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Strong comparison principles for some nonlinear degenerate elliptic equations
Acta Mathematica Scientia 2018,38B(5):1583–1590 http://actams.wipm.ac.cn
STRONG COMPARISON PRINCIPLES FOR SOME NONLINEAR DEGENERATE ELLIPTIC EQUATIONS∗
oññ)
Yanyan LI (
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China; Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd, Piscataway, NJ 08854, USA E-mail : [email protected]
Æ)
Bo WANG (
Corresponding author. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China E-mail : [email protected]
In Memory of Professor Xiaqi DING Abstract In this paper, we obtain the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators of the form ∇2 ψ + L(x, ∇ψ), including the conformal hessian operator. Key words
Hopf lemma; strong comparison principle; degenerate ellipticity; conformal invariance.
2010 MR Subject Classification
1
35J60; 35J70; 35B51; 35B65; 35D40; 53C21; 58J70
Introduction
In this paper, we establish the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators. For a positive integer n ≥ 2, let Ω be an open connected bounded subset of Rn , the ndimensional euclidean space. For any C 2 function u in Ω, we consider a symmetric matrix function F [u] := ∇2 u + L(·, ∇u), (1.1) 0,1 where L ∈ Cloc (Ω × Rn ), is in S n×n , the set of all n × n real symmetric matrices. One such matrix operator is the conformal hessian operator (see e.g. [21, 27] and the references therein), that is,
1 A[u] = ∇2 u + ∇u ⊗ ∇u − |∇u|2 I, 2 where I denotes the n×n identity matrix, and for p, q ∈ Rn , p⊗q denotes the n×n matrix with entries (p ⊗ q)ij = pi qj , i, j = 1, · · · , n. Some comparison principles for this matrix operator ∗ Received April 3, 2018. Li was partially supported by NSF grant DMS-1501004. Wang was partially supported by NSFC (11701027).
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have been studied in [22–25]. Comparison principles for other classes of (degenerate) elliptic operators are available in the literature. See [1–5, 7–20, 26] and the references therein. Let U be an open subset of S n×n , satisfying 0 ∈ ∂U,
U + P ⊂ U,
tU ⊂ U, ∀ t > 0,
(1.2)
where P is the set of all non-negative matrices. Furthermore, in order to conclude that the strong comparison principle holds, we assume Condition Uν , as introduced in [25], for some unit vector ν in Rn : there exists µ = µ(ν) > 0 such that U + Cµ (ν) ⊂ U.
(1.3)
Here Cµ (ν) := {t(ν ⊗ ν + A) : A ∈ S n×n , kAk < µ, t > 0}. Some counter examples for the strong maximum principle were given in [25] to show that the condition (1.3) cannot be simply dropped. Remark 1.1 If U satisfies (1.2), diag{1, 0, · · · , 0} ∈ U, and Ot U O ⊂ U,
∀ O ∈ O(n),
where O(n) denotes the set of n × n orthogonal matrices, then it is easy to see that U satisfies (1.3). 0,1 Let u, v ∈ Cloc (Ω). We say that
F [u] ∈ S n×n \ U
¯ , F [v] ∈ U
in Ω
(1.4)
in the viscosity sense, if for any x0 ∈ Ω, ϕ ∈ C 2 (Ω), (ϕ − u)(x0 ) = 0 ((ϕ − v)(x0 ) = 0) and u−ϕ≥0
(v − ϕ ≤ 0),
near x0 ,
there holds F [ϕ](x0 ) ∈ S n×n \ U
¯ . F [ϕ](x0 ) ∈ U
We have the following strong comparison principle and Hopf Lemma. Theorem 1.2 (strong comparison principle) Let Ω be an open connected subset of Rn , n ≥ 2, U be an open subset of S n×n , satisfying (1.2) and Condition Uν for every unit vector ν 0,1 0,1 in Rn , and F be of the form (1.1) with L ∈ Cloc (Ω × Rn ). Assume that u, v ∈ Cloc (Ω) satisfy (1.4) in the viscosity sense, u ≥ v in Ω. Then either u > v in Ω or u ≡ v in Ω.
Theorem 1.3 (Hopf lemma) Let Ω be an open connected subset of Rn , n ≥ 2, ∂Ω be C 2 near a point xˆ ∈ ∂Ω, and U be an open subset of S n×n , satisfying (1.2) and Condition Uν for 0,1 ν = ν(ˆ x), the interior unit normal of ∂Ω at x ˆ, and F be of the form (1.1) with L ∈ Cloc (Ω×Rn ). 0,1 Assume that u, v ∈ Cloc (Ω∪{ˆ x}) satisfy (1.4) in the viscosity sense, u > v in Ω and u(ˆ x) = v(ˆ x). Then we have (u − v)(ˆ x + sν(ˆ x)) lim inf > 0. s s→0+ Remark 1.4 If u and v ∈ C 2 , then Theorems 1.2 and 1.3 were proved in [25].
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Proof of Theorem 1.2
Proof of Theorem 1.2 We argue by contradiction. Suppose the conclusion is false. 0,1 Since u − v ∈ Cloc (Ω) is non-negative, the set {x ∈ Ω : u = v} is closed. Then there exists an open ball B(x0 , R) ⊂⊂ Ω centered at x0 ∈ Ω with radius R > 0 such that u − v > 0, in B(x0 , R)\{ˆ x}, u(ˆ x) − v(ˆ x) = 0, x ˆ ∈ ∂B(x0 , R). We make use of the standard comparison function 2
2
h(x) := e−α|x−x0 | − e−αR , For i, j = 1, · · · , n, we have hi (x) = and
∀ α > 0, x ∈ Ω.
