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On existence and uniqueness of viscosity solutions to obstacle problems for a class of second order partial differential equations
Accepted Manuscript On Existence and Uniqueness of Viscosity Solutions to Obstacle Problems for a class of Second Order Partial Differential Equations
Sijia Bao, Yuming Xing
PII: DOI: Reference:
S0022-247X(19)30187-8 https://doi.org/10.1016/j.jmaa.2019.02.057 YJMAA 22990
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
11 September 2018
Please cite this article in press as: S. Bao, Y. Xing, On Existence and Uniqueness of Viscosity Solutions to Obstacle Problems for a class of Second Order Partial Differential Equations, J. Math. Anal. Appl. (2019), https://doi.org/10.1016/j.jmaa.2019.02.057
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ON EXISTENCE AND UNIQUENESS OF VISCOSITY SOLUTIONS TO OBSTACLE PROBLEMS FOR A CLASS OF SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS
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SIJIA BAO AND YUMING XING
1 2
Abstract. We prove the existence and uniqueness of the viscosity solutions to the obstacle problems for second order differential partial equations under some conditions. As applications, we show some existence and uniqueness theorems of the viscosity solutions to the obstacle problems for k-Hessian equations, which include Monge-Amp` ere type equations when k = n. Keywords: obstacle problem; viscosity solution; k-Hessian equation
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1. Introduction
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In this paper we study the obstacle problems for second order partial differential equations max {F (x, u, Du, D2 u), u − g} = 0 in Ω (1.1) u=ϕ on ∂Ω,
7
8 9 10
where Ω is a bounded domain in Rn , g is bounded in Ω, g ≥ ϕ ∈ C(∂Ω), F : Ω × R × Rn × S n −→ R is continuous and S n is the set of symmetric n × n matrices. We also assume that F satisfies a fundamental monotonicity condition (see [1]), (1.2)
11
F (x, r, p, X) ≤ F (x, s, p, X) whenever r ≤ s,
and (1.3)
F (x, r, p, X) ≤ F (x, r, p, Y ) whenever Y ≤ X,
16
where r, s ∈ R, x, p ∈ Rn , X, Y ∈ S n and S n is equipped with its usual order. We say that F is degenerate elliptic if it satisfies (1.3). When (1.2) also holds, we will say that F is proper. In [1], Crandall, Ishii and Lions introduced the concept of viscosity solution of second order partial differential equations, see the next section for the definition.
17
Definition 1.1. We say that the comparison principle holds for the equation
12 13 14 15
F (x, u, Du, D2 u) = 0 18 19 20
if whenever u, u ∈ C(Ω) are respectively viscosity subsolution and supersolution of F = 0 and u ≤ u on ∂Ω, then u ≤ u in Ω. By using Perron’s method we will prove Theorem 1.2. Suppose that there is a subsolution u ∈ C(Ω) of F (x, u, Du, D2 u) = 0 that satisfies the boundary condition u(x) = ϕ(x) for x ∈ ∂Ω and u ≤ g. Denote Sϕ = {w : u ≤ w ≤ g and w ∈ C(Ω) is a subsolution of F = 0 with w|∂Ω ≡ ϕ}. 1
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SIJIA BAO AND YUMING XING
If (1.4)
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
35 36 37 38 39
u(x) = sup w(x) w∈Sϕ
is continuous on Ω, then u is a viscosity solution of (1.1). If, in addition, the comparison principle holds for F = 0, the solution is unique. We can see that in order to deduce the existence of solutions of obstacle problems, we need the existence of u and continuity of u, and used the comparison principle to show the uniqueness. Naturally one will ask when u is continuous. In fact, we can prove Theorem 1.3. If there is a subsolution u ∈ C(Ω) of F = 0 that satisfies u|∂Ω ≡ ϕ and u ≤ g, and Sϕ forms an equicontinuous family, then the function u defined in (1.4) is continuous on Ω and u is a viscosity solution of (1.1). Obviously, by Theorem 1.3, we can obtain the following Corollary 1.4. If u exists and Sϕ is a bounded subset of C α (Ω) for some α ∈ (0, 1], then u defined in (1.4) is a viscosity solution of (1.1). If we just have the interior C α regularity of functions in Sϕ , in order to deduce the continuity of u at the boundary ∂Ω should have sufficient regularity. We call a sequence of functions {wi (x)} in C(Ω) a (upper) barrier in Ω relative to F and ϕ at x0 ∈ ∂Ω, if (i) wi is a viscosity supersolution of F = 0 and wi |∂Ω ≥ ϕ, i = 1, 2, · · · ; (ii) wi (x0 ) → ϕ(x0 ) as i → ∞. It is easy to obtain Lemma 1.5. Let comparison principle hold for F = 0. If u exists and there is a barrier at x0 ∈ ∂Ω, then u(x) → ϕ(x0 ) as x → x0 . Then we have Corollary 1.6. Let comparison principle hold for F = 0. Assume that u exists α and there is a barrier at any x0 ∈ ∂Ω. If Sϕ ⊂ Cloc (Ω) for some α ∈ (0, 1] and the α C bounds of functions in Sϕ in any Ω ⊂⊂ Ω are uniform, then u defined in (1.4) is the unique viscosity solution of (1.1). The above Corollary 1.4 and Corollary 1.6 arouse our interest in the C α regularity of viscosity subsolutions of F = 0. In [2], Guy Barles has proved the following C α regularity of viscosity subsolutions: Theorem 1.7. ([2])Let P be a class of functions h : [0, ∞) → [0, ∞) satisfying: (i) h is a C 1 , convex function, (ii) t → h(t) is non-decreasing for t ≥ 1, +∞t2 t (iii) 1 h(t) dt < +∞. Assume that for any R > 0, R (M ) + K1R h(|p|) − K2R ≤ 0 (1.5) F (x, r, p, M ) ≤ 0 ⇒ G for any M ∈ S n , p ∈ Rn , |r| ≤ R, x ∈ Ω, where K1R , K2R are positive constants, h is R is a Lipschitz continuous function satisfying the a function in the class P and G ellipticity condition, which is homogeneous of degree 1. Then, any locally bounded subsolution u of F = 0 is uniformly continuous in Ωδ with a modulus of continuity depending only on δ, Rδ := u L∞ (Ωδ/2 ) and the different constants and functions
OBSTACLE PROBLEMS 1 2 3 4 5 6 7 8 9
3
