REMARKS ON THE STRONG MAXIMUM PRINCIPLE ... - CiteSeerX

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS Volume 3, Number 3, September 2004

Website: http://AIMsciences.org pp. 395–415

REMARKS ON THE STRONG MAXIMUM PRINCIPLE FOR VISCOSITY SOLUTIONS TO FULLY NONLINEAR PARABOLIC EQUATIONS

Francesca Da Lio Dipartimento di Matematica Universit` a di Torino, Via Carlo Alberto 10, 10123, Torino (Italy)

(Communicated by Martino Bardi) Abstract. We prove a Strong Maximum Principle for upper semicontinuous viscosity subsolutions to fully nonlinear degenerate parabolic pde’s. We also describe the set of propagation of maxima in the case of second order HamiltonJacobi-Bellman equations which are either convex or concave with respect to the (u, Du, D2 u) variables and we derive the Strong Maximum Principle in some cases, including a class of nonlinear operators which are not strictly parabolic.

1. Introduction. In this paper we investigate the validity of the Strong Maximum Principle and the Strong Minimum Principle (SMaxP and SMinP in the following) for semicontinuous viscosity subsolutions and supersolutions of fully nonlinear degenerate parabolic equations of the form ut + F (x, t, u, Dx u, Dx2 u) = 0,

in Ω × (0, T ),

(1.1)

N

where Ω ⊆ R is a bounded open set, T > 0, F is a real-valued, locally bounded function in Ω × [0, T ] × R × RN × S N , (S N being the set of real symmetric N × N matrices), it is proper in the sense of [11], that is, F (x, t, r, p, X) ≤ F (x, t, s, p, Y )

if r ≤ s and Y ≤ X,

N

(1.2)

N

for all (x, t) ∈ Ω × [0, T ] r, s ∈ R, p ∈ R \ {0} and X, Y ∈ S where “≤” stands for the usual partial ordering on symmetric matrices. We also say that F is degenerate parabolic if it satisfies F (x, t, r, p, X) ≤ F (x, t, r, p, Y ) for all (x, t) ∈ Ω × [0, T ] r ∈ R, p ∈ R parabolic if there is λ > 0 such that

N

if Y ≤ X ,

\ {0} and X, Y ∈ S

N

and it is uniformly

F (x, t, r, p, X) − F (x, t, r, p, X + Y ) ≥ λTrace(Y ) , N

(1.3)

(1.4)

N

for all (x, t) ∈ Ω × [0, T ], r ∈ R, p ∈ R X, Y ∈ S , Y ≥ 0. The solution u is a scalar function of (x, t) and ut , Dx u, Dx2 u represent respectively the partial derivative with respect to t, the gradient and the Hessian matrix with respect to x of the solution u. 2000Mathematics Subject Classifications. 35B50, 35J60, 35K65, 70H20, 49L25. Key words and phrases. Strong maximum principle, fully nonlinear pde’s, degenerate parabolic equations, Hamilton-Jacobi equations, propagation of maxima, viscosity solutions.

395

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FRANCESCA DA LIO

Moreover we describe the set of propagation of maxima of viscosity subsolutions to fully nonlinear second order parabolic Hamilton-Jacobi-Bellman equations of the form n o α ut + sup −Trace(aα (x, t)Dx2 u) + b (x, t) · Dx u + cα (x, t)u = 0 in Ω × (0, T ) α∈A

(1.5) and ut + inf

α∈A

n o α −Trace(aα (x, t)Dx2 u) + b (x, t) · Dx u + cα (x, t)u = 0 in Ω × (0, T ) ,

(1.6) where A is a closed subset of a normed space, a : Ω × [0, T ] × A → S N , b : Ω × [0, T ] × A → RN and c : Ω × [0, T ] × A → R are continuous functions, with aα (x, t) ≥ 0, cα (x, t) ≥ 0 for all (x, t) ∈ Ω × [0, T ] and for all α ∈ A. More precise assumptions on the data are given later. We recall that such equations appear naturally in stochastic control problems when one is minimizing or maximizing a given payoff. We notice that all the above equations can be considered particular cases of equations of the form 2 G(x, t, u, ut , Dx u, Dxt u) = 0,

in O

(1.7)

2 u denotes the Hessian matrix where O ⊆ RN +1 is of the form O = Ω × (0, T ), Dxt of the second derivatives of u and G is a locally bounded function in Ω × [0, T ] × R × R × RN × S N +1 and is proper. Since degenerate equations of the form (1.7) do not have classical solutions in general, we will work in the framework of viscosity solutions, see, for instance, [1, 7, 2, 11]. In the sequel for every S ⊆ Rk , k ≥ 1, we denote by U SC(S) and LSC(S) the set of respectively upper and lower semicontinuous functions in S. By Strong Maximum and Minimum Principle for the equation (1.7) set in a connected open set Ω × (0, T ) we mean the following properties :

SMaxP: any u ∈ U SC(Ω × [0, T ]) viscosity subsolution of (1.7) that attains a nonnegative maximum in (x0 , t0 ) ∈ Ω × (0, T ] is constant in Ω × [0, t0 ]. SMinP: any v ∈ LSC(Ω × [0, T ]) viscosity supersolution of (1.7) that attains a nonpositive minimum in (x0 , t0 ) ∈ Ω × (0, T ] is constant in Ω × [0, t0 ]. In the first part of this paper we are going to extend to parabolic pde’s the results of local propagation of maxima in the cylindrical region Ω × (0, T ] which have been obtained in [3] for degenerate elliptic equations. We first introduce some notations. We set QT = Ω × (0, T ] and for any point P0 = (x0 , t0 ) ∈ QT , we denote by S(P0 ) the set of all points Q ∈ QT which can be connected to P0 by a simple continuous curve in QT along which the t-coordinate is nondecreasing from Q to P0 and by C(P0 ) we denote the component of Ω × {t = t0 } which contains P0 . We also denote respectively by G∗ and G∗ the upper and lower semicontinuous envelopes of G. We will study separately the propagation of maxima in C(P0 ) and in the region Ω × (0, t0 ) and to this end we will give two different sets of assumptions. More precisely we will show that, given P0 = (x0 , t0 ) ∈ QT , the maximum (resp. the minimum) at P0 of an upper semicontinuous subsolution (resp. lower semicontinuous supersolution) of (1.7) propagates in C(P0 ) if G ∈ LSC(Ω × [0, T ] × R × R ×

STRONG MAXIMUM PRINCIPLE FOR NONLINEAR PARABOLIC EQUATIONS

397

(RN \ {0}) × S N +1 ) (resp. G ∈ U SC(Ω × [0, T ] × R × R × (RN \ {0}) × S N +1 )) and satisfies the following two assumptions : (A1 ) There exists ρ0 > 0 such that, for all 0 < |(s, p)| < ρ0 , the following condition holds for some γ0 ≥ 0 G(x0 , t0 , 0, s, p, I − γ(s, p) ⊗ (s, p)) > 0

∀γ > γ0 ,

(1.8)

∀γ > γ0 ) ,

(1.9)

(resp. G(x0 , t0 , 0, s, p, γ(s, p) ⊗ (s, p) − I) < 0

where s ∈ R, p ∈ RN \ {0}, and for all ν ∈ RN +1 \ {0}, ν ⊗ ν is the matrix defined by (ν ⊗ ν)i,j = νi νj ; (A2 ) For all η > 0, there exist a function ϕ : (0, 1) → (0, +∞), εη > 0 and γ0 ≥ 0 such that for all ε ∈ (0, εη ], γ > γ0 the following condition holds uniformly for all (x, t) ∈ B((x0 , t0 ), η), r ∈ [−1, 0], 0 < |p| ≤ η, |s| ≤ η G∗ (x, t, εr, εs, εp, ε(I − γ(p, s) ⊗ (p, s))) ≥ ϕ(ε)G∗ (x, t, r, s, p, I − γ(p, s) ⊗ (p, s)),

(1.10)

G∗ (x, t, εr, εs, εp, ε(γ(p, s) ⊗ (p, s) − I)) ≤ ϕ(ε)G∗ (x, t, r, s, p, γ(p, s) ⊗ (p, s) − I) ).