2 ∂ h(x) = −2α(xi − (x0 )i )e−α|x−x0 | , ∂xi
1 ∂2 2 −α|x−x0 |2 h(x) = 4α e δij . (xi − (x0 )i )(xj − (x0 )j ) − hij (x) = ∂xi ∂xj 2α
Choose 0 < R′ < x ∈ B(ˆ x, δ),
R 2
(2.1)
(2.2)
(2.3)
such that B(ˆ x, R′ ) ⊂⊂ Ω. For any δ ∈ (0, R′ ), we have that for any −1 ≤ h(x) ≤ 1,
for some C > 0 independent of δ and α. It follows that, for any 0 < εˆ <
|∇h(x)| + |∇2 h(x)| ≤ C min
(B(ˆ x,δ)\B(ˆ x, 12 δ))∩B(x0 ,R)
u − v − εˆh > 0,
1 on B(ˆ x, δ)\B(ˆ x, δ), 2
(2.4)
(u − v),
(u − v − εˆh)(ˆ x) = 0.
(2.5)
Indeed, by (2.4) and the fact that h < 0 outside B(x0 , R), for any 1 x ∈ (B(ˆ x, δ)\B(ˆ x, δ))\B(x0 , R), 2
(u − v)(x) ≥ 0 > εˆh(x);
and for any x ∈ (B(ˆ x, δ)\B(ˆ x, 12 δ)) ∩ B(x0 , R), (u − v)(x) ≥
min
(B(ˆ x,δ)\B(ˆ x, 21 δ))∩B(x0 ,R)
(u − v) > εˆ ≥ εˆh(x).
For any ǫ > 0, we define the ǫ-lower and upper envelope of u and v as n o 1 uǫ (x) := min u(y) + |x − y|2 , ∀x ∈ B(x0 , R) ∪ B(ˆ x, R′ ), ǫ x,R′ ) y∈B(x0 ,R)∪B(ˆ and v ǫ (x) :=
n o 1 v(y) − |x − y|2 , ǫ y∈B(x0 ,R)∪B(ˆ x,R′ ) max
∀x ∈ B(x0 , R) ∪ B(ˆ x, R′ ),
respectively. Then we conclude that there exists ǫ0 = ǫ0 (δ, α, εˆ) such that for 0 < ǫ < ǫ0 , min (uǫ − v ǫ − εˆh) ≤ 0,
B(ˆ x,δ)
1 uǫ − v ǫ − εˆh > 0 on B(ˆ x, δ)\B(ˆ x, δ). 2
(2.6)
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Indeed, the first part of (2.6) follows from the definitions of uǫ and v ǫ , and the fact that h(ˆ x) = 0; ǫ (uǫ − v − εˆh)(ˆ x) ≤ (u − v)(ˆ x) = 0. Now we prove the second part of (2.6). By theorem 5.1 (a) in [6], we have that uǫ − v ǫ ↑ u − v
uniformly on B(x0 , R) ∪ B(ˆ x, R′ ),
as ǫ → 0.
It follows that for any M > 0, there exists ǫ0 (M ) > 0 such that (uǫ − v ǫ − εˆh)(x) >
min
B(ˆ x,δ)\B(ˆ x, 21 δ)
(u − v − εˆh) − M
x, δ)\B(ˆ x, 12 δ). Then by taking 0 < M < for any 0 < ǫ < ǫ0 and any x ∈ B(ˆ
1 2
min
B(ˆ x,δ)\B(ˆ x, 21 δ)
(u −
v − εˆh), (2.6) is obtained. It follows from (2.6) that there exists η¯ = η¯(δ, α, εˆ) > 0 such that for any η ∈ (0, η¯), there exists τ = τ (ǫ, η, δ, α, εˆ) ∈ R1 such that 1 uǫ − v ǫ − εˆh − τ > 0 on B(ˆ x, δ)\B(ˆ x, δ). 2
min (uǫ − v ǫ − εˆh − τ ) = −η,
B(ˆ x,δ)
(2.7)
Let ξǫ := uǫ − v ǫ − εˆh − τ, and Γξǫ− denote the convex envelope of ξǫ− := − min{ξǫ , 0} on B(ˆ x, δ). Then by (20) in [24] and (2.4), we have 4 1 ∇2 ξǫ ≤ I + C εˆI a.e. in B(ˆ x, δ). ǫ 2 And by lemma 3.5 in [6], we have Z det(∇2 Γξǫ− ) > 0, {ξǫ =Γξ− } ǫ
which implies that the Lebesgue measure of {ξǫ = Γξǫ− } is positive. Then there exists xǫ,η ∈ {ξǫ = Γξǫ− } ∩ B(ˆ x, 12 δ) such that both of v ǫ and uǫ are punctually second order differentiable at xǫ,η , 0 > ξǫ (xǫ,η ) ≥ −η, (2.8) |∇ξǫ (xǫ,η )| ≤ Cη,
(2.9)
∇2 ξǫ (xǫ,η ) = ∇2 (uǫ − v ǫ − εˆh)(xǫ,η ) ≥ 0.