appearing in (1.5) with R = Rδ . Moreover, if h(t) = tm (m > 2), Ω is a C 1,1 domain and either u is bounded on Ω or K1R , K2R can be chosen independent of R, then u ∈ C α (Ω). Consequently we have Theorem 1.8. Let comparison principle and (1.5) hold for F = 0. Assume that u exists. If either there is a barrier at any x0 ∈ ∂Ω or h(t) = tm , Ω is a C 1,1 -domain then there is a (unique) viscosity solution of (1.1). Let us end the Introduction with some examples satisfying condition (1.5). The main example is from [2] where Theorem 1.7 was proved. Let F1 (x, r, p, M ) = −tr(a(x, r)M ) + H(x, p) + c(x)u − f (x), x ∈ Ω ⊂ Rn
10 11 12
be a viscous Hamilton-Jacobi Equation, where a : Ω × R → S n is continuous and satisfies a(x, r) = {aij (x, r)} ≥ 0 for all x ∈ Ω and r ∈ R, H : Ω × Rn → R is continuous and there exists constants K1 , K2 > 0 and m > 2 such that H(x, p) ≥ K1 |p|m − K2 ,
13 14 15
16 17 18
for all x ∈ Ω and p ∈ Rn ,
and c, f : Ω → R are continuous. In fact, there exists a constant ΛR > 0 depending on R such that aij (x, r)ξi ξj ≤ ΛR |ξ|2 for all ξ ∈ Rn and for any x ∈ Ω and |r| ≤ R. Define GR (M ) = −ΛR tr(M ) and K2R = K2 + R supΩ |c| + supΩ |f |. It is easy to find that F1 satisfies (1.5) for such GR , K1 and K2R . We can also consider some fully nonlinear cases. For example, let 1/k
F2 (x, r, p, M ) = −Sk (M ) + ψ(x, r, p), 19 20
where the operator Sk is defined as in (4.1), ψ is a positive continuous function which may depend on x, r and p and satisfies ψ(x, r, p) ≥ K1R h(|p|) − K2R
32
whenever |r| ≤ R and x ∈ Ω, here h is a function in the class P. Let G(M ) = 1/k −Sk (M ). Obviously, F2 satisfies (1.5). However, G is not elliptic for all M ∈ S n . Instead, G is elliptic when restricted to a subset of S n , which is {M ∈ S n : λ(M ) ∈ Γk } (see (4.2) for the definition of Γk ), where λ(M ) denotes the eigenvalues of M . See Section 4 for more details. It would be an interesting problem that whether we can obtain similar results as Theorem 1.7 for the equation F2 (x, u, Du, D2 u) = 0. It will also be interested to considering such equations on manifolds. The plan of this paper is as follows. In the next section we introduce some preliminaries which may be useful. In section 3, we prove the existence and uniqueness of viscosity solutions. Finally, in section 4, we show some existence and uniqueness theorems which is viscosity solution of the obstacle problem for the k-Hessian equations as applications.
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2. Preliminaries
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We first recall the notions of viscosity subsolutions, supersolutions, and solutions in [1].
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SIJIA BAO AND YUMING XING
Definition 2.1. Let F be proper. A viscosity subsolution of F = 0 in Ω is an upper semicontinuous function u : Ω → R such that (2.1)
F (x, u(x), p, X) ≤ 0 f or all x ∈ Ω and (p, X) ∈ JΩ2,+ u(x),
where JΩ2,+ u(x) is the set of the (p, X) ∈ Rn × S n such that 1 < X(y − x), y − x > +o(|y − x|2 ) as y → x. 2 Similarly, a viscosity supersolution of F = 0 in Ω is lower semicontinuous function u : Ω → R such that u(y) ≤ u(x)+ < p, y − x > +
3 4
(2.2) 5 6 7
F (x, u(x), p, X) ≥ 0 f or all x ∈ Ω and (p, X) ∈ JΩ2,− u(x),
where JΩ2,− u(x) = −JΩ2,+ (−u)(x). Finally, u is a viscosity solution of F = 0 in Ω if it is both a viscosity subsolution and a viscosity supersolution of F = 0 in Ω. It is easy to see that JΩ2,+ u(x) = {(Dφ(x), D2 φ(x)) : φ is C 2 and u − φ has a local maximum at x}. Definition 2.2. If u : Ω → [−∞, ∞], then we say u∗ (x) = lim sup {u(y) : y ∈ Ω and |y − x| ≤ r} r→0
the upper semicontinuous envelope of u; it is the smallest upper semicontinuous function (with values in [−∞, ∞]) satisfying u ≤ u∗ . Similarly, we say u∗ (x) = lim inf {u(y) : y ∈ Ω and |y − x| ≤ r} r→0
8 9 10 11
the lower semicontinuous envelope of u. Definition 2.3. We say a set Ω ⊂ Rn locally compact if each point x ∈ Ω has a compact neighbourhood with respect to the topology of Ω. We need the Lemma 4.2 in [1]:
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¯× Lemma 2.4. Let Ω ⊂ Rn be locally compact, F be lower semicontinuous in Ω n n R × R × S , and F be a family of solutions of F ≤ 0 in Ω. Let w(x) = sup {u(x) : u ∈ F} and assume that w∗ (x) < ∞ for x ∈ Ω. Then w∗ is a solution of F ≤ 0 in Ω.
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3. Existence and uniqueness of viscosity solution
17
We are ready to prove theorem 1.2. Proof of theorem 1.2 It is easy to see that u = ϕ on ∂Ω. By lemma 2.4 u∗ is a subsolution of F = 0 and hence, because of the continuity of u,u = u∗ and u is a subsolution of F = 0 and u ≤ g. So u is a subsolution of (1.1). Next we prove that u is a supersolution of F = 0 in E = {x ∈ Ω : u(x) < g(x)}. Actually, if not, there exists a point x ∈ E such that
12 13 14
18 19 20
x). F ( x, u( x), p, X) < 0 f or some (p, X) ∈ JΩ2,− u( We may assume x = 0. Then by continuity uδ,γ = u(0) + δ+ < p, x > +
1 < Xx, x > −γ|x|2 2
OBSTACLE PROBLEMS
5
is a classical solution of F ≤ 0 in Br = {x : |x| < r} and uδ,γ (x) ≤ g(x) for small r, δ, γ > 0. Since u(x) ≥ u(0)+ < p, x > +
1 < Xx, x > +o(|x|2 ), 2
if we choose δ = (r2 /8)γ then u(x) > uδ,γ (x) for r/2 ≤ |x| ≤ r if r is sufficiently small and then, by lemma 2.4, the function max {u(x), uδ,γ (x)} if |x| < r U (x) = u(x) otherwise, 1 2 3 4
is a viscosity solution of F ≤ 0 in Ω. Note that U (x) ≤ g(x), U = ϕ on ∂Ω for small r and U (0) ≥ uδ,γ (0) = u(0) + δ > u(0), this contradicts the definition of u. Thus u is a supersolution of F = 0 in E. Then, obviously, u is a supersolution of (1.1). So u is a viscosity solution of (1.1). Now if the comparison principle holds for F = 0, we assume u1 , u2 are two (viscosity) solutions to (1.1). Suppose there exists a point x0 ∈ Ω, such that u1 (x0 ) < u2 (x0 ). Let G ⊂ Ω be a connected domain containing x0 , such that u1 (x) < u2 (x) in G,
u1 = u2 on ∂G.
Since u2 ≤ g in Ω, u1 < g in G. Thus u1 is a solution of F = 0 in G. On the other hand, u2 is a subsolution of F = 0. Then by comparison principle, we have u2 ≤ u1 in G, 5
which is a contradiction, then we get the uniqueness. We complete the proof.