(1.11)

(resp.

As far as the propagation of maxima in Ω×(0, t0 ) is concerned we will show that the maximum (resp. the minimum) at P0 = (x0 , t0 ) ∈ QT of an upper semicontinuous subsolution (resp. lower semicontinuous supersolution) of (1.7) locally propagates in the region Ω × (0, t0 ) if G ∈ LSC(Ω × [0, T ] × R × R × (RN \ {0}) × S N +1 ) (resp. G ∈ U SC(Ω × [0, T ] × R × R × (RN \ {0}) × S N +1 )) and satisfies the following two properties : (A3 ) For any (x0 , t0 ) ∈ QT there exists δ > 0 such that e > 0, ∀0 < δ ≤ δ, G∗ (x0 , t0 , 0, 1, 0, δ I)

(1.12)

e < 0, ∀0 < δ ≤ δ ), G∗ (x0 , t0 , 0, −1, 0, −δ I)

(1.13)

(resp. where Ie denotes the (N + 1) × S N +1 matrix   1 ··· 0 0  .. . .  .  . .. 0  Ie :=  .  ;  0 ··· 1 0  0 ··· 0 0 (A4 ) For all η > 0, there exist a function ϕ : (0, 1) → (0, ∞), εη > 0 such that ∀K > 0 and ∀ε ∈ (0, εη ], the following condition holds uniformly for all −η ≤ r ≤ 0, and for all (x, t) ∈ B((x0 , t0 ), η) e ≥ ϕ(ε)G∗ (x, t, r, 1, 2K(x − x0 ), 2K I), e G∗ (x, t, εr, ε, 2Kε(x − x0 ), 2KεI)

(1.14)

(resp. ∗

e ≤ ϕ(ε)G∗ (x, t, r, 1, 2K(x − x0 ), −2K I) e ). G (x, t, εr, ε, −2Kε(x − x0 ), −2KεI) (1.15)

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FRANCESCA DA LIO

We observe that the assumptions (A1 ) and (A3 ) represent some kinds of nondegeneracy conditions of the operator G which are trivially satisfied if G is uniformly parabolic, while the assumptions (A2 ) and (A4 ) can be seen as scaling properties which are clearly verified if G is positively homogeneous and in particular if it is linear. We remark that the homogeneity of G can be avoided if G can be approximated e satisfying the previous assumptions in the sense that, with operators G e for some η > 0, there exists ψ : (0, 1) → R such that limε→0+ ψ(ε) = 0 and (A) e t, εr, εs, εp, εX) + ϕ(ε)ψ(ε), G(x, t, εr, εs, εp, εX) ≥ G(x,

(1.16)

(resp. ≤) holds uniformly for (x, t) ∈ B((x0 , t0 ), η), r ∈ [−1, 0], |s|, |p|, ||X|| ≤ η, with the same ϕ in (1.11) and (1.15). In the case that the maximum propagates in C(P0 ) we say that there is a horizontal propagation of maxima , whereas if the maximum locally propagates in the region Ω × (0, t0 ) then we say that there is a local vertical propagation of maxima. Whenever the operator G satisfies the above set of assumptions, we get that if u ∈ U SC(QT ) is a viscosity subsolution of (1.7) that attains a nonnegative maximum at P0 = (x0 , t0 ) ∈ Ω × (0, T ], then u is constant in S(P0 ), in particular if Ω is a connected set then u is constant in Ω × [0, t0 ]. In this way we extend to viscosity subsolutions the results obtained in [23, 16, 17] for classical solutions to linear parabolic equations. We test our assumptions on several examples. In particular we consider as model case an equation of the form ut + c(x, t)|u|k−1 u − a(x, t)F (Dx u, Dx2 u) = 0,

(1.17)

where F : RN \ {0} × S N → R is positively homogeneous of degree α ∈ R, k ≥ 1, the coefficients c, a are continuous and satisfy c ≥ 0, a > 0. We show that the SMaxP holds for (1.17)when F is the Laplace operator, or a Pucci extremal operator (which are both well-known results for classical solutions), or it is the mean curvature operator for graphs. Whereas when F is either the m-Laplacian, with m 6= 2, or the ∞-Laplacian we get partial results and we refer to Section 3 for more details. We apply our methods also to the Levi equation and we get the propagation of maxima along C(P0 ) of viscosity subsolutions of such an equation, a result previously obtained for classical solutions in [24, 10], and the SMaxP in the case the Levi curvature is strictly positive. In the second part of the paper we focus our attention to equations of the form (1.5) and (1.6). We first observe that both equations (1.5) and (1.6) trivially satisfy the two scaling properties (1.10), (1.14), being linear with respect to (u, ut , Du, D2 u), and the nondegeneracy condition (1.12). Thus for this kind of equations we always have the local propagation of maxima along the region Ω × (0, t0 ). Our aim is to describe the set of propagation of maxima of subsolutions to either (1.5) or (1.6) in terms of the vector fields appearing in the operator, thus extending the results obtained by Bardi and the author in [4] and [5] for respectively fully nonlinear degenerate elliptic convex and concave Hamilton-Jacobi operators. From the properties of the propagation set we deduce a Strong Maximum Principle in some cases including also a class of convex and concave nonlinear operators which are not strictly parabolic. These results represents a generalization to convex and concave operators of the Bony’s SMaxP for linear hypoelliptic operators (see [8])


399

. In [4] and [5] the authors give also a characterization of the propagation set in terms of a controlled stochastic system. For sake of simplicity we are not going to analyze this aspect in the present paper, being the aim of a future work. Before concluding we briefly mention some results in the literature related to the Strong Maximum Principle for parabolic equations (for a more complete descriptions of the results in the elliptic case we refer to [3]). We recall that the Strong Maximum Principle for linear elliptic and parabolic equations goes back to the work of E. Hopf in the 20s and to the paper of Nirenberg [23] for parabolic equations. For weak solutions of quasilinear parabolic equations of m-Laplacian type we refer e.g. to [12, 13, 21] where typically it is studied the existence of a dead core, namely the set where the solution vanishes. With this regard the SMinP obtained in this note can be viewed as an extension to viscosity solutions of fully nonlinear parabolic equations of the results of nonexistence of a dead core. For related topics concerning mean curvature type equations we refer e.g to [14, 15, 18]. The SMaxP obtained in this paper for semicontinuous viscosity solutions is new (to the best of our knowledge) also in the case of uniformly parabolic equations. This paper is organized as follows. In Section 2 we prove a SMaxP for fully nonlinear parabolic equations. Section 3 is devoted to provide some examples. In Section 4 we give a characterization of the set of propagation of maxima for a class of convex and concave Hamilton- Jacobi operators of the form (1.5) and (1.6). 2. The Strong Maximum Principle. In this Section we consider partial differential equations of the form 2 G(x, t, u, ut , Dx u, Dxt u) = 0

in Ω × (0, T ) ,

(2.1)