(2.10)
and For xǫ,η ∈ Ω, by the definitions of uǫ and v ǫ , there exist (xǫ,η )∗ and (xǫ,η )∗ ∈ Ω such that 1 uǫ (xǫ,η ) = u((xǫ,η )∗ ) + |(xǫ,η )∗ − xǫ,η |2 , ǫ
and
1 v ǫ (xǫ,η ) = v((xǫ,η )∗ ) − |(xǫ,η )∗ − xǫ,η |2 . ǫ 0,1 Since u and v ∈ Cloc (Ω), by (2.6) and (2.7) in [23], we have
|(xǫ,η )∗ − xǫ,η | + |(xǫ,η )∗ − xǫ,η | ≤ C1 ǫ,
(2.11)
|∇uǫ (xǫ,η )| + |∇v ǫ (xǫ,η )| ≤ C2 ,
(2.12)
and
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where C1 and C2 are two universal positive constant independent of ǫ and η. Since uǫ is punctually second order differentiable at xǫ,η , we have 1 uǫ (xǫ,η + z) ≥ uǫ (xǫ,η ) + ∇uǫ (xǫ,η ) · z + z T ∇2 uǫ (xǫ,η )z + o(|z|2 ), 2 By the definition of uǫ , we have 1 uǫ (xǫ,η + z) ≤ u((xǫ,η )∗ + z) + |(xǫ,η )∗ − xǫ,η |2 , ǫ and therefore, in view of (2.13),
as z → 0.
(2.13)
1 u((xǫ,η )∗ + z) ≥ uǫ (xǫ,η + z) − |(xǫ,η )∗ − xǫ,η |2 ǫ ≥ Pǫ ((xǫ,η )∗ + z) + o(|z|2 ), as z → 0, where Pǫ is a quadratic polynomial with 1 Pǫ ((xǫ,η )∗ ) = uǫ (xǫ,η ) − |(xǫ,η )∗ − xǫ,η |2 = u(xǫ,η )∗ ), ǫ ∇Pǫ ((xǫ,η )∗ ) = ∇uǫ (xǫ,η ),
∇2 Pǫ ((xǫ,η )∗ ) = ∇2 uǫ (xǫ,η ).
Since u satisfies (1.4) in the viscosity sense, we thus have ∇2 uǫ (xǫ,η ) + L((xǫ,η )∗ , ∇uǫ (xǫ,η )) = F [Pǫ ]((xǫ,η )∗ ) ∈ S n×n \ U.
(2.14)
0,1 On the other hand, in view of (2.11), (2.12) and the fact that L ∈ Cloc (Ω × Rn ),
L(xǫ,η , ∇uǫ (xǫ,η )) − L((xǫ,η )∗ , ∇uǫ (xǫ,η )) ≤ C|xǫ,η − (xǫ,η )∗ |I ≤ a1 ǫI,
(2.15)
where C and a1 > 0 are universal constants. It follows from (1.2), (2.14) and (2.15) that F [uǫ ](xǫ,η ) − a1 ǫI ∈ S n×n \U.
(2.16)
Analogusly, we can obtain ¯ F [v ǫ ](xǫ,η ) + a2 ǫI ∈ U for some universal constants a2 > 0. 0,1 By (2.9), (2.10), (2.12) and the fact that L ∈ Cloc (Ω × Rn ), F [uǫ ](xǫ,η ) ≥ ∇2 (v ǫ + εˆh)(xǫ,η ) + L(xǫ,η , ∇uǫ (xǫ,η ))
= F [v ǫ + εˆh](xǫ,η ) + L(xǫ,η , ∇uǫ (xǫ,η )) − L(xǫ,η , ∇v ǫ (xǫ,η ))
≥ F [v ǫ + εˆh](xǫ,η ) − C|∇(uǫ − v ǫ )(xǫ,η )|
≥ F [v ǫ + εˆh](xǫ,η ) − C(η + εˆ|∇h(xǫ,η )|)I.
(2.17)
0,1 By (2.4), (2.12) and the fact that L ∈ Cloc (Ω × Rn ), we have
F [v ǫ + εˆh](xǫ,η ) = F [v ǫ ](xǫ,η ) + εˆ∇2 h(xǫ,η ) + L(xǫ,η , ∇(v ǫ + εˆh)(xǫ,η )) − L(xǫ,η , ∇v ǫ (xǫ,η )) ≥F [v ǫ ](xǫ,η ) + εˆ[∇2 h(xǫ,η ) − C|∇h(xǫ,η )|I].
Then by (2.2), (2.3) and the fact |xǫ,η − x0 | < 2R, ∇2 h(xǫ,η ) − C|∇h(xǫ,η )|I
(2.18)
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1 C I− |xǫ,η − x0 |I 2α 4α 2 C ≥ 4α2 e−α|xǫ,η −x0 | (xǫ,η − x0 ) ⊗ (xǫ,η − x0 ) − I α 2 C ≥ 4α2 e−α|xǫ,η −x0 | [(ˆ x − x0 ) ⊗ (ˆ x − x0 ) − CδRI − I] α C x ˆ − x0 x ˆ − x0 2 2 −α|xǫ,η −x0 |2 ⊗ − CδI − I = 4R α e R R α 2 x ˆ − x0 C xˆ − x0 ≥ 4R2 α2 e−4R α ⊗ − CδI − I . R R α = 4α2 e−α|xǫ,η −x0 |
2
(xǫ,η − x0 ) ⊗ (xǫ,η − x0 ) −
Vol.38 Ser.B
(2.19)
Inserting (2.19) into (2.18), we have F [v ǫ + εh](xǫ,η ) ǫ
2
2 −4R2 α
≥ F [v ](xǫ,η ) + 4R εˆα e
x ˆ − x0 xˆ − x0 C ⊗ − CδI − I . R R α
(2.20)
It follows from (2.17) and (2.20) that F [uǫ ](xǫ,η ) − a1 ǫI
≥ F [v ǫ ](xǫ,η ) + a2 ǫI " 2
2 −4R2 α
+ 4R εˆα e
x ˆ−x 0
R
⊗
xˆ − x 0
R
# 2 C e4R α − CδI − I − C (ǫ + η)I . α εˆα2
(2.21)
We can firstly fix the value of small δ > 0 and a large α > 1, then fix the value of small εˆ > 0, and lastly fix the value of small ǫ and η > 0 such that
4R2 α
1 xˆ − x0
CδI + C I + C e
< µ (ǫ + η)I ,
2 α εˆα2 R where µ is obtained from condition (1.3). Therefore, by (1.3) and (2.21), we have that
F [uǫ ](xǫ,η ) − a1 ǫI ∈ U, which is a contradiction with (2.16). Theorem 1.2 is proved.