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Now we deal with theorem 1.3. Proof of theorem 1.3 Firstly since g is bounded u is well defined. Since u is the supremum of continuous functions, it is lower semicontinuous. Let {yn }∞ n=1 ⊂ Ω be dense on Ω. For each n, there exists a sequence {vn,m } ⊂ Sϕ such that lim vn,m (yn ) = u(yn ).
m→∞
By virtue of lemma 2.4, the maximum of any finite set of subsolutions is also a subsolution. Therefore, we may assume that for each n, {vn,m } is a monotonically nondecreasing sequence. Next we will construct a monotonically nondecreasing sequence {um } in Sϕ that will converge to u at each point yn . Let u1 = v1,1 . Let u2 (x) = max {u1 (x), v1,2 (x), v2,2 (x)}. Note that u2 ∈ Sϕ , and u2 ≥ u1 . Having defined ui , define ui+1 by ui+1 (x) = max {ui (x), v1,i+1 (x), v2,i+1 (x), . . . vi+1,i+1 (x)}. Then ui+1 ∈ Sϕ , and ui+1 ≥ ui ≥ · · · ≥ u1 . We also have that each ui is continuous on Ω. Since for any n and any j ≥ n, u(yn ) ≥ uj (yn ) ≥ vn,j (yn ), we have that limj→∞ uj (yn ) = u(yn ). Because of the equicontinuity of Sϕ , {uj } is equicontinuous on Ω. Since {uj } ⊂ Sϕ , {uj } is uniformly bounded. Therefore, by Arzela-Ascoli’s theorem, a subsequence of {uj } converges uniformly on Ω to a function U ∈ C(Ω). Thus, by the monotonicity of {uj }, we obtain that uj → U .
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SIJIA BAO AND YUMING XING
Then U (yn ) = u(yn ). By density, for any x ∈ Ω, there exists a sequence {zj } ⊂ {yn } such that zj → x. Then u(x) ≤ lim inf u(zj ) = lim U (zj ) = U (x) j→∞
j→∞
4
On the other hand, since U ∈ Sϕ , U ≤ u on Ω. Therefore, u agrees with the continuous function U on Ω, and hence it is continuous. Therefore u is a viscosity solution of (1.1).
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4. Applications to k-Hessian equations
1 2 3
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In this section, we apply the preceding ideas to the k-Hessian operators (4.1)
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Sk [u] = Sk [D2 u] = σk (λ),
where 1 ≤ k ≤ n, λ = (λ1 , · · · , λn ) are the eigenvalues of the Hessian matrix D2 u, and σk (λ) = λ i1 · · · λ i k 1≤i1 <···
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is the k-th elementary symmetric polynomial, i.e., Sk [D2 u] = [D2 u]k , where [A]k is the sum of the k × k principal minors of A ∈ S n . We also define S0 ≡ 1. Obviously, the k-Hessian operators include the Laplacian operator(as k = 1) and the Monge-Amp`ere operator(as k = n). These operators are not elliptic for all smooth functions. However, they are elliptic when restricted to a certain subset of C 2 (Ω), which we now introduce. A function u ∈ C 2 (Ω) is called k-convex if λ(D2 u) ∈ Γk , where Γk is an open symmetric convex cone in Rn , with vertex at the origin, given by (4.2)
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Γk = {λ = (λ1 , · · · , λn ) ∈ Rn : σj (λ) > 0, ∀j = 1, · · · , k}.
The cone Γk may also be equivalently defined as the component {λ ∈ Rn : σk (λ) > 0} containing the vector (1, . . . , 1). In particular, Γk can be equivalently characterized as (see Section 2 of [4]) Γk = {λ ∈ Rn : 0 < σk (λ) ≤ σk (λ + η), for all η = (η1 , . . . , ηn ) with each ηi ≥ 0} = {λ ∈ Rn : σk (λ + η) ≥ 0, for all η = (η1 , . . . , ηn ) with each ηi ≥ 0}.
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Note that 1-convex functions are subharmonic, and n-convex functions are convex in the usual sense. In [4], Trudinger and Wang generalized this notion to upper semicontinuous functions. An upper semicontinuous function u : Ω → [−∞, ∞) is called k-convex in Ω if Sk [q] ≥ 0 for all quadratic polynomials q for which the difference u − q has a finite local maximum in Ω i.e., u is a viscosity subsolution of −Sk [u] = 0. Since the k-Hessian operators are not elliptic on all of C 2 (Ω), the definition of viscosity solutions of k-Hessian type equations (4.3)
−(Sk [u] − h(x, u, Du)) = 0
needs to be modified somewhat (see [7]), where h is non-negative. An upper (lower) semicontinuous function u : Ω → R is said to be a viscosity subsolution (supersolution) of (4.3) for any k-convex function φ ∈ C 2 (Ω) and any local maximum (minimum) x0 ∈ Ω of u − φ, we have Sk [φ(x0 )] ≥ h(x0 , u(x0 ), Dφ(x0 ))(≤ h(x0 , u(x0 ), Dφ(x0 )))
OBSTACLE PROBLEMS
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Equivalently, an upper (lower) semicontinuous function u : Ω → R is said a viscosity subsolution (supersolution) of (4.3) if Sk [X] ≥ h(x, u(x), p)(≤ h(x, u(x), p))
1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16
for all x ∈ Ω and (p, X) ∈ JΩ2,+ u(x) = JΩ2,+ u(x)∩(Rn ×Skn ) (JΩ2,− u(x) = JΩ2,− u(x)∩ (Rn × Skn )), where Skn = {X ∈ S n : [X]j ≥ 0, ∀j = 1, · · · , k}. This modification is necessary since an example in [6] shows that a classical solution may be not a viscosity solution without this modification. In [7], Urbas has proved Theorem 4.1. If u is an admissible (k-convex) classical solution of (4.3), then u is a viscosity solution. Conversely, if u is a viscosity solution of (4.3), h is positive, and u is of class C 2 , then u is an admissible classical solution. In this section, we consider the obstacle problem ⎧ ⎨ max {−(Sk [u] − h(x, u, Du)), u − g} = 0 u is k-convex in Ω, (4.4) ⎩ u=ϕ
in Ω, on ∂Ω,
where g is bounded and g ≥ ϕ ∈ C(∂Ω). In the proof of Theorem1.2, if JΩ2,+ u(x) is replaced by JΩ2,+ u(x) and JΩ2,− u(x) by JΩ2,− u(x) and a little modification is made in the proof of Lemma 2.4 in [1], we can obtain a similar result with respect to the k-Hessian operators as follows Theorem 4.2. Let comparison principle holds for (4.3), i.e., if u, v ∈ C(Ω) are respectively viscosity subsolution and supersolution of (4.3) and u ≤ v on ∂Ω, then u ≤ v in Ω. Suppose that there is a subsolution u ∈ C(Ω) of (4.3) with u|∂Ω ≡ ϕ and u ≤ g. If (4.5)
u(x) = sup {w(x)} w∈A
is continuous on Ω, where A = {w ∈ C(Ω) : u ≤ w ≤ g is a k-convex subsolution of (4.3) with w|∂Ω ≡ ϕ}, 17
then u is the unique viscosity solution of (4.4).
