcan be understood as the complement of the result proved in the previ-ous section. For simplicity we do not present the local (in Ω) stabilityresult but presnet here only the Neuman elliptic and parabolic prob-lem. Moreover, since the proof for the elliptic and the parabolic caseare almost the same and we discussed the differences in the previoussection, we give only the statement for both cases but present only theproof for the elliptic one. So the main stability results of this sectionare the following.

Lemma 3.1 (Neumann-elliptic-sum). Let Ω ⊂ Rn be an open boundedLipschitz domain. Assume that there is a sequence umm∈N and asequence Hm

ν such that um u weakly in H1(Ω;RN), ‖um‖∞ ≤ Kfor all m ∈ N, Hamiltonians Hm

ν satisfy (8)–(9), (12)–(16), and (30).Assume that um solve

(73)

∫Ω

n∑i,j=1

aijDiumν Djψ + λνu

mν ψ dx =

∫Ω

Hmν (·,um,∇um)ψ dx

for all m ∈ N, all ψ ∈ L∞∩H1(Ω) and all ν = 1, . . . , N. Then u satisfy

(74)N∑ν=1

∫Ω

n∑i,j=1

aijDiuνDjψ + λνuνψ dx ≥N∑ν=1

∫Ω

Hν(·,u,∇u)ψ dx

for all ν = 1, . . . , N and all nonnegative ψ ∈ L∞ ∩H1(Ω).

Lemma 3.2 (Neumann-parabolic-sum). Let Ω ⊂ Rn be an open boundedLipschitz domain and T > 0 be arbitrary. Assume that there is a se-quence umm∈N and a sequence Hm

ν such that um u weakly inH1(Ω;RN), ‖um‖∞ ≤ K for all m ∈ N, Hamiltonians Hm

ν satisfy(8)–(9), (12)–(16) and (30). Assume that um solves

−∫Q

umν Dtψ dx dt+

∫Q

n∑i,j=1

aijDiumν Djψ + λνu

mν ψ dx dt

=

∫Q

Hmν (·,um,∇um)ψ dx dt

(75)


for all m ∈ N, all ψ ∈ W 1,10 (0, T ;L∞ ∩ H1(Ω)) and all ν = 1, . . . , N.

Then u satisfies

−∫Q

∑ν

uνDtψ dx dt+N∑ν=1

∫Q

n∑i,j=1

aijDiuνDjψ + λνuνψ dx dt

≥N∑ν=1

∫Q

Hν(·,u,∇u)ψ dx dt

(76)

for all ν = 1, . . . , N and all nonnegative ψ ∈ W 1,10 (0, T ;L∞ ∩H1(Ω)).

The proof of Lemma 3.1 relies on the following renormalized prop-erty.

Lemma 3.3 (Neumann-renormalized-sum). Let all assumptions of Lemma 3.1be satisfied. Assume that σ is arbitrary and satisfies

(77) σ ≥ K2

c1c0

.

Moreover, let v be defined as

(78) v :=N∑ν=1

uν .

Then v and u satisfyN∑ν=1

∫Ω

n∑i,j=1

aijDjuνDi

(ϕe−σv

)+ ϕλνuνe

−σv dx

≥N∑ν=1

∫Ω

ϕHν(·,u,∇u)e−σv dx

(79)


Based on this Lemma we can directly prove Lemma 3.1. We do notprovide the proof of Lemma 3.1 since it is basically the same.

Proof of Lemma 3.1. We fix σ := K2

c1c0, so we see that (77) holds and

we observe that u satisfies (79). Then setting ϕ := eσvψ in (79), whichis possible since u ∈ L∞ ∩H1 we directly obtain (74).

We end this section with the proof of Lemma 3.3.

Proof of Lemma 3.3. We use a procedure very similar to that given inthe proof of Lemma 2.4. So, we define

vm := vm(um) :=N∑ν=1

umν ,


set ψ := ϕe−σvm

in (73) and take the sum over ν = 1, . . . , N to get theidentity

0 =N∑ν=1

−∫

Ω

Diϕn∑

i,j=1

Djumν e−σvm dx

−∫

Ω

ϕλνumν e−σvm dx

+ σ

∫Ω

ϕn∑

i,j=1

aijDjumν Div

me−σvm

dx

+

∫Ω

ϕHmν e−σvm dx

=:

4∑`=1

∫Ω

E`(um).

(80)

Similarly as before, we see that to prove (79), it is enough to show that

(81) lim infm→∞

4∑`=1

∫Ω

E`(um) ≥

4∑`=1

∫Ω

E`(u).