3
Proof of Theorem 1.3
Proof of Theorem 1.3 Since ∂Ω is C 2 near x ˆ, there exists an open ball B(x0 , R) ⊂ Ω x} and such that B(x0 , R) ∩ ∂Ω = {ˆ u − v > 0, in B(x , R)\{ˆ x}, 0 u(ˆ x) − v(ˆ x) = 0. Let h be defined as in (2.1). We work in the domain
Aδ := B(ˆ x, δ) ∩ B(x0 , R). It is easy to see that u − v ≥ εˆh,
on ∂Aδ
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Y.Y. Li & B. Wang: STRONG COMPARISON PRINCIPLES
for any 0 < δ <
R 2
and 0 < εˆ <
min ∂B(ˆ x,δ)∩B(x0 ,R)
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(u − v).
We claim that for εˆ small enough, u − v ≥ εˆh,
on Aδ .
Once the claim is proved, then we have that lim inf s→0+
2 (u − v)(ˆ x + sν(ˆ x)) h(ˆ x + sν(ˆ x)) ≥ εˆ lim inf = 2ˆ εαRe−αR > 0. + s s s→0
Therefore, in order to finish the proof of Theorem 1.3, we only need to prove the above claim. Suppose the contrary, that is, ζ = ζ(ˆ ε, α, δ) := min(u − v − εˆh) < 0. Aδ
It follows that min(u − v − εˆh − ζ) = 0, Aδ
u − v − εˆh − ζ ≥ −ζ > 0 on ∂Aδ .
Now we can follow the argument as in the proof of Theorem 1.2 to get a contradiction. Theorem 1.3 is proved. References [1] Amendola M E, Galise G, Vitolo A. Riesz capacity, maximum principle, and removable sets of fully nonlinear second-order elliptic operators. Differential Integral Equations, 2013, 26: 845–866 [2] Bardi M, Da Lio F. On the strong maximum principle for fully nonlinear degenerate elliptic equations. Arch Math (Basel), 1999, 73: 276–285 [3] Berestycki H, Capuzzo Dolcetta I, Porretta A, Rossi L. Maximum principle and generalized principal eigenvalue for degenerate elliptic operators. J Math Pures Appl, 2015, 103(5): 1276–1293 [4] Birindelli I, Demengel F. Comparison principle and Liouville type results for singular fully nonlinear operators. Ann Fac Sci Toulouse Math, 2004, 13(6): 261–287 [5] Birindelli I, Demengel F. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Commun Pure Appl Anal, 2007, 6: 335–366 [6] Caffarelli L, Cabr´ e X. Fully Nonlinear Elliptic Equations. Vol 43 of American Mathematical Society Colloquium Publications. Providence, RI: Amer Math Soc, 1995 [7] Caffarelli L, Li Y Y, Nirenberg L. Some remarks on singular solutions of nonlinear elliptic equations III: viscosity solutions including parabolic operators. Comm Pure Appl Math, 2013, 66: 109–143 [8] Crandall M G, Ishii H, Lions P -L. User’s guide to viscosity solutions of second order partial differential equations. Bull Amer Math Soc (NS), 1992, 27: 1–67 [9] Dolcetta I C, Vitolo A. On the maximum principle for viscosity solutions of fully nonlinear elliptic equations in general domains. Matematiche (Catania), 2007, 62: 69–91 [10] Dolcetta I C, Vitolo A. The weak maximum principle for degenerate elliptic operators in unbounded domains. preprint [11] Harvey F R, Lawson Jr H B. Existence, uniqueness and removable singularities for nonlinear partial differential equations in geometry//Surveys in Differential Geometry. Geometry and Topology. Vol 18 of Surv Differ Geom. Somerville, MA: International Press, 2013: 103–156 [12] Harvey F R, Lawson Jr H B. Characterizing the strong maximum principle for constant coefficient subequations. Rend Mat Appl, 2016, 37(1/2): 63–104 [13] Ishii H. On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs. Comm Pure Appl Math, 1989, 42: 15–45 [14] Ishii H, Lions P -L. Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J Differential Equations, 1990, 83: 26–78 [15] Jensen R. The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch Rational Mech Anal, 1988, 101: 1–27
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[16] Kawohl B, Kutev N. Strong maximum principle for semicontinuous viscosity solutions of nonlinear partial differential equations. Arch Math (Basel), 1998, 70: 470–478 [17] Kawohl B, Kutev N. Comparison principle and Lipschitz regularity for viscosity solutions of some classes of nonlinear partial differential equations. Funkcial Ekvac, 2000, 43: 241–253 [18] Kawohl B, Kutev N. Comparison principle for viscosity solutions of fully nonlinear, degenerate elliptic equations. Comm Partial Differential Equations, 2007, 32: 1209–1224 [19] Koike S, Kosugi T. Remarks on the comparison principle for quasilinear PDE with no zeroth order terms. Commun Pure Appl Anal, 2015, 14: 133–142 [20] Koike S, Ley O. Comparison principle for unbounded viscosity solutions of degenerate elliptic PDEs with gradient superlinear terms. J Math Anal Appl, 2011, 381: 110–120 [21] Li A, Li Y Y. On some conformally invariant fully nonlinear equations. Comm Pure Appl Math, 2003, 56: 1416–1464 [22] Li Y Y. Degenerate conformally invariant fully nonlinear elliptic equations. Arch Rational Mech Anal, 2007, 186: 25–51 [23] Li Y Y. Local gradient estimates of solutions to some conformally invariant fully nonlinear equations. Comm Pure Appl Math, 2009, 62: 1293–1326; C R Math Acad Sci Paris, 2006, 343(4): 249–252 [24] Li Y Y, Nguyen L, Wang B. Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations. http://arxiv.org/abs/1612.09418v1. To appear in Calc Var Partial Differential Equations [25] Li Y Y, Nirenberg L. A miscellany//Percorsi incrociati (in ricordo di Vittorio Cafagna). Collana Scientifica di Ateneo, Universita di Salerno, 2010: 193–208. http://arxiv.org/abs/0910.0323 [26] Trudinger N S. Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations. Rev Mat Iberoamericana, 1988, 4: 453–468 [27] Viaclovsky J. Conformal geometry, contact geometry, and the calculus of variations. Duke Math J, 2000, 101: 283–316
STRONG COMPARISON PRINCIPLES FOR SOME NONLINEAR DEGENERATE ELLIPTIC EQUATIONS∗
oññ)
Yanyan LI (
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China; Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd, Piscataway, NJ 08854, USA E-mail : [email protected]
Æ)
Bo WANG (
Corresponding author. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China E-mail : [email protected]
In Memory of Professor Xiaqi DING Abstract In this paper, we obtain the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators of the form ∇2 ψ + L(x, ∇ψ), including the conformal hessian operator. Key words
Hopf lemma; strong comparison principle; degenerate ellipticity; conformal invariance.
2010 MR Subject Classification
1
35J60; 35J70; 35B51; 35B65; 35D40; 53C21; 58J70
Introduction
In this paper, we establish the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators. For a positive integer n ≥ 2, let Ω be an open connected bounded subset of Rn , the ndimensional euclidean space. For any C 2 function u in Ω, we consider a symmetric matrix function F [u] := ∇2 u + L(·, ∇u), (1.1) 0,1 where L ∈ Cloc (Ω × Rn ), is in S n×n , the set of all n × n real symmetric matrices. One such matrix operator is the conformal hessian operator (see e.g. [21, 27] and the references therein), that is,
1 A[u] = ∇2 u + ∇u ⊗ ∇u − |∇u|2 I, 2 where I denotes the n×n identity matrix, and for p, q ∈ Rn , p⊗q denotes the n×n matrix with entries (p ⊗ q)ij = pi qj , i, j = 1, · · · , n. Some comparison principles for this matrix operator ∗ Received April 3, 2018. Li was partially supported by NSF grant DMS-1501004. Wang was partially supported by NSFC (11701027).
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have been studied in [22–25]. Comparison principles for other classes of (degenerate) elliptic operators are available in the literature. See [1–5, 7–20, 26] and the references therein. Let U be an open subset of S n×n , satisfying 0 ∈ ∂U,
U + P ⊂ U,
tU ⊂ U, ∀ t > 0,
(1.2)
where P is the set of all non-negative matrices. Furthermore, in order to conclude that the strong comparison principle holds, we assume Condition Uν , as introduced in [25], for some unit vector ν in Rn : there exists µ = µ(ν) > 0 such that U + Cµ (ν) ⊂ U.
(1.3)
Here Cµ (ν) := {t(ν ⊗ ν + A) : A ∈ S n×n , kAk < µ, t > 0}. Some counter examples for the strong maximum principle were given in [25] to show that the condition (1.3) cannot be simply dropped. Remark 1.1 If U satisfies (1.2), diag{1, 0, · · · , 0} ∈ U, and Ot U O ⊂ U,
∀ O ∈ O(n),
where O(n) denotes the set of n × n orthogonal matrices, then it is easy to see that U satisfies (1.3). 0,1 Let u, v ∈ Cloc (Ω). We say that
F [u] ∈ S n×n \ U
¯ , F [v] ∈ U
in Ω
(1.4)
in the viscosity sense, if for any x0 ∈ Ω, ϕ ∈ C 2 (Ω), (ϕ − u)(x0 ) = 0 ((ϕ − v)(x0 ) = 0) and u−ϕ≥0
(v − ϕ ≤ 0),
near x0 ,
there holds F [ϕ](x0 ) ∈ S n×n \ U
¯ . F [ϕ](x0 ) ∈ U
We have the following strong comparison principle and Hopf Lemma. Theorem 1.2 (strong comparison principle) Let Ω be an open connected subset of Rn , n ≥ 2, U be an open subset of S n×n , satisfying (1.2) and Condition Uν for every unit vector ν 0,1 0,1 in Rn , and F be of the form (1.1) with L ∈ Cloc (Ω × Rn ). Assume that u, v ∈ Cloc (Ω) satisfy (1.4) in the viscosity sense, u ≥ v in Ω. Then either u > v in Ω or u ≡ v in Ω.
Theorem 1.3 (Hopf lemma) Let Ω be an open connected subset of Rn , n ≥ 2, ∂Ω be C 2 near a point xˆ ∈ ∂Ω, and U be an open subset of S n×n , satisfying (1.2) and Condition Uν for 0,1 ν = ν(ˆ x), the interior unit normal of ∂Ω at x ˆ, and F be of the form (1.1) with L ∈ Cloc (Ω×Rn ). 0,1 Assume that u, v ∈ Cloc (Ω∪{ˆ x}) satisfy (1.4) in the viscosity sense, u > v in Ω and u(ˆ x) = v(ˆ x). Then we have (u − v)(ˆ x + sν(ˆ x)) lim inf > 0. s s→0+ Remark 1.4 If u and v ∈ C 2 , then Theorems 1.2 and 1.3 were proved in [25].