N

where Ω ⊆ R is an open set, T > 0 and G is a locally bounded function in Ω × [0, T ] × R × R × RN × S N +1 → R. Since we want to consider also the case of singular equations, the nonlinearity G cannot be assumed to be continuous and we reduce the possible discontinuity of G in the following way. (A0 ) The function G ∈ LSC(Ω × [0, T ] × R × R × RN \ {0} × S N +1 ) and it is proper. The aim of this Section is to prove some results about the local propagation of maxima of viscosity subsolutions of (2.1) in the cylindrical region QT . More precisely we will see that if a viscosity subsolution attains a nonnegative maximum at P0 = (x0 , t0 ) ∈ QT , then such a maximum propagates both in the component C(P0 ) of Ω × {t = t0 } containing P0 and in the set S(P0 ) of all points Q ∈ QT which can be connected to P0 by a simple continuous curve in QT along which the t-coordinate is nondecreasing from Q to P0 . In the following Theorem we will prove that if u ∈ U SC(QT ) is a viscosity subsolution of (2.1) that attains a nonnegative maximum in a point P0 = (x0 , t0 ) ∈ QT , then such a maximum propagates in C(P0 ). On the analogy of elliptic case this result is based on the nondegeneracy property (1.8) and on the scaling property (1.10) of the operator G. Theorem 2.1. Let u ∈ U SC(QT ) be a viscosity subsolution of (2.1) that attains a nonnegative maximum at P0 = (x0 , t0 ) ∈ QT . Assume G satisfies (A0 ) − (A2 ), e with these properties and satisfying (A). e Then u is or alternatively, there exists G constant in C(P0 ).

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FRANCESCA DA LIO

Proof. We follow the same line of argument as in the classic parabolic case ([23]). Suppose by contradiction there exists a point P1 = (x1 , t0 ) such that u(P1 ) < u(P0 ). By standard geometric arguments and the upper semicontinuity of u, we can find an ellipsoid Eλ0 |x − x|2 + λ0 |t − t0 |2 ≤ ε2 λ0 , with ε > 0 small enough, (x, t0 ) ∈ C(P0 ), such that u < u(P0 ) in the interior of Eλ0 and a P ∗ = (x∗ , t∗ ) ∈ ∂Eλ0 such that u(P ∗ ) = u(P0 ), with x 6= x∗ and P ∗ ∈ / ∂Ω×(0, T ). Let B := B((˜ x, t˜), R) be such that B ⊆ Eλ0 , and ∂B∩∂Eλ0 = {(x∗ , t∗ )}. It is not restrictive to suppose that R < ρ0 , (ρ0 being the constant appearing in (A1 )), otherwise we can change the center of the ball along the direction of the vector (˜ x − x∗ , t˜−t∗ ). Let y = (x, t) and y˜ = (˜ x, t˜), introduce the following auxiliary function 2 2 v(y) := e−γR − e−γ|y−˜y| , (2.2) where γ is a positive constant yet to be determined. Note that v(x∗ , t∗ ) = 0 and −1 < v(x, t) < 0, for |x − x ˜|2 + |t − t˜|2 < R2 . Direct calculations give 2

Dv(y)

=

2γe−γ|y−˜y| (y − y˜),

D2 v(y)

=

2γe−γ|y−˜y| I − 4γ 2 e−γ|y−˜y| (y − y˜) ⊗ (y − y˜).

2

2

Thus if G satisfies (A0 ) − (A2 ) we have G(x∗ , t∗ , v(x∗ , t∗ ), vt (x∗ , t∗ ), Dx v(x∗ , t∗ ), D2 v(x∗ , t∗ )) 2

= G(x∗ , t∗ , 0, 2γe−γR (t∗ − t˜), 2

2

2γe−γR (x∗ − x ˜), 2γe−γR (I − 2γ(y ∗ − y˜) ⊗ (y ∗ − y˜))) 2

≥ ϕ(2γe−γR )G(x∗ , t∗ , 0, t∗ − t˜, x∗ − x ˜, (I − 2γ(y ∗ − y˜) ⊗ (y ∗ − y˜))) > 0, for all γ > γ0 . By the lower semicontinuity of G there exists r ∈ (0, R) such that, for all (x, t) ∈ X := B((x∗ , t∗ ), r) ∩ B((˜ x, t˜), R), Dx v(x, t) does not vanish and G(x, t, v(x, t), vt (x, t), Dx v(x, t), D2 v(x, t)) ≥ C > 0.

(2.3)

Moreover by (A2 ) there exists ε > 0 such that ∀ε ≤ ε, εv is a strict supersolution of (2.1) in X as well. Now we claim that εv is a strict supersolution of (2.1) in X e as above. Indeed we have also in the case that G satisfies (1.16) with G G(x, t, εv(x, t), εvt (x, t), εDx v(x, t), εD2 v(x, t)) e t, v(x, t), vt (x, t), Dx v(x, t), D2 v(x, t)) + o(ϕ(ε)), ≥ ϕ(ε)G(x, e we get as ε → 0+ , thus by using the inequality (2.3) for G C , ∀ε ≤ ε. (2.4) 2 Let ε ≤ ε be such that u(x, t) − u(x∗ , t∗ ) ≤ εv(x, t), ∀(x, t) ∈ ∂X. We claim that u(x, t) − u(x∗ , t∗ ) ≤ εv(x, t), ∀(x, t) ∈ X. Suppose by contradiction that there exists (x0 , t0 ) ∈ X such that u(x0 , t0 ) − u(x∗ , t∗ ) − εv(x0 , t0 ) = maxX (u − u(x∗ , t∗ ) − εv) > 0. Since εv ∈ C ∞ (RN × R+ ), we can use the definition of viscosity subsolution, the assumptions that u(x∗ , t∗ ) ≥ 0, G is proper and the fact that Dx v(x, t) 6= 0 in X to get G(x, t, εv(x, t), vt (x, t)εDv(x, t), εD2 v(x, t)) ≥ ϕ(ε)

G(x0 , t0 , εv(x0 , t0 ), εvt (x0 , t0 ), εDv(x0 , t0 ), εD2 v(x0 , t0 )) ≤ G(x0 , t0 , u(x0 , t0 ), εvt (x0 , t0 ), εDv(x0 , t0 ), εD2 v(x0 , t0 )) ≤ 0,


401

that contradicts (2.3) and proves the claim. Now consider in B((x∗ , t∗ ), r) the function Φ(x, t) := u(x, t) − εv(x, t), and note that Φ has a maximum at (x∗ , t∗ ). Thus using again the fact that u is a subsolution of (2.1) in B((x∗ , t∗ ), r), v(x∗ , t∗ ) = 0 ≤ u(x∗ , t∗ ), Dx v(x∗ , t∗ ) 6= 0 and G is proper we get G(x∗ , t∗ , εv(x∗ , t∗ ), vt (x∗ , t∗ ), εDx v(x∗ , t∗ ), εD2 v(x∗ , t∗ )) ≤ 0, which is in contradiction with (2.3) and we can conclude. Now we show the local vertical propagation of the maxima of a viscosity subsolutions of (2.1), namely we prove that if u ∈ U SC(QT ) is a viscosity subsolution of (2.1) that attains a nonnegative maximum at P0 = (x0 , t0 ) ∈ QT , then such a maximum propagates locally in Ω × (0, t0 ). More precisely let us consider the following rectangle < := {(x, t) : xi0 − ai ≤ xi ≤ xi0 + ai , t0 − a0 ≤ t ≤ t0 },