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By Theorem 4.2, in order to deduce the existence and uniqueness of the solutions of the obstacle problems, we need the existence of u, the continuity of u and the comparison principle. Actually, in [7], Urbas has proved the following
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Lemma 4.3. Let Ω be a bounded domain in Rn and suppose that h(x, z, p) is positive and uniformly continuous on Ω × R × Rn and Lipschitz continuous with respect to the z and p variables with (4.6)
25
26
in Ω × R × Rn
and (4.7)
27
hz ≥ 0 sup
Ω×R×Rn
|hp | ≤ μ < ∞.
If u, v ∈ C(Ω) are respectively viscosity subsolution and supersolution of (4.3) and u ≤ v on ∂Ω and u ≤ v in Ω.
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3
In the special case, h(x, z, p) = h(x), the classical solutions of the Dirichlet problems for k-Hessian equations are studied by Caffarelli, Nirenberg and Spruck in [8] and Trudinger in [9]:
4
Theorem 4.4. Assume that Ω is (k − 1)-convex, ∂Ω ∈ C 3,1 , h ∈ C 1,1 (Ω), and h ≥ h0 > 0. Then there is a unique k-convex solution u ∈ C 3,α (Ω) to the Dirichlet problem Sk [u] = h(x) in Ω u=ϕ on ∂Ω, where ϕ ∈ C 3,1 (∂Ω).
1 2
5 6 7 8 9 10
A domain Ω ⊂ Rn with boundary ∂Ω ∈ C 2 is called k-convex (uniformly kconvex), k = 1, · · · , n−1, if its j-mean curvatures Hj [∂Ω] satisfy Hj [∂Ω] = σj (κ) ≥ 0 (> 0), j = 1, · · · , k, where κ = (κ1 , · · · , κn−1 ) denote the principal curvatures of ∂Ω with respect to its inner normal. In [10], Trudinger also defined the weak solutions to the k-Hessian equations by approximation. Definition 4.5. A function u ∈ C(Ω) is called an admissible weak solution of equation Sk [u] = h(x) in the domain Ω, if there exists a sequence {um } ⊂ C 2 (Ω) of k-convex functions such that um → u in C(Ω)
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Sk [D2 um ] → h in L1loc (Ω).
He also gave an existence and uniqueness theorem: Theorem 4.6. ([10])Let Ω be a uniformly (k − 1)-convex domain in Rn , k = 2, · · · , n, ϕ ∈ C(Ω) and h ≥ 0, ∈ Lp (Ω), for p > n/2k. Then there exists a unique admissible weak solution u ∈ C(Ω) of equation Sk [u] = h in Ω satisfying u = ϕ on ∂Ω. Furthermore u ∈ C α (Ω) for any exponent α < 1, satisfying α ≤ 2 − n/kp. Trudinger and Wang extent it to Hessian measures (see [3, 4, 5]). It is easy to see that the definition of weak solutions coincides with the definition of viscosity solutions restricted to continuous h (see [4]). By using a similar idea to Theorem 1.3 we can obtain the following theorem 4.7: Theorem 4.7. If whenever u exists and A forms an equicontinuous family, then the function u defined in (4.5) is continuous on Ω. Hence it is a viscosity solution of (4.4). We also have Theorem 4.8. If whenever u exists and A is a bounded subset of C α (Ω) for some α ∈ (0, 1], then u defined in (4.5) is a viscosity solution of (4.4). and Theorem 4.9. Let comparison principle hold for (4.3). Assume that whenever u α exists and there is a barrier at any x0 ∈ ∂Ω. If A ⊂ Cloc (Ω) for some α ∈ (0, 1] α and the C bounds of functions in A in any Ω ⊂⊂ Ω are uniform, then u defined in (4.5) is the unique viscosity solution of (4.4). α In [4], they have proved that for k > n/2, any k-convex function u ∈ Cloc (Ω) for α = 2 − n/k and for any Ω ⊂⊂ Ω,
(n) |u|0,α;Ω ≤ C |u|, Ω
OBSTACLE PROBLEMS 1
9
where C = C(k, n). Then we can prove Theorem 4.10. Let Ω be a uniformly (k − 1)-convex domain in Rn , k > n/2, ϕ ∈ C(Ω) and there exists positive constants h0 and h1 such that 0 < h0 ≤ h(x, r, p) ≤ h1 for any (x, r, p) ∈ Ω×R×Rn . By theorem 4.6, there is a viscosity solution u ∈ C(Ω) of the Dirichlet problem Sk [u] = h1 in Ω on ∂Ω. u=ϕ
2 3 4
5
If u ≤ g and the comparison principle holds for (4.3), then there exists a unique viscosity solution of (4.4). Proof Obviously u is a subsolution of (4.3) with u|∂Ω ≡ ϕ and u ≤ g. On the other hand, let u be the viscosity solution of in Ω Sk [u] = h0 u=ϕ on ∂Ω. Then {wi }, which wi = u for any i, is a barrier at any x0 ∈ ∂Ω. Furthermore, for any w ∈ A and any Ω ⊂⊂ Ω, we have
(n) |w|0,α;Ω ≤ C |w| ≤ M, Ω
7
where M depends only on k, n, Ω and the upper bounds of |g|. Then, by theorem 4.9, u defined in (4.5) is the unique viscosity solution of (4.4).
8
6
9
Therefore, by lemma 4.3, we have
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Corollary 4.11. Let Ω be a uniformly (k − 1)-convex domain in Rn , k > n/2, ϕ ∈ C(Ω) and h satisfy the condition as in lemma 4.3. Assume that there exists positive constants h0 and h1 such that 0 < h0 ≤ h(x, r, p) ≤ h1 for any (x, r, p) ∈ Ω × R × Rn . If u ≤ g, where u is as in theorem 4.10, then there exists a unique viscosity solution of (4.4).