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Proof of Theorem 1.2
Proof of Theorem 1.2 We argue by contradiction. Suppose the conclusion is false. 0,1 Since u − v ∈ Cloc (Ω) is non-negative, the set {x ∈ Ω : u = v} is closed. Then there exists an open ball B(x0 , R) ⊂⊂ Ω centered at x0 ∈ Ω with radius R > 0 such that u − v > 0, in B(x0 , R)\{ˆ x}, u(ˆ x) − v(ˆ x) = 0, x ˆ ∈ ∂B(x0 , R). We make use of the standard comparison function 2
2
h(x) := e−α|x−x0 | − e−αR , For i, j = 1, · · · , n, we have hi (x) = and
∀ α > 0, x ∈ Ω.
2 ∂ h(x) = −2α(xi − (x0 )i )e−α|x−x0 | , ∂xi
1 ∂2 2 −α|x−x0 |2 h(x) = 4α e δij . (xi − (x0 )i )(xj − (x0 )j ) − hij (x) = ∂xi ∂xj 2α
Choose 0 < R′ < x ∈ B(ˆ x, δ),
R 2
(2.1)
(2.2)
(2.3)
such that B(ˆ x, R′ ) ⊂⊂ Ω. For any δ ∈ (0, R′ ), we have that for any −1 ≤ h(x) ≤ 1,
for some C > 0 independent of δ and α. It follows that, for any 0 < εˆ <
|∇h(x)| + |∇2 h(x)| ≤ C min
(B(ˆ x,δ)\B(ˆ x, 12 δ))∩B(x0 ,R)
u − v − εˆh > 0,
1 on B(ˆ x, δ)\B(ˆ x, δ), 2
(2.4)
(u − v),
(u − v − εˆh)(ˆ x) = 0.
(2.5)
Indeed, by (2.4) and the fact that h < 0 outside B(x0 , R), for any 1 x ∈ (B(ˆ x, δ)\B(ˆ x, δ))\B(x0 , R), 2
(u − v)(x) ≥ 0 > εˆh(x);
and for any x ∈ (B(ˆ x, δ)\B(ˆ x, 12 δ)) ∩ B(x0 , R), (u − v)(x) ≥
min
(B(ˆ x,δ)\B(ˆ x, 21 δ))∩B(x0 ,R)
(u − v) > εˆ ≥ εˆh(x).
For any ǫ > 0, we define the ǫ-lower and upper envelope of u and v as n o 1 uǫ (x) := min u(y) + |x − y|2 , ∀x ∈ B(x0 , R) ∪ B(ˆ x, R′ ), ǫ x,R′ ) y∈B(x0 ,R)∪B(ˆ and v ǫ (x) :=
n o 1 v(y) − |x − y|2 , ǫ y∈B(x0 ,R)∪B(ˆ x,R′ ) max
∀x ∈ B(x0 , R) ∪ B(ˆ x, R′ ),
respectively. Then we conclude that there exists ǫ0 = ǫ0 (δ, α, εˆ) such that for 0 < ǫ < ǫ0 , min (uǫ − v ǫ − εˆh) ≤ 0,
B(ˆ x,δ)
1 uǫ − v ǫ − εˆh > 0 on B(ˆ x, δ)\B(ˆ x, δ). 2
(2.6)
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Indeed, the first part of (2.6) follows from the definitions of uǫ and v ǫ , and the fact that h(ˆ x) = 0; ǫ (uǫ − v − εˆh)(ˆ x) ≤ (u − v)(ˆ x) = 0. Now we prove the second part of (2.6). By theorem 5.1 (a) in [6], we have that uǫ − v ǫ ↑ u − v
uniformly on B(x0 , R) ∪ B(ˆ x, R′ ),
as ǫ → 0.
It follows that for any M > 0, there exists ǫ0 (M ) > 0 such that (uǫ − v ǫ − εˆh)(x) >
min
B(ˆ x,δ)\B(ˆ x, 21 δ)
(u − v − εˆh) − M
x, δ)\B(ˆ x, 12 δ). Then by taking 0 < M < for any 0 < ǫ < ǫ0 and any x ∈ B(ˆ
1 2
min
B(ˆ x,δ)\B(ˆ x, 21 δ)
(u −
v − εˆh), (2.6) is obtained. It follows from (2.6) that there exists η¯ = η¯(δ, α, εˆ) > 0 such that for any η ∈ (0, η¯), there exists τ = τ (ǫ, η, δ, α, εˆ) ∈ R1 such that 1 uǫ − v ǫ − εˆh − τ > 0 on B(ˆ x, δ)\B(ˆ x, δ). 2
min (uǫ − v ǫ − εˆh − τ ) = −η,
B(ˆ x,δ)
(2.7)
Let ξǫ := uǫ − v ǫ − εˆh − τ, and Γξǫ− denote the convex envelope of ξǫ− := − min{ξǫ , 0} on B(ˆ x, δ). Then by (20) in [24] and (2.4), we have 4 1 ∇2 ξǫ ≤ I + C εˆI a.e. in B(ˆ x, δ). ǫ 2 And by lemma 3.5 in [6], we have Z det(∇2 Γξǫ− ) > 0, {ξǫ =Γξ− } ǫ
which implies that the Lebesgue measure of {ξǫ = Γξǫ− } is positive. Then there exists xǫ,η ∈ {ξǫ = Γξǫ− } ∩ B(ˆ x, 12 δ) such that both of v ǫ and uǫ are punctually second order differentiable at xǫ,η , 0 > ξǫ (xǫ,η ) ≥ −η, (2.8) |∇ξǫ (xǫ,η )| ≤ Cη,
(2.9)
∇2 ξǫ (xǫ,η ) = ∇2 (uǫ − v ǫ − εˆh)(xǫ,η ) ≥ 0.