(2.5)

i

with a , a0 small enough, and denote by 0 such that for all (x, t) ∈ B((x0 , t0 ), r) ⊆ QT , G(x, t, h(x, t), ht (x, t), Dx h(x, t), D2 h(x, t)) ≥ C > 0. T Consider the set X := B((x0 , t0 ), r) {(x, t) : h(x, t) < 0}. Observe that by (A2 ) there exists ε > 0 such that ∀ε ≤ ε, εh is a strict supersolution of (2.1) in X as well. It can be seen that εh is a strict supersolution of (2.1) also in the case that e as above. Choose ε ≤ ε so that u(x, t) − u(x0 , t0 ) ≤ G satisfies (1.16) with G εh(x, t), ∀(x, t) ∈ ∂X. Now using the same arguments of Theorem 2.1 we can prove that u(x, t) − u(x0 , t0 ) ≤ εh(x, t), ∀(x, t) ∈ X and get a contradiction.

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FRANCESCA DA LIO

If we assume (A0 ) − (A4 ), from Theorem 2.1 and Proposition 2.2 it follows that if a viscosity subsolution of (2.1) attains a nonnegative maximum at an interior point (x0 , t0 ) ∈ QT then u is constant in any rectangle < contained in Ω × [0, t0 ]. Corollary 2.3. Let u ∈ U SC(QT ) be a viscosity subsolution of (2.1) that attains a nonnegative maximum at P0 = (x0 , t0 ) ∈ QT . Assume that G satisfies (A0 ) − (A4 ) e with these properties and satisfying (A). e Then u is or alternatively there exists G i i i i i constant in any rectangle < := {(x, t) : x0 − a ≤ x ≤ x0 + a , t0 − a0 ≤ t ≤ t0 } ⊆ Ω × [0, t0 ]. Proof. Let < = {(x, t) : xi0 − ai ≤ xi ≤ xi0 + ai , t0 − a0 ≤ t ≤ t0 } be a rectangle contained in Ω × [0, t0 ] and suppose by contradiction there exists Q ∈ < such that u(Q) < u(P0 ). It is not restrictive to suppose that Q lies on t = t0 − a0 and u < u(P0 ) on the straight line γ connecting Q to P0 . Since for every point Q0 ∈ 0, p ∈ R, X ∈ S N , k ≥ 1, the coefficients c, a are continuous and satisfy c ≥ 0, a > 0. Example 2.6. The m-Laplacian operator. F (p, X) := |p|m−2 TraceX + (m − 2)|p|m−4 p · Xp In this case F is defined for all p ∈ R if m ≥ 2 and for p 6= 0 if 1 < m < 2. It is positively homogeneous of degree α = m − 1 and it coincides with the usual Laplacian if m = 2. We consider here the case m ≥ 2. Both the nondegeneracy conditions (1.8) and (1.12) are always satisfied if a(x) > 0. Now we check the two scaling properties (1.10) and (1.14). As far the condition (1.10) is concerned, we have εs + c(x, t)|εr|k−1 εr − a(x, t)F (εν, ε(I − γν ⊗ ν)) = εs + εk c(x, t)|r|k−1 r + εm−1 a(x, t)|ν|m−2 [γ(m − 1)|ν|2 − (N + m − 2)].


403

Observe that if γ > 0 is large enough, then γ(m − 1)|ν|2 − (N + m − 2) > 0, hence the condition (1.10) is satisfied only if m = 2. As far as the condition (1.14) is concerned, given x0 ∈ R and η > 0, for any r < 0 and x ∈ B(x0 , η), we have ε + c(x, t)|εr|k−1 εr − a(x)F (ε2K(x − x0 ), ε2KI) = ε + εk c(x, t)|r|k−1 r − εm−1 a(x)(N + m − 2)(2K)m−1 |x − x0 |m−2 . Thus if m ≥ 2 the condition (1.14) is always satisfied. From the results of Section 2, it follows that the SMaxP holds if m = 2, which is a well known result for classical solutions. On the other hand in [13] it is proved the existence of solutions of (2.8) with compact support when m > 2 and c(x, t) ≡ 0, (see also [21]) and therefore the SMaxP cannot hold if m > 2. Example 2.7. The ∞-Laplacian operator. F (p, X) := p · Xp . We recall that −∆∞ u = 0 can be formally considered as the Euler equation associated to the problem of minimizing the L∞ norm of the gradient of u (see e.g. [20]). We observe that it is homogeneous of degree α = 3. In this case the assumption (A2 ) is not satisfied, whereas both hypotheses (A3 ) − (A4 ) are verified. Indeed for all δ > 0 we have G(x, t, 0, 1, 0, δI) = 1 and for all r ≤ 0 we have ε + c(x, t)εk |r|k−1 r − a(x, t)(2Kε)3 |x − x0 |2 ≥ ε(1 + |r|k−1 r − a(x, t)(2K)3 |x − x0 |2 ) . Thus from the results obtained in the previous section we can deduce that we have a local vertical propagation of maxima for equation (2.8) when F is the ∞-Laplacian. Example 2.8. The mean curvature operator for graphs. p⊗p X F (p, X) = Trace I− 1 + |p|2

(2.9)

Such an operator arises in the study of motion of graphs with normal velocity given by the mean curvature, see e.g [11, 15, 18]. In this case the equation (2.8) satisfies e because the assumption (A) εs + cεk |r|k−1 r

− aF (εp, εX) = εs + cεk |r|k−1 r − aεTraceX + O(ε3 ), as ε → 0+ ,

for s, p, X in a bounded set. Moreover one can see that the operator Fe(x, t, r, s, p, X) := s + c(x, t)|r|k−1 r − a(x)TraceX satisfies the assumptions (A1 ) − (A4 ) and thus the SMaxP holds for equation (2.8) when F is the mean curvature operator for graphs. This result is new in the framework of viscosity solutions’ theory. Now consider the mean curvature operator for hypersurfaces p⊗p X . (2.10) F (p, X) = Trace I− |p|2

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FRANCESCA DA LIO

In this case F is homogeneous of degree 1, hence the equation (2.8) satisfies both the scaling properties (1.10) and (1.14), moreover it trivially satisfies (1.12), but it does not verify (1.8) since −a(x, t)F (p, I − γp ⊗ p) = −a(x, t)N + a(x, t) ≤ 0. It is an open problem if the SMaxP holds when F is given by (2.10) even for classical solutions. Example 2.9. The Pucci minimal and maximal operators. Fm (X) = λTraceX + (1 − N λ) min{eigenvalues of X}; FM (X) = λTraceX + (1 − N λ) max{eigenvalues of X},

(2.11) (2.12)

where 0 < λ ≤ 1/N is fixed, see e.g. [11]. They are positive homogeneous of degree 1. In both cases the SMaxP holds for equation (2.8). Example 2.10. Uniformly parabolic equation. ut + c(x, t)|u|k−1 u − ∆u + b(x, t)|Du|m = 0.