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References
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[1] M. Crandall, H. Ishii and P. Lions. User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 1992,27:1-67 [2] G. Barles. A short proof of the C 0,α -regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications. Nonlinear Anal. 2010,73:31-47 [3] N.S. Trudinger and X.-J. Wang. Hessian measures I. Topol. Methods Nonlin. Anal. 1997,10:225-239 [4] N.S. Trudinger and X.-J. Wang. Hessian measures II. Ann. Math. 1999,150:579-604 [5] N.S. Trudinger and X.-J. Wang. Hessian measures III. J. Funct. Anal. 2002,193:1-23 [6] A. Colesanti and P. Salani. Hessian equations in non-smooth domains. Nonlinear Anal. 1999,38:803-812 [7] J. Urbas. On the existence of nonclassial solutions for two class of fully nonlinear elliptic equations. Indiana Univ. Math. J. 1990,39:355-382 [8] L. Caffarelli, L. Nirenberg, J. Spruck. Dirichlet problem for nonlinear second order elliptic equations III, Functions of the eigenvalues of the Hessian. Acta Math. 1985,155:261-301 [9] N.S. Trudinger. On the Dirichlet problem for Hessian equations. Acta Math. 1995,175:151-164 [10] N.S. Trudinger. Weak solutions of Hessian equations. Comm. Partial Differential Equations. 1997,22:1251-1261
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SIJIA BAO AND YUMING XING
Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China E-mail address: [email protected] Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China E-mail address: [email protected]
Sijia Bao, Yuming Xing
PII: DOI: Reference:
S0022-247X(19)30187-8 https://doi.org/10.1016/j.jmaa.2019.02.057 YJMAA 22990
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
11 September 2018
Please cite this article in press as: S. Bao, Y. Xing, On Existence and Uniqueness of Viscosity Solutions to Obstacle Problems for a class of Second Order Partial Differential Equations, J. Math. Anal. Appl. (2019), https://doi.org/10.1016/j.jmaa.2019.02.057
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3
ON EXISTENCE AND UNIQUENESS OF VISCOSITY SOLUTIONS TO OBSTACLE PROBLEMS FOR A CLASS OF SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS
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SIJIA BAO AND YUMING XING
1 2
Abstract. We prove the existence and uniqueness of the viscosity solutions to the obstacle problems for second order differential partial equations under some conditions. As applications, we show some existence and uniqueness theorems of the viscosity solutions to the obstacle problems for k-Hessian equations, which include Monge-Amp` ere type equations when k = n. Keywords: obstacle problem; viscosity solution; k-Hessian equation
5
1. Introduction
6
In this paper we study the obstacle problems for second order partial differential equations max {F (x, u, Du, D2 u), u − g} = 0 in Ω (1.1) u=ϕ on ∂Ω,
7
8 9 10
where Ω is a bounded domain in Rn , g is bounded in Ω, g ≥ ϕ ∈ C(∂Ω), F : Ω × R × Rn × S n −→ R is continuous and S n is the set of symmetric n × n matrices. We also assume that F satisfies a fundamental monotonicity condition (see [1]), (1.2)
11
F (x, r, p, X) ≤ F (x, s, p, X) whenever r ≤ s,
and (1.3)
F (x, r, p, X) ≤ F (x, r, p, Y ) whenever Y ≤ X,
16
where r, s ∈ R, x, p ∈ Rn , X, Y ∈ S n and S n is equipped with its usual order. We say that F is degenerate elliptic if it satisfies (1.3). When (1.2) also holds, we will say that F is proper. In [1], Crandall, Ishii and Lions introduced the concept of viscosity solution of second order partial differential equations, see the next section for the definition.
17
Definition 1.1. We say that the comparison principle holds for the equation
12 13 14 15
F (x, u, Du, D2 u) = 0 18 19 20
if whenever u, u ∈ C(Ω) are respectively viscosity subsolution and supersolution of F = 0 and u ≤ u on ∂Ω, then u ≤ u in Ω. By using Perron’s method we will prove Theorem 1.2. Suppose that there is a subsolution u ∈ C(Ω) of F (x, u, Du, D2 u) = 0 that satisfies the boundary condition u(x) = ϕ(x) for x ∈ ∂Ω and u ≤ g. Denote Sϕ = {w : u ≤ w ≤ g and w ∈ C(Ω) is a subsolution of F = 0 with w|∂Ω ≡ ϕ}. 1
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SIJIA BAO AND YUMING XING
If (1.4)
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
35 36 37 38 39
u(x) = sup w(x) w∈Sϕ
is continuous on Ω, then u is a viscosity solution of (1.1). If, in addition, the comparison principle holds for F = 0, the solution is unique. We can see that in order to deduce the existence of solutions of obstacle problems, we need the existence of u and continuity of u, and used the comparison principle to show the uniqueness. Naturally one will ask when u is continuous. In fact, we can prove Theorem 1.3. If there is a subsolution u ∈ C(Ω) of F = 0 that satisfies u|∂Ω ≡ ϕ and u ≤ g, and Sϕ forms an equicontinuous family, then the function u defined in (1.4) is continuous on Ω and u is a viscosity solution of (1.1). Obviously, by Theorem 1.3, we can obtain the following Corollary 1.4. If u exists and Sϕ is a bounded subset of C α (Ω) for some α ∈ (0, 1], then u defined in (1.4) is a viscosity solution of (1.1). If we just have the interior C α regularity of functions in Sϕ , in order to deduce the continuity of u at the boundary ∂Ω should have sufficient regularity. We call a sequence of functions {wi (x)} in C(Ω) a (upper) barrier in Ω relative to F and ϕ at x0 ∈ ∂Ω, if (i) wi is a viscosity supersolution of F = 0 and wi |∂Ω ≥ ϕ, i = 1, 2, · · · ; (ii) wi (x0 ) → ϕ(x0 ) as i → ∞. It is easy to obtain Lemma 1.5. Let comparison principle hold for F = 0. If u exists and there is a barrier at x0 ∈ ∂Ω, then u(x) → ϕ(x0 ) as x → x0 . Then we have Corollary 1.6. Let comparison principle hold for F = 0. Assume that u exists α and there is a barrier at any x0 ∈ ∂Ω. If Sϕ ⊂ Cloc (Ω) for some α ∈ (0, 1] and the α C bounds of functions in Sϕ in any Ω ⊂⊂ Ω are uniform, then u defined in (1.4) is the unique viscosity solution of (1.1). The above Corollary 1.4 and Corollary 1.6 arouse our interest in the C α regularity of viscosity subsolutions of F = 0. In [2], Guy Barles has proved the following C α regularity of viscosity subsolutions: Theorem 1.7. ([2])Let P be a class of functions h : [0, ∞) → [0, ∞) satisfying: (i) h is a C 1 , convex function, (ii) t → h(t) is non-decreasing for t ≥ 1, +∞t2 t (iii) 1 h(t) dt < +∞. Assume that for any R > 0, R (M ) + K1R h(|p|) − K2R ≤ 0 (1.5) F (x, r, p, M ) ≤ 0 ⇒ G for any M ∈ S n , p ∈ Rn , |r| ≤ R, x ∈ Ω, where K1R , K2R are positive constants, h is R is a Lipschitz continuous function satisfying the a function in the class P and G ellipticity condition, which is homogeneous of degree 1. Then, any locally bounded subsolution u of F = 0 is uniformly continuous in Ωδ with a modulus of continuity depending only on δ, Rδ := u L∞ (Ωδ/2 ) and the different constants and functions
OBSTACLE PROBLEMS 1 2 3 4 5 6 7 8 9
3
appearing in (1.5) with R = Rδ . Moreover, if h(t) = tm (m > 2), Ω is a C 1,1 domain and either u is bounded on Ω or K1R , K2R can be chosen independent of R, then u ∈ C α (Ω). Consequently we have Theorem 1.8. Let comparison principle and (1.5) hold for F = 0. Assume that u exists. If either there is a barrier at any x0 ∈ ∂Ω or h(t) = tm , Ω is a C 1,1 -domain then there is a (unique) viscosity solution of (1.1). Let us end the Introduction with some examples satisfying condition (1.5). The main example is from [2] where Theorem 1.7 was proved. Let F1 (x, r, p, M ) = −tr(a(x, r)M ) + H(x, p) + c(x)u − f (x), x ∈ Ω ⊂ Rn
10 11 12
be a viscous Hamilton-Jacobi Equation, where a : Ω × R → S n is continuous and satisfies a(x, r) = {aij (x, r)} ≥ 0 for all x ∈ Ω and r ∈ R, H : Ω × Rn → R is continuous and there exists constants K1 , K2 > 0 and m > 2 such that H(x, p) ≥ K1 |p|m − K2 ,
13 14 15
16 17 18
for all x ∈ Ω and p ∈ Rn ,
and c, f : Ω → R are continuous. In fact, there exists a constant ΛR > 0 depending on R such that aij (x, r)ξi ξj ≤ ΛR |ξ|2 for all ξ ∈ Rn and for any x ∈ Ω and |r| ≤ R. Define GR (M ) = −ΛR tr(M ) and K2R = K2 + R supΩ |c| + supΩ |f |. It is easy to find that F1 satisfies (1.5) for such GR , K1 and K2R . We can also consider some fully nonlinear cases. For example, let 1/k
F2 (x, r, p, M ) = −Sk (M ) + ψ(x, r, p), 19 20
where the operator Sk is defined as in (4.1), ψ is a positive continuous function which may depend on x, r and p and satisfies ψ(x, r, p) ≥ K1R h(|p|) − K2R
32
whenever |r| ≤ R and x ∈ Ω, here h is a function in the class P. Let G(M ) = 1/k −Sk (M ). Obviously, F2 satisfies (1.5). However, G is not elliptic for all M ∈ S n . Instead, G is elliptic when restricted to a subset of S n , which is {M ∈ S n : λ(M ) ∈ Γk } (see (4.2) for the definition of Γk ), where λ(M ) denotes the eigenvalues of M . See Section 4 for more details. It would be an interesting problem that whether we can obtain similar results as Theorem 1.7 for the equation F2 (x, u, Du, D2 u) = 0. It will also be interested to considering such equations on manifolds. The plan of this paper is as follows. In the next section we introduce some preliminaries which may be useful. In section 3, we prove the existence and uniqueness of viscosity solutions. Finally, in section 4, we show some existence and uniqueness theorems which is viscosity solution of the obstacle problem for the k-Hessian equations as applications.