(2.10)
and For xǫ,η ∈ Ω, by the definitions of uǫ and v ǫ , there exist (xǫ,η )∗ and (xǫ,η )∗ ∈ Ω such that 1 uǫ (xǫ,η ) = u((xǫ,η )∗ ) + |(xǫ,η )∗ − xǫ,η |2 , ǫ
and
1 v ǫ (xǫ,η ) = v((xǫ,η )∗ ) − |(xǫ,η )∗ − xǫ,η |2 . ǫ 0,1 Since u and v ∈ Cloc (Ω), by (2.6) and (2.7) in [23], we have
|(xǫ,η )∗ − xǫ,η | + |(xǫ,η )∗ − xǫ,η | ≤ C1 ǫ,
(2.11)
|∇uǫ (xǫ,η )| + |∇v ǫ (xǫ,η )| ≤ C2 ,
(2.12)
and
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where C1 and C2 are two universal positive constant independent of ǫ and η. Since uǫ is punctually second order differentiable at xǫ,η , we have 1 uǫ (xǫ,η + z) ≥ uǫ (xǫ,η ) + ∇uǫ (xǫ,η ) · z + z T ∇2 uǫ (xǫ,η )z + o(|z|2 ), 2 By the definition of uǫ , we have 1 uǫ (xǫ,η + z) ≤ u((xǫ,η )∗ + z) + |(xǫ,η )∗ − xǫ,η |2 , ǫ and therefore, in view of (2.13),
as z → 0.
(2.13)
1 u((xǫ,η )∗ + z) ≥ uǫ (xǫ,η + z) − |(xǫ,η )∗ − xǫ,η |2 ǫ ≥ Pǫ ((xǫ,η )∗ + z) + o(|z|2 ), as z → 0, where Pǫ is a quadratic polynomial with 1 Pǫ ((xǫ,η )∗ ) = uǫ (xǫ,η ) − |(xǫ,η )∗ − xǫ,η |2 = u(xǫ,η )∗ ), ǫ ∇Pǫ ((xǫ,η )∗ ) = ∇uǫ (xǫ,η ),
∇2 Pǫ ((xǫ,η )∗ ) = ∇2 uǫ (xǫ,η ).
Since u satisfies (1.4) in the viscosity sense, we thus have ∇2 uǫ (xǫ,η ) + L((xǫ,η )∗ , ∇uǫ (xǫ,η )) = F [Pǫ ]((xǫ,η )∗ ) ∈ S n×n \ U.
(2.14)
0,1 On the other hand, in view of (2.11), (2.12) and the fact that L ∈ Cloc (Ω × Rn ),
L(xǫ,η , ∇uǫ (xǫ,η )) − L((xǫ,η )∗ , ∇uǫ (xǫ,η )) ≤ C|xǫ,η − (xǫ,η )∗ |I ≤ a1 ǫI,
(2.15)
where C and a1 > 0 are universal constants. It follows from (1.2), (2.14) and (2.15) that F [uǫ ](xǫ,η ) − a1 ǫI ∈ S n×n \U.
(2.16)
Analogusly, we can obtain ¯ F [v ǫ ](xǫ,η ) + a2 ǫI ∈ U for some universal constants a2 > 0. 0,1 By (2.9), (2.10), (2.12) and the fact that L ∈ Cloc (Ω × Rn ), F [uǫ ](xǫ,η ) ≥ ∇2 (v ǫ + εˆh)(xǫ,η ) + L(xǫ,η , ∇uǫ (xǫ,η ))
= F [v ǫ + εˆh](xǫ,η ) + L(xǫ,η , ∇uǫ (xǫ,η )) − L(xǫ,η , ∇v ǫ (xǫ,η ))
≥ F [v ǫ + εˆh](xǫ,η ) − C|∇(uǫ − v ǫ )(xǫ,η )|
≥ F [v ǫ + εˆh](xǫ,η ) − C(η + εˆ|∇h(xǫ,η )|)I.
(2.17)
0,1 By (2.4), (2.12) and the fact that L ∈ Cloc (Ω × Rn ), we have
F [v ǫ + εˆh](xǫ,η ) = F [v ǫ ](xǫ,η ) + εˆ∇2 h(xǫ,η ) + L(xǫ,η , ∇(v ǫ + εˆh)(xǫ,η )) − L(xǫ,η , ∇v ǫ (xǫ,η )) ≥F [v ǫ ](xǫ,η ) + εˆ[∇2 h(xǫ,η ) − C|∇h(xǫ,η )|I].