(2.13)

Both the nondegeneracy conditions (1.8) and (1.12) are satisfied, since for γ large enough we have G(x, t, 0, s, p, I − γν ⊗ ν) ≥ −N + γ|p|2 > 0 and if δ > 0 is small we have e = 1 − δN > 0. G(x, t, 0, 1, 0, δ I) For c ≥ 0 the scaling properties (1.10) and (1.14) are satisfied if either b ≥ 0 and m ≤ 1, or b ≤ 0 and m ≥ 1,

(2.14)

either c ≡ 0 or k ≥ 1.

(2.15)

and Example 2.11. The Levi operator. In this case G : R3 × R3 × S(3) → R is defined as follows G(x, y, t, s, p, X) := −TraceA(s, p)X + k(x, y, t)(1 + |p|2 )3/2 , where k is a nonnegative continuous function and  1 + s2 0 0 1 + s2 A(s, p) :=  −sp1 + p2 −sp2 − p1

 −sp1 + p2 −sp2 − p1  p21 + p22

Observe that G(x, y, t, εp, εX)

e − ε2 Trace(A1 (s, p)X) = −εTrace(IX) − ε3 Trace(A2 (s, p)X) + k(x, y, t)(1 + |εp|2 )3/2 e + ε(k(x, y, t)(1 + |p|2 )) + O(ε2 ), as ε → 0, ≥ −εTrace(IX)

for 0 < |p| ≤ 1, (x, y, t), X bounded, where  1 0 Ie :=  0 1 0 0

 0 0  0


s2  0 A1 (s, p) := −sp1 

0 s2 −sp2

  −sp1 0 −sp2  A2 (s, p) :=  0 p21 + p22 p2

0 0 −p1

405

 p2 −p1  0

The operator e + k(x, y, t)(1 + |p|2 ) Fe(x, y, t, p, X) := −Trace(IX) satisfies the assumptions (A2 ), (A4 ) since for all 0 < |p| ≤ 1 and for ε small enough, we have e e + k(x, y, t)(1 + |p|2 )]. −Trace(IεX) + k(x, y, t)(1 + |εp|2 )3/2 ≥ ε[−Trace(IX) The nondegeneracy conditions (A1 ), (A3 ) are satisfied provided k > 0. Indeed for all γ > 0 and for all p ∈ R3 we have Fe(x, y, t, p, I − γp ⊗ p) ≥ −2 + γ|p|2 + k(x, y, t)(1 + |p|2 ) > 0 and for all p = (0, 0, p3 ) and for all δ < δ0 := inf QT k/2 > 0 the following estimate holds ˜ = −2 + δk(x, y, t)(1 + |p3 |2 ) > 0 . Fe(x, y, t, 0, 0, 0, p3 , δ I) Thus we extend to viscosity subsolutions the results obtained in [24] where it is shown the horizontal propagation of maxima of classical subsolutions to the Levi equation. We deduce that if k > 0 no viscosity subsolutions of the Levi equation can have local maxima in QT . Indeed by Corollary 2.4, it would be a constant and thus k ≡ 0 which is a contradiction. In particular if k > 0 the Levi equation cannot have solution with compact support. On the other hand it is well known that if k ≡ 0 the SMaxP does not hold for the Levi equation, since any function of the time only satisfies this equation. It is an open question what happens if k ≥ 0 and k 6≡ 0 and it is the aim of a future work to investigate the mechanism of maxima propagation in this case. 3. Propagation of Maxima for Convex and Concave Hamilton-Jacobi Equations. In this Section we are given an open set Ω ⊆ RN , T > 0 and functions G1 , G2 : Ω × [0, T ] × R × R × RN × S N → R defined respectively by G1 (x, t, r, s, p, X)

α

:= s + sup {−Trace(aα (x, t)X) + b (x, t) · p + cα (x, t)r} ; α∈A

G2 (x, t, r, s, p, X)

α

:= s + inf {−Trace(aα (x, t)X) + b (x, t) · p + cα (x, t)r} . α∈A

Let us consider the partial differential equations corresponding respectively to G1 and G2 G1 (x, t, u, ut , Dx u, Dx2 u) = 0 in Ω × (0, T ), (3.1) and G2 (x, t, u, ut , Dx u, Dx2 u) = 0 in Ω × (0, T ).

(3.2)

Our aim is to extend to nonlinear parabolic equations of the form (3.1) and (3.2) the results of propagation of maxima that Bardi and the author obtained in [4] and [5] for elliptic Hamilton-Jacobi equations. The basic assumptions we will use in this Section are: (H0 ) A is a compact subset of a normed space;

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FRANCESCA DA LIO

(H1 ) a : Ω × [0, T ] × A → S N is a continuous function such that aα (x, t) ≥ 0, for all (x, t) ∈ Ω × [0, T ], for all α ∈ A and satisfies for some C > 0 ||aα (x, t) − aα (y, s)|| ≤ C(|x − y| + |t − s|),

for all (x, t), (y, s) ∈ Ω × [0, T ], α ∈ A; (3.3) (H2 ) c : Ω × [0, T ] × A → R is a continuous function such that cα (x, t) ≥ 0 for all (x, t) ∈ Ω × [0, T ] and for all α ∈ A; b : Ω × [0, T ] × A → RN is continuous and satisfies for some C > 0 α

α

|b (x, t) − b (y, s)| ≤ C(|x − y| + |t − s|),

for all (x, t), (y, s) ∈ Ω × [0, T ], α ∈ A; (3.4) (H3 ) for all α ∈ A, (x, t) ∈ Ω×[0, T ], aα (x, t) = 12 σ α (x, t)(σ α )T (x, t), where σ α (x, t) is a continuous N × M matrix satisfying for some C > 0 ||σ α (x, t) − σ α (y, s)|| ≤ C(|x − y| + |t − s|), where

T

for all (x, t), (y, s) ∈ Ω × [0, T ], α ∈ A, (3.5)

denotes the transpose matrix.

We remark that the results in this section still hold under weaker hypotheses on data. Yet for simplicity here we do not look for the most general assumptions and we refer the reader to the papers [4, 5] for remarks and comments in this directions. We start by listing some preliminary results that we use in this Section and have been obtained in [4] and [5] about the propagation of maxima for elliptic pde’s respectively of the form sup −Trace(aα (x)Dx2 u) + bα (x) · Dx u + cα (x)u = 0 in O, (3.6) α∈A

and inf

α∈A

−Trace(aα (x)Dx2 u) + bα (x) · Dx u + cα (x)u = 0 in O,

(3.7)

where O ⊆ Rk is an open set, with k ≥ 1, (for simplicity of notations we take k = N ). We start by defining precisely the sets where an interior maximum of a subsolution of (3.6) (or (3.7)) propagates and we recall the notion of subunit vector for the matrix aα (x). Definition 3.1. For x ∈ O and u ∈ U SC(O) viscosity subsolution of (3.6) (resp. (3.7)) with 0 ≤ supO u = u(x), P rop(x, u) is the set of points y ∈ O such that u(y) = u(x). The propagation set P rop(x) of a maximum at x is the intersection of all the sets P rop(x, u) as u varies among all subsolutions of (3.6) (resp. (3.7)) attaining a nonnegative maximum at x. Note that P rop(x, u) and P rop(x) are relatively closed in O for all x and u. We ∂2 recall that Z ∈ RN is a subunit vector of the operator −aα ij (·) ∂xi ∂xj at x, or of the matrix aα (x), if aα (x) − Z ⊗ Z ≥ 0, i.e., 2 ξ T aα (x)ξ = aα ij (x)ξi ξj ≥ (Z · ξ) ,

∀ξ ∈ RN .