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2. Preliminaries
34
We first recall the notions of viscosity subsolutions, supersolutions, and solutions in [1].
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SIJIA BAO AND YUMING XING
Definition 2.1. Let F be proper. A viscosity subsolution of F = 0 in Ω is an upper semicontinuous function u : Ω → R such that (2.1)
F (x, u(x), p, X) ≤ 0 f or all x ∈ Ω and (p, X) ∈ JΩ2,+ u(x),
where JΩ2,+ u(x) is the set of the (p, X) ∈ Rn × S n such that 1 < X(y − x), y − x > +o(|y − x|2 ) as y → x. 2 Similarly, a viscosity supersolution of F = 0 in Ω is lower semicontinuous function u : Ω → R such that u(y) ≤ u(x)+ < p, y − x > +
3 4
(2.2) 5 6 7
F (x, u(x), p, X) ≥ 0 f or all x ∈ Ω and (p, X) ∈ JΩ2,− u(x),
where JΩ2,− u(x) = −JΩ2,+ (−u)(x). Finally, u is a viscosity solution of F = 0 in Ω if it is both a viscosity subsolution and a viscosity supersolution of F = 0 in Ω. It is easy to see that JΩ2,+ u(x) = {(Dφ(x), D2 φ(x)) : φ is C 2 and u − φ has a local maximum at x}. Definition 2.2. If u : Ω → [−∞, ∞], then we say u∗ (x) = lim sup {u(y) : y ∈ Ω and |y − x| ≤ r} r→0
the upper semicontinuous envelope of u; it is the smallest upper semicontinuous function (with values in [−∞, ∞]) satisfying u ≤ u∗ . Similarly, we say u∗ (x) = lim inf {u(y) : y ∈ Ω and |y − x| ≤ r} r→0
8 9 10 11
the lower semicontinuous envelope of u. Definition 2.3. We say a set Ω ⊂ Rn locally compact if each point x ∈ Ω has a compact neighbourhood with respect to the topology of Ω. We need the Lemma 4.2 in [1]:
15
¯× Lemma 2.4. Let Ω ⊂ Rn be locally compact, F be lower semicontinuous in Ω n n R × R × S , and F be a family of solutions of F ≤ 0 in Ω. Let w(x) = sup {u(x) : u ∈ F} and assume that w∗ (x) < ∞ for x ∈ Ω. Then w∗ is a solution of F ≤ 0 in Ω.
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3. Existence and uniqueness of viscosity solution
17
We are ready to prove theorem 1.2. Proof of theorem 1.2 It is easy to see that u = ϕ on ∂Ω. By lemma 2.4 u∗ is a subsolution of F = 0 and hence, because of the continuity of u,u = u∗ and u is a subsolution of F = 0 and u ≤ g. So u is a subsolution of (1.1). Next we prove that u is a supersolution of F = 0 in E = {x ∈ Ω : u(x) < g(x)}. Actually, if not, there exists a point x ∈ E such that
12 13 14
18 19 20
x). F ( x, u( x), p, X) < 0 f or some (p, X) ∈ JΩ2,− u( We may assume x = 0. Then by continuity uδ,γ = u(0) + δ+ < p, x > +
1 < Xx, x > −γ|x|2 2
OBSTACLE PROBLEMS
5
is a classical solution of F ≤ 0 in Br = {x : |x| < r} and uδ,γ (x) ≤ g(x) for small r, δ, γ > 0. Since u(x) ≥ u(0)+ < p, x > +
1 < Xx, x > +o(|x|2 ), 2
if we choose δ = (r2 /8)γ then u(x) > uδ,γ (x) for r/2 ≤ |x| ≤ r if r is sufficiently small and then, by lemma 2.4, the function max {u(x), uδ,γ (x)} if |x| < r U (x) = u(x) otherwise, 1 2 3 4
is a viscosity solution of F ≤ 0 in Ω. Note that U (x) ≤ g(x), U = ϕ on ∂Ω for small r and U (0) ≥ uδ,γ (0) = u(0) + δ > u(0), this contradicts the definition of u. Thus u is a supersolution of F = 0 in E. Then, obviously, u is a supersolution of (1.1). So u is a viscosity solution of (1.1). Now if the comparison principle holds for F = 0, we assume u1 , u2 are two (viscosity) solutions to (1.1). Suppose there exists a point x0 ∈ Ω, such that u1 (x0 ) < u2 (x0 ). Let G ⊂ Ω be a connected domain containing x0 , such that u1 (x) < u2 (x) in G,
u1 = u2 on ∂G.
Since u2 ≤ g in Ω, u1 < g in G. Thus u1 is a solution of F = 0 in G. On the other hand, u2 is a subsolution of F = 0. Then by comparison principle, we have u2 ≤ u1 in G, 5
which is a contradiction, then we get the uniqueness. We complete the proof.