First, using the weak convergence of um in H1(Ω;RN) and the L∞

bound, it is standard to show that

(82) lim infm→∞

∫Ω

E1(um) + E2(um) dx =

∫Ω

E1(u) + E2(u) dx.

Next, using (54) and (55), we can also deduce that

(83) E3(um) + E4(um)→ E3(u) + E4(u) a.e. in Ω.

Thus, similarly as in the preceding section, we can use Fatou’s lemmato deduce (81) provided that we are able to find K > 0 such that forall m ∈ N

(84) E3(um) + E4(um) ≥ −K a.e. in Ω.

Thus, it remains to show (84). Hence, for any w ∈ H1(Ω;RN) suchthat |w| ≤ K in Ω we get by using the assumption (6) we get that

E3(w) = σϕe−σv(w)

n∑i,j=1

aijDiv(w)Djv(w)

≥ C1σϕe−σv(w)|∇v(w)|2 =: V1(w).

(85)


Next, using the structure assumption (12) we have

E4(w) = ϕe−σv(w)

(N∑ν=1

(H0ν (x,w,∇w) + fν(x))

+ ∇v(w) ·G(x,w,∇,w))

=: U1(w) + U2(w) + U3(w).

(86)

First, using (15) and the fact that |w| ≤ K, we see that

(87) U2(w) ≥ −K(K, σ,N, ϕ).

Next, using (16) and the L∞ bound, we get

(88) U1(w) ≥ C0ϕe−σv(w)|B(x,∇w)|2 − K(K, σ,N, ϕ).

Finally, using (14) and the Young inequality we deduce that

U3(w) ≥ −ϕe−σv(w)|∇v(w)||G(x,w,∇w)|

≥ −V1(w)− ϕe−σv(w)

σc1

|G(x,w,∇w)|2

≥ −V1(w)− K(K, σ, ϕ,N, c1)− ϕe−σv(w)K2

σc1

|B(x,∇w)|2.

(89)

Consequently, using (85)–(89) and the assumption on σ (77) we obtain

E3(w) + E4(w) ≥ −K + ϕe−σv(w)|B(x,∇w)|2(c0 −

K2

c1σ

)≥ K,

which completes the proof of (84) and also the proof of Lemma 3.3.

4. Proof of the main theorems

Here we give only the proof of Theorem 1.1. The proof of Theo-rem 1.2 is almost the same. The only difference consists in the factthe we need to use the theory for parabolic elliptic equations to deducecompactness of ∇u in L1 and to deal with the initial condition. Butsince such theory is nowadays standard we do not discuss it here.

Hence to prove Theorem 1.1, we use an approximative problem (10)and denote um its solution. Following [10] it can be deduced that (forthis (7) is used. Note that it differs for the elliptic and the paraboliccase)

(90) ‖um‖L∞∩H1(Ω;RN ) ≤ C(N, n, λ1, . . . , λN , K, c0, c1).

Therefore we can extract a not-relabeled subsequence such that

(91) um ∗ u weakly∗ in L∞ ∩H1(Ω;RN).


Thus, defining for any nonnegative ψ ∈ L∞ ∩ H1(Ω) and any ν =1, . . . , N

(92) Iν :=

∫Ω

n∑i,j=1

aijDiuνDjψ + λνuνψ −Hν(·,u,∇u)ψ dx,

we can use Lemma 2.1 and Lemma 3.1 to get that

Iν ≤ 0 for all ν = 1, . . . , N,(93)

N∑ν=1

Iν ≥ 0.(94)

Consequently,

0(94)

≤N∑ν=1

Iν = I1 +N∑ν=2

Iν(93)

≤ I1

(93)

≤ 0.

Thus, repeating this procedure for any ν we finally observe that

I1 = · · · = IN = 0

and the proof of Theorem 1.1 is finished.

References

[1] A. Bensoussan and J. Frehse. Nonlinear elliptic systems in stochastic gametheory. J. Reine Angew. Math., 350:23–67, 1984.

[2] A. Bensoussan and J. Frehse. Cα-regularity results for quasilinear parabolicsystems. Comment. Math. Univ. Carolin., 31(3):453–474, 1990.

[3] A. Bensoussan and J. Frehse. Regularity results for nonlinear elliptic sys-tems and applications, volume 151 of Applied Mathematical Sciences. Springer-Verlag, Berlin, 2002.

[4] A. Bensoussan and J. Frehse. Smooth solutions of systems of quasilinear par-abolic equations. ESAIM Control Optim. Calc. Var., 8:169–193 (electronic),2002. A tribute to J. L. Lions.

[5] A. Bensoussan and J. Frehse. Systems of Bellman equations to stochastic differ-ential games with discount control. Boll. Unione Mat. Ital. (9), 1(3):663–681,2008.