The main examples of subunit vectors are the rescaled columns of aα and the columns of a square root of aα . For all α ∈ A we consider subunit vectors Zjα (y) corresponding to the matrix aα (y) appearing in (3.6) (resp. (3.7)) and the controlled deterministic system


y 0 (t) =

M X

α(t)

βj (t)Zj

(y(t)),

407

(3.8)

j=1

with control functions βj and α taking values, respectively, in [−1, 1] and A. In the convex case studied in [4] the two controllers α and β cooperate to drive the system from x to as many points as possible and all reachable points through the trajectories of (3.8) belong to P rop(x, u). Instead, in the concave case studied in [5] there is a conflict between the two controllers, α plays the role of a disturbance, and P rop(x, u) contains, roughly speaking, the points that β can reach under the worst possible behavior of α. In particular this last result is obtained within the theory of deterministic differential games by using a viability theorem proved in [6] and [9]. We introduce some preliminary notations and definitions. Nota 3.2. For all α ∈ A, h = 1, . . . , N, j = 1, . . . , M, we denote by Yhα , Xjα respectively the h-th column and j-th column of the matrix aα and σ α . If the α coefficient aα ij are differentiable, we denote by X0 the following vector field   N N α X X ∂a ij  ∂ . bα (3.9) X0α := i + ∂x ∂xi j i=1 j=1 Nota 3.3. Let Z1 , . . . Zk be vector fields. We denote by L(Z1 , . . . , Zk ) the Lie Algebra generated by Z1 , . . . , Zk . The following notion of generalized exterior normal goes back to Bony [8] and it is sometimes called proximal normal, see, for instance, [1]. Definition 3.4. Let O be an open subset of RN and ∅ = 6 K ⊆ O be a relatively closed set. A vector ν ∈ RN \ {0} is an exterior normal of K at y ∈ K (in short: ν ⊥ K at y) if for some δ > 0 \ B(y + δν, |δν|) K = ∅ . The following Proposition gives a necessary and sufficient condition under which a vector ν ∈ RN \ {0} is an exterior normal of a closed set K at a point y ∈ K. Proposition 3.5. Let K be a closed subset of RN . A vector ν ∈ RN \ {0} is an exterior normal of K at x ∈ K if and only if there exists C > 0 such that ν · (y − x) ≤ C|y − x|2 , as K 3 y → x.

(3.10)

The proof is elementary and we omit it. Now we recall the definitions of invariant and reachable set from x in O for a given controlled dynamical system. Definition 3.6. Let O be a connected open subset of RN , X : O × A → RN be a controlled vector field and ∅ 6= K ⊆ O. We say that K is invariant for if for all x ∈ K, for all α ∈ A := {α : [0, ∞) → A measurable}, and for all τ > 0 such that the solution yx (·, α) of the system ( y 0 (t) = X(y(t), α(t)) (S) y(0) = x ∈ K exists on [0, τ ), we have that yx (t, α) ∈ K for all t ∈ [0, τ ).

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FRANCESCA DA LIO

Definition 3.7. The reachable set from x in O for the controlled dynamical system (S) is RS (x)

:= {z ∈ Ω : ∃t > 0, ∃α ∈ A such that yx (t, α) = z, where yx (·, α) is the trajectory of (S) } .

Note that RS (x) is the smallest invariant set for X containing x. Next Theorem gives a sufficient condition for the invariance of a closed set with respect to a given vector field. It is a generalization of the Nagumo Theorem (see [22]) for controlled dynamical systems. We refer the reader to [4] for its proof. Theorem 3.8. Let X : O × A → RN be a vector field satisfying either a local Lipschitz condition with respect to the state uniformly in the control, i.e. for any compact K ⊆ O there exists LK > 0 such that |X(x, α) − X(y, α)| ≤ LK |x − y|, ∀x, y ∈ K, ∀α ∈ A;

(3.11)

or for some constant L > 0 (X(x, α) − X(y, α)) · (x − y) ≤ L|x − y|2 ,

∀x, y ∈ O, ∀α ∈ A.

(3.12)

Let ∅ = 6 S ⊆ O be a relatively closed set such that for all x ∈ S, for all ν ⊥ S at x and for any control α ∈ A we have X(x, α) · ν ≤ 0.

(3.13)

Then S is invariant for X. To give a more precise idea of the mechanism of propagation of maxima, we consider the following controlled dynamical system ( Pk αh (t) α0 (t) y 0 (t) = h=1 Zh (y(t))β h (t) − X0 (y(t))w(t) ˜ (S) y(0) = x where the control is (α0 . . . αk , β 1 . . . β k , w)(·) ∈ Ak × B k × W with B := {β : [0, ∞) → [−1, 1], measurable}, W := {w : [0, ∞) → [0, 1], measurable} and for all h ∈ {1 . . . k}, Zh : O × A → RN is a continuous subunit vector for the operator (3.6) satisfying either (3.11) or (3.12). In [4] (Theorem 3.1) the authors prove that if u ∈ U SC(O) is a viscosity subsolution of (3.6) that attains a nonnegative maximum at a some point x0 ∈ O, h then RS˜ (x0 ) ⊆ P rop(x0 ). In particular this result holds if the Zhα ’s are either the h αh columns of the matrix aα h or, if (H3 ) holds, of the matrix σh . In the case of concave operators (3.7) the propagation of maxima is more complicated and it is described in terms of suitable differential games. In order to avoid further notations and definitions we are not going to enter in details and we refer the reader to [5]. We remark that the characterization of the propagation set in term of the vector field X0α (x) has been obtained only in the convex case and it is still an open problem in the concave case. By using the properties of the set of propagation of maxima, in [4] and in [5] the authors prove the SMaxP for nonlinear operators that are not uniformly elliptic. In particular they show the SMaxP for strictly elliptic operators and for operators