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Now we deal with theorem 1.3. Proof of theorem 1.3 Firstly since g is bounded u is well defined. Since u is the supremum of continuous functions, it is lower semicontinuous. Let {yn }∞ n=1 ⊂ Ω be dense on Ω. For each n, there exists a sequence {vn,m } ⊂ Sϕ such that lim vn,m (yn ) = u(yn ).
m→∞
By virtue of lemma 2.4, the maximum of any finite set of subsolutions is also a subsolution. Therefore, we may assume that for each n, {vn,m } is a monotonically nondecreasing sequence. Next we will construct a monotonically nondecreasing sequence {um } in Sϕ that will converge to u at each point yn . Let u1 = v1,1 . Let u2 (x) = max {u1 (x), v1,2 (x), v2,2 (x)}. Note that u2 ∈ Sϕ , and u2 ≥ u1 . Having defined ui , define ui+1 by ui+1 (x) = max {ui (x), v1,i+1 (x), v2,i+1 (x), . . . vi+1,i+1 (x)}. Then ui+1 ∈ Sϕ , and ui+1 ≥ ui ≥ · · · ≥ u1 . We also have that each ui is continuous on Ω. Since for any n and any j ≥ n, u(yn ) ≥ uj (yn ) ≥ vn,j (yn ), we have that limj→∞ uj (yn ) = u(yn ). Because of the equicontinuity of Sϕ , {uj } is equicontinuous on Ω. Since {uj } ⊂ Sϕ , {uj } is uniformly bounded. Therefore, by Arzela-Ascoli’s theorem, a subsequence of {uj } converges uniformly on Ω to a function U ∈ C(Ω). Thus, by the monotonicity of {uj }, we obtain that uj → U .
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SIJIA BAO AND YUMING XING
Then U (yn ) = u(yn ). By density, for any x ∈ Ω, there exists a sequence {zj } ⊂ {yn } such that zj → x. Then u(x) ≤ lim inf u(zj ) = lim U (zj ) = U (x) j→∞
j→∞
4
On the other hand, since U ∈ Sϕ , U ≤ u on Ω. Therefore, u agrees with the continuous function U on Ω, and hence it is continuous. Therefore u is a viscosity solution of (1.1).
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4. Applications to k-Hessian equations
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In this section, we apply the preceding ideas to the k-Hessian operators (4.1)
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Sk [u] = Sk [D2 u] = σk (λ),
where 1 ≤ k ≤ n, λ = (λ1 , · · · , λn ) are the eigenvalues of the Hessian matrix D2 u, and σk (λ) = λ i1 · · · λ i k 1≤i1 <···
9 10 11 12 13 14 15 16
is the k-th elementary symmetric polynomial, i.e., Sk [D2 u] = [D2 u]k , where [A]k is the sum of the k × k principal minors of A ∈ S n . We also define S0 ≡ 1. Obviously, the k-Hessian operators include the Laplacian operator(as k = 1) and the Monge-Amp`ere operator(as k = n). These operators are not elliptic for all smooth functions. However, they are elliptic when restricted to a certain subset of C 2 (Ω), which we now introduce. A function u ∈ C 2 (Ω) is called k-convex if λ(D2 u) ∈ Γk , where Γk is an open symmetric convex cone in Rn , with vertex at the origin, given by (4.2)
17 18 19
Γk = {λ = (λ1 , · · · , λn ) ∈ Rn : σj (λ) > 0, ∀j = 1, · · · , k}.
The cone Γk may also be equivalently defined as the component {λ ∈ Rn : σk (λ) > 0} containing the vector (1, . . . , 1). In particular, Γk can be equivalently characterized as (see Section 2 of [4]) Γk = {λ ∈ Rn : 0 < σk (λ) ≤ σk (λ + η), for all η = (η1 , . . . , ηn ) with each ηi ≥ 0} = {λ ∈ Rn : σk (λ + η) ≥ 0, for all η = (η1 , . . . , ηn ) with each ηi ≥ 0}.
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Note that 1-convex functions are subharmonic, and n-convex functions are convex in the usual sense. In [4], Trudinger and Wang generalized this notion to upper semicontinuous functions. An upper semicontinuous function u : Ω → [−∞, ∞) is called k-convex in Ω if Sk [q] ≥ 0 for all quadratic polynomials q for which the difference u − q has a finite local maximum in Ω i.e., u is a viscosity subsolution of −Sk [u] = 0. Since the k-Hessian operators are not elliptic on all of C 2 (Ω), the definition of viscosity solutions of k-Hessian type equations (4.3)
−(Sk [u] − h(x, u, Du)) = 0
needs to be modified somewhat (see [7]), where h is non-negative. An upper (lower) semicontinuous function u : Ω → R is said to be a viscosity subsolution (supersolution) of (4.3) for any k-convex function φ ∈ C 2 (Ω) and any local maximum (minimum) x0 ∈ Ω of u − φ, we have Sk [φ(x0 )] ≥ h(x0 , u(x0 ), Dφ(x0 ))(≤ h(x0 , u(x0 ), Dφ(x0 )))
OBSTACLE PROBLEMS
7
Equivalently, an upper (lower) semicontinuous function u : Ω → R is said a viscosity subsolution (supersolution) of (4.3) if Sk [X] ≥ h(x, u(x), p)(≤ h(x, u(x), p))
1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16
for all x ∈ Ω and (p, X) ∈ JΩ2,+ u(x) = JΩ2,+ u(x)∩(Rn ×Skn ) (JΩ2,− u(x) = JΩ2,− u(x)∩ (Rn × Skn )), where Skn = {X ∈ S n : [X]j ≥ 0, ∀j = 1, · · · , k}. This modification is necessary since an example in [6] shows that a classical solution may be not a viscosity solution without this modification. In [7], Urbas has proved Theorem 4.1. If u is an admissible (k-convex) classical solution of (4.3), then u is a viscosity solution. Conversely, if u is a viscosity solution of (4.3), h is positive, and u is of class C 2 , then u is an admissible classical solution. In this section, we consider the obstacle problem ⎧ ⎨ max {−(Sk [u] − h(x, u, Du)), u − g} = 0 u is k-convex in Ω, (4.4) ⎩ u=ϕ
in Ω, on ∂Ω,
where g is bounded and g ≥ ϕ ∈ C(∂Ω). In the proof of Theorem1.2, if JΩ2,+ u(x) is replaced by JΩ2,+ u(x) and JΩ2,− u(x) by JΩ2,− u(x) and a little modification is made in the proof of Lemma 2.4 in [1], we can obtain a similar result with respect to the k-Hessian operators as follows Theorem 4.2. Let comparison principle holds for (4.3), i.e., if u, v ∈ C(Ω) are respectively viscosity subsolution and supersolution of (4.3) and u ≤ v on ∂Ω, then u ≤ v in Ω. Suppose that there is a subsolution u ∈ C(Ω) of (4.3) with u|∂Ω ≡ ϕ and u ≤ g. If (4.5)
u(x) = sup {w(x)} w∈A
is continuous on Ω, where A = {w ∈ C(Ω) : u ≤ w ≤ g is a k-convex subsolution of (4.3) with w|∂Ω ≡ ϕ}, 17
then u is the unique viscosity solution of (4.4).