Then by (2.2), (2.3) and the fact |xǫ,η − x0 | < 2R, ∇2 h(xǫ,η ) − C|∇h(xǫ,η )|I
(2.18)
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1 C I− |xǫ,η − x0 |I 2α 4α 2 C ≥ 4α2 e−α|xǫ,η −x0 | (xǫ,η − x0 ) ⊗ (xǫ,η − x0 ) − I α 2 C ≥ 4α2 e−α|xǫ,η −x0 | [(ˆ x − x0 ) ⊗ (ˆ x − x0 ) − CδRI − I] α C x ˆ − x0 x ˆ − x0 2 2 −α|xǫ,η −x0 |2 ⊗ − CδI − I = 4R α e R R α 2 x ˆ − x0 C xˆ − x0 ≥ 4R2 α2 e−4R α ⊗ − CδI − I . R R α = 4α2 e−α|xǫ,η −x0 |
2
(xǫ,η − x0 ) ⊗ (xǫ,η − x0 ) −
Vol.38 Ser.B
(2.19)
Inserting (2.19) into (2.18), we have F [v ǫ + εh](xǫ,η ) ǫ
2
2 −4R2 α
≥ F [v ](xǫ,η ) + 4R εˆα e
x ˆ − x0 xˆ − x0 C ⊗ − CδI − I . R R α
(2.20)
It follows from (2.17) and (2.20) that F [uǫ ](xǫ,η ) − a1 ǫI
≥ F [v ǫ ](xǫ,η ) + a2 ǫI " 2
2 −4R2 α
+ 4R εˆα e
x ˆ−x 0
R
⊗
xˆ − x 0
R
# 2 C e4R α − CδI − I − C (ǫ + η)I . α εˆα2
(2.21)
We can firstly fix the value of small δ > 0 and a large α > 1, then fix the value of small εˆ > 0, and lastly fix the value of small ǫ and η > 0 such that
4R2 α
1 xˆ − x0
CδI + C I + C e
< µ (ǫ + η)I ,
2 α εˆα2 R where µ is obtained from condition (1.3). Therefore, by (1.3) and (2.21), we have that
F [uǫ ](xǫ,η ) − a1 ǫI ∈ U, which is a contradiction with (2.16). Theorem 1.2 is proved.
3
Proof of Theorem 1.3
Proof of Theorem 1.3 Since ∂Ω is C 2 near x ˆ, there exists an open ball B(x0 , R) ⊂ Ω x} and such that B(x0 , R) ∩ ∂Ω = {ˆ u − v > 0, in B(x , R)\{ˆ x}, 0 u(ˆ x) − v(ˆ x) = 0. Let h be defined as in (2.1). We work in the domain
Aδ := B(ˆ x, δ) ∩ B(x0 , R). It is easy to see that u − v ≥ εˆh,
on ∂Aδ
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Y.Y. Li & B. Wang: STRONG COMPARISON PRINCIPLES
for any 0 < δ <
R 2
and 0 < εˆ <
min ∂B(ˆ x,δ)∩B(x0 ,R)
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(u − v).
We claim that for εˆ small enough, u − v ≥ εˆh,
on Aδ .
Once the claim is proved, then we have that lim inf s→0+
2 (u − v)(ˆ x + sν(ˆ x)) h(ˆ x + sν(ˆ x)) ≥ εˆ lim inf = 2ˆ εαRe−αR > 0. + s s s→0
Therefore, in order to finish the proof of Theorem 1.3, we only need to prove the above claim. Suppose the contrary, that is, ζ = ζ(ˆ ε, α, δ) := min(u − v − εˆh) < 0. Aδ
It follows that min(u − v − εˆh − ζ) = 0, Aδ
u − v − εˆh − ζ ≥ −ζ > 0 on ∂Aδ .
Now we can follow the argument as in the proof of Theorem 1.2 to get a contradiction. Theorem 1.3 is proved. References [1] Amendola M E, Galise G, Vitolo A. Riesz capacity, maximum principle, and removable sets of fully nonlinear second-order elliptic operators. Differential Integral Equations, 2013, 26: 845–866 [2] Bardi M, Da Lio F. On the strong maximum principle for fully nonlinear degenerate elliptic equations. Arch Math (Basel), 1999, 73: 276–285 [3] Berestycki H, Capuzzo Dolcetta I, Porretta A, Rossi L. Maximum principle and generalized principal eigenvalue for degenerate elliptic operators. J Math Pures Appl, 2015, 103(5): 1276–1293 [4] Birindelli I, Demengel F. Comparison principle and Liouville type results for singular fully nonlinear operators. Ann Fac Sci Toulouse Math, 2004, 13(6): 261–287 [5] Birindelli I, Demengel F. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Commun Pure Appl Anal, 2007, 6: 335–366 [6] Caffarelli L, Cabr´ e X. Fully Nonlinear Elliptic Equations. Vol 43 of American Mathematical Society Colloquium Publications. Providence, RI: Amer Math Soc, 1995 [7] Caffarelli L, Li Y Y, Nirenberg L. Some remarks on singular solutions of nonlinear elliptic equations III: viscosity solutions including parabolic operators. Comm Pure Appl Math, 2013, 66: 109–143 [8] Crandall M G, Ishii H, Lions P -L. User’s guide to viscosity solutions of second order partial differential equations. Bull Amer Math Soc (NS), 1992, 27: 1–67 [9] Dolcetta I C, Vitolo A. On the maximum principle for viscosity solutions of fully nonlinear elliptic equations in general domains. Matematiche (Catania), 2007, 62: 69–91 [10] Dolcetta I C, Vitolo A. The weak maximum principle for degenerate elliptic operators in unbounded domains. preprint [11] Harvey F R, Lawson Jr H B. Existence, uniqueness and removable singularities for nonlinear partial differential equations in geometry//Surveys in Differential Geometry. Geometry and Topology. Vol 18 of Surv Differ Geom. Somerville, MA: International Press, 2013: 103–156 [12] Harvey F R, Lawson Jr H B. Characterizing the strong maximum principle for constant coefficient subequations. Rend Mat Appl, 2016, 37(1/2): 63–104 [13] Ishii H. On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs. Comm Pure Appl Math, 1989, 42: 15–45 [14] Ishii H, Lions P -L. Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J Differential Equations, 1990, 83: 26–78 [15] Jensen R. The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch Rational Mech Anal, 1988, 101: 1–27
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