409

satisfying a sort of H¨ ormander conditions thus extending the Bony’s Strong Maximum Principle for linear hypoelliptic operators (see [8]) to the nonlinear convex and concave case. More precisely the conditions under which the SMaxP holds for elliptic convex pde’s are given by the following two Theorems. Theorem 3.9. ([4]) Assume (H0 )−(H3 ). Let u ∈ U SC(O) be a viscosity subsolution of (3.6) that achieves a nonnegative maximum in O. If for any x ∈ O there exists αx ∈ A such that aαx (x) is positive definite, then P rop(x) = O. Theorem 3.10. ([4]) Assume (H0 ) − (H3 ). Let u ∈ U SC(O) be a viscosity subsolution of (3.6) that achieves a nonnegative maximum in O. Suppose there are k ≥ 1 subunit vector fields Zhα ∈ C ∞ (RN ) associated to the operator (3.6) satisfying the following property : for every x ∈ O and for every h = 1, . . . , k there exists αh := αh (x) such that rkL(Z1α1 . . . Zkαk ) = N (3.14) at the point x. Then P rop(x) = O. On the other hand for elliptic concave pde’s we have the following two results. Theorem 3.11. ([5]) Assume (H0 ) − (H3 ). Let u ∈ U SC(O) be a viscosity subsolution of (3.7) that achieves a nonnegative maximum in O. Assume that for all α ∈ A and x ∈ O the matrix aα (x) is positive definite. Then P rop(x) = O. Theorem 3.12. ([5]) Assume (H0 ) − (H3 ). Let u ∈ U SC(O) be a viscosity subsolution of (3.7) that achieves a nonnegative maximum in O. Suppose there are k ≥ 1 vector fields Zh ∈ C ∞ (RN ), which are independent of α and subunit for the operator (3.7), such that the Lie algebra they generate has full rank N at every point x ∈ O. Then P rop(x) = O. In order to extend the previous results to the parabolic equations (3.1) and (3.2) we consider them as a particular case respectively of (3.6) and (3.7) where aα and bα are given respectively by  α  0 a11 (x, t) · · · aα 1N (x, t)  .. .. ..  ..  . . . .  aα (x, t) :=   α  aα aN N (x, t) 0  N 1 (x, t) · · · 0 ··· 0 0 α

bα (x, t) := (b (x, t), 1). We first remark that for parabolic pde’s of the form (3.1) and (3.2) we always have the local vertical propagation of the maxima in the sense of Lemma 2.1. Indeed one can see that they satisfy the conditions (1.8) and (1.12). Thus by Theorem 2.1 and Corollary 2.2 the SMaxP holds for (3.1) (resp. (3.2)) if the operator is strictly parabolic, namely the coefficients aα (x, t) satisfy the hypotheses of Theorem 3.9 (resp. Theorem 3.11), in this case the nondegeneracy condition (1.8) being trivially verified. Next we are going to give a slight different proof of the SMaxP for (3.1) (resp.(3.2)) in the case where the coefficients aα (x, t) satisfy the hypotheses either of Theorem 3.9 or Theorem 3.10 (resp. Theorem 3.11 and Theorem 3.12), by using directly the properties of the set of propagation of the maxima of subsolutions to (3.1) and

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FRANCESCA DA LIO

(3.2). In this way we are able to get the SMaxP also for operators which are not strictly parabolic. The main results of this Section are the following two Theorems where we prove the SMaxP respectively for convex and concave parabolic operators. Theorem 3.13. [SMaxP convex case] Assume (H0 ) − (H3 ) and either the hypotheses of Theorem 3.9 or of Theorem 3.10. Let u ∈ U SC(QT ) be a viscosity subsolution of (3.1) that attains a nonnegative maximum at some point P0 := (x0 , t0 ) ∈ QT . Then u is a constant in S(P0 ). Theorem 3.14. [SMaxP concave case] Assume (H0 ) − (H3 ) and either the hypotheses of Theorem 3.11 or of Theorem 3.12. Let u ∈ U SC(QT ) be a viscosity subsolution of (3.2) that attains a nonnegative maximum at some point P0 := (x0 , t0 ) ∈ QT . Then u is a constant in S(P0 ). Before proving Theorem 3.13 and 3.14 we first show the horizontal propagation of maxima for (3.1) and (3.2). Proposition 3.15. Assume (H0 ) − (H3 ) and either the hypotheses of Theorem 3.9 or of Theorem 3.10. Let u ∈ U SC(QT ) be a viscosity subsolution of (3.1) that attains a nonnegative maximum at some point P0 := (x0 , t0 ) ∈ QT . Then u is a constant in C(P0 ). Proposition 3.16. Assume (H0 ) − (H3 ) and either the hypotheses of Theorem 3.11 or of Theorem 3.12. Let u ∈ U SC(QT ) be a viscosity subsolution of (3.2) that attains a nonnegative maximum at some point P0 := (x0 , t0 ) ∈ QT . Then u is a constant in C(P0 ). We are going to prove only Proposition 3.15 (the proof of Proposition 3.16 being similar). To this end we premise the following Lemma. Lemma 3.17. Assume (H0 ) − (H3 ). Let u ∈ U SC(QT ) be a viscosity subsolution of (3.1) (resp. (3.2)) that attains a nonnegative maximum at some point P0 := (x0 , t0 ) ∈ QT . Then, for any (x, t) ∈ ∂P rop((x0 , t0 ), u) ∩ QT and ν ⊥ P rop((x0 , t0 ), u) at (x, t), for all α ∈ A (resp. there exists α ∈ A such that), for all subunit vectors Z of the matrix aα (x, t) we have Z · ν = 0. Proof of Lemma 3.17. We consider only the case when u is a subsolution of (3.1) (the case when u is a subsolution of (3.2) being proved in a similar way). Fix (x0 , t0 ) ∈ QT and set K = P rop((x0 , t0 ), u) and let (x, t) ∈ ∂K, ν ⊥ K at (x, t). Suppose by contradiction that there exists α ∈ A and a subunit vector Z of the matrix aα (x, t) such that Z · ν 6= 0. This implies ν · aα (x, t)ν > 0. By definition of ν there exists (˜ x, t˜) ∈ QT and R > 0 such that B((˜ x, t˜), R) ⊆ QT \ K ˜ and B((˜ x, t), R) ∩ K = {(x, t)}. We set for simplicity y = (x, t) and y˜ = (˜ x, t˜) and consider the function defined by 2

2

v(y) := e−γR − e−γ|y−˜y| ,

(3.15)

where γ is positive constant to be determined. Then one proceeds exactly as in the proof of Theorem 2.1 and get a contradiction. Next we prove Proposition 3.15.


411

Proof of Proposition 3.15. Suppose by contradiction there exists a point P1 = (x1 , t0 ) such that u(P1 ) < u(P0 ). By standard geometric arguments and the upper semicontinuity of u, we can find an ellipsoid Eλ0 |x − x|2 + λ0 |t − t0 |2 ≤ ε2 λ0 , with (x, t0 ) ∈ C(P0 ), such that u < u(P0 ) in the interior of Eλ0 and a P ∗ = / ∂Ω × (0, T ). Let (x∗ , t∗ ) ∈ ∂Eλ0 such that u(P ∗ ) = u(P0 ), with x 6= x∗ and P ∗ ∈ B := B((˜ x, t˜), R) be such that B ⊆ Eλ0 , and ∂B ∩ ∂Eλ0 = {(x∗ , t∗ )}. We observe that the vector ν = (˜ x − x∗ , t˜ − t∗ ) is ⊥ P rop(x∗ , t∗ ) at (x∗ , t∗ ). We first assume that aα satisfies the hypotheses of Theorem 3.9. In this case there exists α = α(x∗ , t∗ ) such that the first N columns of aα are linearly independent. Moreover by Lemma 3.17 for all h = 1, . . . , N, we have Yhα (x∗ , t∗ ) · ν = 0.