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By Theorem 4.2, in order to deduce the existence and uniqueness of the solutions of the obstacle problems, we need the existence of u, the continuity of u and the comparison principle. Actually, in [7], Urbas has proved the following
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Lemma 4.3. Let Ω be a bounded domain in Rn and suppose that h(x, z, p) is positive and uniformly continuous on Ω × R × Rn and Lipschitz continuous with respect to the z and p variables with (4.6)
25
26
in Ω × R × Rn
and (4.7)
27
hz ≥ 0 sup
Ω×R×Rn
|hp | ≤ μ < ∞.
If u, v ∈ C(Ω) are respectively viscosity subsolution and supersolution of (4.3) and u ≤ v on ∂Ω and u ≤ v in Ω.
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In the special case, h(x, z, p) = h(x), the classical solutions of the Dirichlet problems for k-Hessian equations are studied by Caffarelli, Nirenberg and Spruck in [8] and Trudinger in [9]:
4
Theorem 4.4. Assume that Ω is (k − 1)-convex, ∂Ω ∈ C 3,1 , h ∈ C 1,1 (Ω), and h ≥ h0 > 0. Then there is a unique k-convex solution u ∈ C 3,α (Ω) to the Dirichlet problem Sk [u] = h(x) in Ω u=ϕ on ∂Ω, where ϕ ∈ C 3,1 (∂Ω).
1 2
5 6 7 8 9 10
A domain Ω ⊂ Rn with boundary ∂Ω ∈ C 2 is called k-convex (uniformly kconvex), k = 1, · · · , n−1, if its j-mean curvatures Hj [∂Ω] satisfy Hj [∂Ω] = σj (κ) ≥ 0 (> 0), j = 1, · · · , k, where κ = (κ1 , · · · , κn−1 ) denote the principal curvatures of ∂Ω with respect to its inner normal. In [10], Trudinger also defined the weak solutions to the k-Hessian equations by approximation. Definition 4.5. A function u ∈ C(Ω) is called an admissible weak solution of equation Sk [u] = h(x) in the domain Ω, if there exists a sequence {um } ⊂ C 2 (Ω) of k-convex functions such that um → u in C(Ω)
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Sk [D2 um ] → h in L1loc (Ω).
He also gave an existence and uniqueness theorem: Theorem 4.6. ([10])Let Ω be a uniformly (k − 1)-convex domain in Rn , k = 2, · · · , n, ϕ ∈ C(Ω) and h ≥ 0, ∈ Lp (Ω), for p > n/2k. Then there exists a unique admissible weak solution u ∈ C(Ω) of equation Sk [u] = h in Ω satisfying u = ϕ on ∂Ω. Furthermore u ∈ C α (Ω) for any exponent α < 1, satisfying α ≤ 2 − n/kp. Trudinger and Wang extent it to Hessian measures (see [3, 4, 5]). It is easy to see that the definition of weak solutions coincides with the definition of viscosity solutions restricted to continuous h (see [4]). By using a similar idea to Theorem 1.3 we can obtain the following theorem 4.7: Theorem 4.7. If whenever u exists and A forms an equicontinuous family, then the function u defined in (4.5) is continuous on Ω. Hence it is a viscosity solution of (4.4). We also have Theorem 4.8. If whenever u exists and A is a bounded subset of C α (Ω) for some α ∈ (0, 1], then u defined in (4.5) is a viscosity solution of (4.4). and Theorem 4.9. Let comparison principle hold for (4.3). Assume that whenever u α exists and there is a barrier at any x0 ∈ ∂Ω. If A ⊂ Cloc (Ω) for some α ∈ (0, 1] α and the C bounds of functions in A in any Ω ⊂⊂ Ω are uniform, then u defined in (4.5) is the unique viscosity solution of (4.4). α In [4], they have proved that for k > n/2, any k-convex function u ∈ Cloc (Ω) for α = 2 − n/k and for any Ω ⊂⊂ Ω,
(n) |u|0,α;Ω ≤ C |u|, Ω
OBSTACLE PROBLEMS 1
9
where C = C(k, n). Then we can prove Theorem 4.10. Let Ω be a uniformly (k − 1)-convex domain in Rn , k > n/2, ϕ ∈ C(Ω) and there exists positive constants h0 and h1 such that 0 < h0 ≤ h(x, r, p) ≤ h1 for any (x, r, p) ∈ Ω×R×Rn . By theorem 4.6, there is a viscosity solution u ∈ C(Ω) of the Dirichlet problem Sk [u] = h1 in Ω on ∂Ω. u=ϕ
2 3 4
5
If u ≤ g and the comparison principle holds for (4.3), then there exists a unique viscosity solution of (4.4). Proof Obviously u is a subsolution of (4.3) with u|∂Ω ≡ ϕ and u ≤ g. On the other hand, let u be the viscosity solution of in Ω Sk [u] = h0 u=ϕ on ∂Ω. Then {wi }, which wi = u for any i, is a barrier at any x0 ∈ ∂Ω. Furthermore, for any w ∈ A and any Ω ⊂⊂ Ω, we have
(n) |w|0,α;Ω ≤ C |w| ≤ M, Ω
7
where M depends only on k, n, Ω and the upper bounds of |g|. Then, by theorem 4.9, u defined in (4.5) is the unique viscosity solution of (4.4).
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6
9
Therefore, by lemma 4.3, we have
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Corollary 4.11. Let Ω be a uniformly (k − 1)-convex domain in Rn , k > n/2, ϕ ∈ C(Ω) and h satisfy the condition as in lemma 4.3. Assume that there exists positive constants h0 and h1 such that 0 < h0 ≤ h(x, r, p) ≤ h1 for any (x, r, p) ∈ Ω × R × Rn . If u ≤ g, where u is as in theorem 4.10, then there exists a unique viscosity solution of (4.4).
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References
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[1] M. Crandall, H. Ishii and P. Lions. User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 1992,27:1-67 [2] G. Barles. A short proof of the C 0,α -regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications. Nonlinear Anal. 2010,73:31-47 [3] N.S. Trudinger and X.-J. Wang. Hessian measures I. Topol. Methods Nonlin. Anal. 1997,10:225-239 [4] N.S. Trudinger and X.-J. Wang. Hessian measures II. Ann. Math. 1999,150:579-604 [5] N.S. Trudinger and X.-J. Wang. Hessian measures III. J. Funct. Anal. 2002,193:1-23 [6] A. Colesanti and P. Salani. Hessian equations in non-smooth domains. Nonlinear Anal. 1999,38:803-812 [7] J. Urbas. On the existence of nonclassial solutions for two class of fully nonlinear elliptic equations. Indiana Univ. Math. J. 1990,39:355-382 [8] L. Caffarelli, L. Nirenberg, J. Spruck. Dirichlet problem for nonlinear second order elliptic equations III, Functions of the eigenvalues of the Hessian. Acta Math. 1985,155:261-301 [9] N.S. Trudinger. On the Dirichlet problem for Hessian equations. Acta Math. 1995,175:151-164 [10] N.S. Trudinger. Weak solutions of Hessian equations. Comm. Partial Differential Equations. 1997,22:1251-1261
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SIJIA BAO AND YUMING XING
Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China E-mail address: [email protected] Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China E-mail address: [email protected]