[6] A. Bensoussan and J. Frehse. Diagonal elliptic Bellman systems to stochasticdifferential games with discount control and noncompact coupling. Rend. Mat.Appl. (7), 29(1):1–16, 2009.

[7] A. Bensoussan, J. Frehse, and J. Vogelgesang. On a class of nonlinear ellip-tic systems with applications to Stackelberg and Nash differential games. toappear in Chin. Ann. Math., 2010.

[8] A. Bensoussan, J. Frehse, and J. Vogelgesang. Systems of Bellman equationsto stochastic differential games with non-compact coupling. Discrete Contin.Dyn. Syst., 27(4):1375–1389, 2010.


[9] A. Bensoussan and J.-L. Lions. Impulse control and quasivariational inequal-ities. µ. Gauthier-Villars, Montrouge, 1984. Translated from the French by J.M. Cole.

[10] M. Bulıcek and J. Frehse. On nonlinear elliptic bellman systems for a classof stochastic differential games in arbitrary dimension. Math. Models MethodsAppl. Sci., 21(1):215–240, 2011.

[11] W. H. Fleming and R. W. Rishel. Deterministic and stochastic optimal control.Springer-Verlag, Berlin, 1975. Applications of Mathematics, No. 1.

[12] J. Frehse. A discontinuous solution of a mildly nonlinear elliptic system. Math.Z., 134:229–230, 1973.

[13] J. Frehse. Existence and perturbation theorems for nonlinear elliptic systems.In Nonlinear partial differential equations and their applications. College deFrance Seminar, Vol. IV (Paris, 1981/1982), volume 84 of Res. Notes inMath., pages 87–111. Pitman, Boston, MA, 1983.

[14] J. Frehse. A refinement of Rellich’s theorem. Rend. Mat. (7), 5(3-4):229–242(1988), 1985.

[15] J. Frehse. Remarks on diagonal elliptic systems. In Partial differential equa-tions and calculus of variations, volume 1357 of Lecture Notes in Math., pages198–210. Springer, Berlin, 1988.

[16] J. Frehse. Bellman systems of stochastic differential games with three players.In Optimal control and partial differential equation. Conference, pages 3–22,2001.

[17] Avner Friedman. Stochastic differential equations and applications. Vol. 2.Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1976.Probability and Mathematical Statistics, Vol. 28.

[18] D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of secondorder. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the1998 edition.

[19] S. Hildebrandt. Nonlinear elliptic systems and harmonic mappings. In Proceed-ings of the 1980 Beijing Symposium on Differential Geometry and DifferentialEquations, Vol. 1, 2, 3 (Beijing, 1980), pages 481–615, Beijing, 1982. SciencePress.

[20] O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural′ceva. Linear and quasi-linear equations of parabolic type. Translated from the Russian by S. Smith.Translations of Mathematical Monographs, Vol. 23. American MathematicalSociety, Providence, R.I., 1967.

[21] O. A. Ladyzhenskaya and N. N. Ural′tseva. Linear and quasilinear ellipticequations. Translated from the Russian by Scripta Technica, Inc. Translationeditor: Leon Ehrenpreis. Academic Press, New York, 1968.

[22] R. Landes. On the existence of weak solutions of perturbated systems withcritical growth. J. Reine Angew. Math., 393:21–38, 1989.

[23] W. von Wahl and M. Wiegner. Uber die Holderstetigkeit schwacher Losungensemilinearer elliptischer Systeme mit einseitiger Bedingung. ManuscriptaMath., 19(4):385–399, 1976.

[24] M. Wiegner. Ein optimaler Regularitatssatz fur schwache Losungen gewisserelliptischer Systeme. Math. Z., 147(1):21–28, 1976.


[25] M. Wiegner. das Existenz- und Regularitatsproblem bei Systemen nichtlinearerelliptischer Differentialgleichungen. phdthesis, University of Bochum, 1977.Habilitation thesis.

School of Management, International Center for Decision and RiskAnalysis, University of Texas at Dallas, 800 W. Campbell Rd, SM30Richardson, TX 75080-3021, USA

Department of Logistics and Maritime Studies, The Hong KongPolytechnic University, Hung Hom, Kowloon, Hong Kong

E-mail address: [email protected]

Mathematical Institute, Faculty of Mathematics and Physics, CharlesUniversity, Sokolovska 83, 186 75 Praha 8, Czech Republic


Institute for Applied Mathematics, Department of Applied Analy-sis, Endenicher Allee 60, 53115 Bonn, Germany


existence and compactness for weak solutions to bellman...

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