(3.16) α

∗

∗

Since the columns and the subunit vectors of the matrix a (x , t ) have the last component equal to 0, the condition (3.16) implies that the first N components of ν are zero, and this contradicts the fact that x ˜ 6= x∗ . Suppose the hypotheses of Theorem 3.10 hold. In this case for every h = 1, . . . , k, there exists αh := αh (x∗ , t∗ ) such that rkL(Z1α1 . . . Zkαk ) = N

(3.17)

at the point (x∗ , t∗ ). We need the following Lemma whose proof is postponed. Lemma 3.18. For every vector field Z ∈ L(Z1α1 . . . Zkαk ), we have Z · ν = 0. From the condition (3.17) and Lemma 3.18 it follows that there are N vector fields Wj ∈ L(Z1α1 . . . Zkαk ), j = 1, . . . , N, which are linearly independent and satisfy Wj · ν = 0. Since (Wj )N +1 = 0, the previous condition of orthogonality of Wj with respect to ν implies that the first N components of ν are zero, that contradicts again the fact that x ˜ 6= x∗ . Thus we can conclude the proof of Proposition 3.15. Next we give the proof of Lemma 3.18. Proof of Lemma 3.18. The proof is very similar to the one of Corollary 3.2 in [4] and we give it for the reader’s convenience. Suppose by contradiction that there exists Z = Z(x∗ , t∗ ) ∈ L(Z1α1 . . . Zkαk ) such that Z · ν = r > 0. Set y ∗ = (x∗ , t∗ ), and consider the following system ( Pk h y 0 (τ ) = h=1 Zhα (y(τ ))β h (τ ); (S 0 ) y(0) = y ∗ , where, for all h ∈ {1 . . . k}, β h (·) ∈ B, whose trajectories are also trajectories of ˜ Since the system (S 0 ) is symmetric, by a known result of geometric control (S). theory (see, e.g. [1] and references therein), there exist C > 0 and, for any τ > 0, a 1 k piecewise constant control (β . . . β )(·) generating a trajectory of the form y(τ ) = y ∗ + CZ(y ∗ )τ p+1 + o(τ p+1 ), as τ → 0,

(3.18)

where p is the number of bracket operations necessary to generate Z. We recall that by the results in [4] we have RS 0 (x∗ , t∗ ) ⊆ P rop(x∗ , t∗ ). Thus from Proposition 3.5 it follows that for some C 0 , C 00 > 0 we have (y(τ ) − y ∗ ) · ν ≤ C 0 |yx0 (τ ) − x0 |2 ≤ C 00 τ 2p+2 as τ → 0.

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FRANCESCA DA LIO

On the other hand (y(τ ) − y ∗ ) · ν

= CZ(y ∗ ) · ντ p+1 + o(τ p+1 ) = rτ

p+1

+ o(τ

p+1

(3.19)

) as τ → 0.

This gives the desired contradiction and complete the proof. Now we prove Theorem 3.13, namely the SMaxP for convex equations of the form (3.1). The proof of Theorem 3.14 is similar and we omit it. Proof of Theorem 3.13. We divide the proof in two steps. Step 1. Define the set K := {(x, t) ∈ QT : u(x, t) = max u} . QT

N +1

We consider in R the vector b0 = (0, . . . , 0, 1). We claim that for any (x, t) ∈ K, ν ⊥ K at (x, t) we have b0 · ν ≥ 0, namely, by Theorem 3.8, K is invariant for the vector −b0 . Suppose by contradiction that there exist (x0 , t0 ) ∈ QT and ν ⊥ K at (x0 , t0 ) such that b0 · ν < 0. For simplicity of notations we suppose that (x0 , t0 ) = (0, 0). We also observe that we may assume ν = (0, . . . , 0, νN +1 ). Indeed by the same arguments of Proposition 3.15, under the assumptions of either Theorem 3.9 or of Theorem 3.10, one can find N vector fields Wj which are linearly independent and satisfy Wj · ν = 0. This last conditions implies that the first N components of ν are zero. By definition of ν, we have for some r > 0, B(((0, 0), rν), r|ν|) ⊆ (QT \ K) ∪ {(0, 0)}). Now for 0 < δ ≤ r and γ > 0 we consider the ellipsoid E := {(x, t) ∈ RN +1 : γ 2 |x|2 + (t − δνN +1 )2 ≤ δ 2 |ν|2 } . δ We observe that for δ and small enough E is contained in QT . Moreover we also γ have E ⊆ (QT \ K) ∪ {(0, 0)}. Indeed if there is a point (¯ x, t¯) ∈ K ∩ E, then by Proposition 3.15, such a maximum would propagate in B(((0, 0), δν), δν) ∩ {t = t¯} . Let us introduce the auxiliary function v(x, t) := e−δ

2

|ν|2

− e−(γ

2

|x|2 +(t−δνN +1 )2 )

.

(3.20)

Note that v(0, 0) = 0 and −1 < v(x, t) < 0 in E. For all α ∈ A we set Lα v := vt − Trace(aα D2 v) + bα · Dv + cα v.

(3.21)

Direct calculations give Dv(x, t) 2

D v(x) vt (x, t)

= = =

2γ 2 e−(γ

2

|x|2 +(t−δνN +1 )2 )

2 −(γ 2 |x|2 +(t−δνN +1 )2 )

2γ e

2e−(γ

2

2

2

|x| +(t−δνN +1 ) )

x I − 4γ 4 e−(γ

2

|x|2 +(t−δνN +1 )2 )

(t − δνN +1 ) .

Thus we have Lα v(x, t)

:= 2e−(γ

2

|x|2 +(t−δνN +1 )2 )

−(γ 2 |x|2 +(t−δνN +1 )2 )

(t − δνN +1 )

{4γ 4 aα ij (x, t)xi xj

+

e

+

α α 2γ 2 [−aα ii (x, t) + bi (x, t)xi ] − c (x, t)}

+

cα (x, t)e−δ

Notice that Lα v(0, 0) = e−δ

2

|ν|2

2

|ν|2

.

(−2δνN +1 + 2γ 2 (−aα ii (0, 0))).

(x ⊗ x),


413

Now let 0 ≤ M = supα∈A aα (0, 0) < +∞. Since −νN +1 > 0, if we choose

δ > γ2

M then for all α ∈ A we get Lα v(0, 0) ≥ C > 0 and consequently −νN +1 G1 (0, 0, v(0, 0), vt (0, 0), Dv(0, 0), D2 v(0, 0)) ≥ C.

By (H0 ), there exists µ > 0 such that G1 (x, t, v(x, t), vt (x, t), Dv(x, t), D2 v(x, t)) ≥ C, ∀(x, t) ∈ B((0, 0), µ)

(3.22)

Define the set X := B((0, 0), µ) ∩ E and let us choose ε > 0 small enough so that u(x, t)−u(0, 0) ≤ εv(x, t), ∀(x, t) ∈ ∂X. Note that εv is a strict supersolution of (3.1) in X as well. We claim that u(x, t) − u(0, 0) ≤ εv(x, t), ∀(x, t) ∈ X. In fact suppose by contradiction that there exists (x, t) ∈ X such that u(x, t¯) − u(0, 0) − εv(x, t¯) = maxX u(x, t) − u(0, 0) − εv(x, t) > 0. Since εv ∈ C ∞ (R), we can use the fact that u − u(0, 0) is a viscosity subsolution of (3.1) as well to get G1 (x, t, εv(x, t), εvt (x, t), εDv(x, t), εD2 v(x, t)) ≤ 0

(3.23)

which contradicts (3.22) and we conclude the first step. Step 2. We prove that u is constant in any rectangle < := {(x, t) : xi0 − ai ≤ xi ≤ xi0 + ai , t0 − a0 ≤ t ≤ t0 } ⊆ Ω × [0, t0 ]. Suppose by contradiction there exists Q := (x, t) ∈