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Existence of multi-valued solutions with asymptotic behavior of Hessian equations
Nonlinear Analysis 74 (2011) 3261–3268
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Existence of multi-valued solutions with asymptotic behavior of Hessian equations Limei Dai School of Mathematics and Information Science, Weifang University, Shandong 261061, People’s Republic of China
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Article history: Received 27 November 2009 Accepted 5 February 2011
In this paper, we use the Perron method to prove the existence of multi-valued solutions with asymptotic behavior at infinity of Hessian equations. © 2011 Elsevier Ltd. All rights reserved.
Keywords: Hessian equations Multi-valued solutions Asymptotic behavior
1. Introduction In this paper, we study the multi-valued solutions of the Hessian equation
σk (λ(D2 u)) = 1,
(1.1)
where σk (λ) denotes the kth elementary symmetric function of λ = (λ1 , . . . , λn ), which is defined by
σk (λ) =
−
λi1 · · · λik ,
k = 1, . . . , n,
i1 <···
λ = λ(D2 u) are the eigenvalues of the Hessian matrix D2 u. For k = 1 (1.1) is the Poisson equation 1u = 1, and for k = n (1.1) is the Monge–Ampère equation det(D2 u) = 1. From the theory of analytic functions, we know that the typical two dimensional examples of multi-valued harmonic functions are
1
u1 (z ) = Re z k
u2 (z ) = Arg(z ),
,
z ∈ C \ {0}, z ∈ C \ {0},
and
u3 (z ) = Re( (z − 1)(z + 1)),
z ∈ C \ {±1}.
By 1970s, Almgren [1] had realized that a minimal variety near a multiplicity-k disc could be well approximated by the graph of a multi-valued function minimizing a suitable analog of the ordinary Dirichlet integral. Many facts about harmonic functions are also true for these Dirichlet minimizing multi-valued functions. Evans [2–4], Levi [5] and Caffarelli [6,7] studied the multi-valued harmonic functions. Evans [3] proved that the conductor potential of a surface with minimal capacity was a double-valued harmonic function. In [7], Caffarelli proved the Hölder continuity of the multi-valued harmonic functions.
E-mail address: [email protected]. 0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.02.004
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L. Dai / Nonlinear Analysis 74 (2011) 3261–3268
At the beginning of this century, the multi-valued solutions of the eikonal equation were considered in [8,9], respectively. Later, Jin et al. provided a level set method for the computation of multi-valued geometric solutions to general quasilinear PDEs and multi-valued physical observables to the semiclassical limit of the Schrödinger equations; see [10,11]. Recently, Caffarelli and Li have investigated the multi-valued solutions of the Monge–Ampère equation in [12] where they first introduced the geometric situation of the multi-valued solutions and then obtained the existence, boundedness, regularity and the asymptotic behavior at infinity of the multi-valued viscosity solutions. The multi-valued solutions for the Dirichlet problem of the Monge–Ampère equation on exterior planar domains are discussed by Ferrer, Martínez and Milán in [13] using complex variable methods. In [14], the author and Bao studied the multi-valued solutions of Hessian equation where we obtained the existence and boundedness of the multi-valued solutions. The geometric situation of the multi-valued functions is given in [12]. Let n ≥ 2, D ⊂ Rn be a bounded domain with smooth boundary ∂ D, and let Σ ⊂ D be homeomorphic in Rn to an n − 1 dimensional closed disc, i.e., there exists a homeomorphism ψ : Rn → Rn such that ψ(Σ ) is an n − 1 dimensional closed disc. Let Γ = ∂ Σ , the boundary of Σ . Thus Γ is homeomorphic to an n − 2 dimensional sphere for n ≥ 3. Let Z be the set of integers and M = (D \ Γ ) × Z denote a covering of D \ Γ with the following standard parameterization: fixing an x∗ ∈ D \ Γ , and connecting x∗ by a smooth curve in D \ Γ to a point x in D \ Γ . If the curve goes through Σ m ≥ 0 times in the positive direction (fixing such a direction), then we arrive at (x, m) in M. If the curve goes through Σ m ≥ 0 times in the negative direction, then we arrive at (x, −m) in M. For k = 2, 3, . . . , we introduce an equivalence relation ‘‘∼ k’’ on M as follows: (x, m) and (y, j) in M are ‘‘∼ k’’ equivalent if x = y and m − j is an integer multiple of k. We let Mk := M / ∼ k denote the k-sheet cover of D \ Γ , and let
∂ ′ Mk :=
k
(∂ D × {m}).
m =1
For n = 2, we can understand the covering space Mk more clearly from the above example u3 . In this example, Γ = {1, −1} and Σ is the interval (−1, 1). Each time the point z goes around −1 or 1, it crosses the interval (−1, 1) one time. We define a distance in Mk as follows: for any (x, m), (y, j) ∈ Mk , let l((x, m), (y, j)) denote a smooth curve in Mk which connects (x, m) and (y, j), and let |l((x, m), (y, j))| denote its length. Define d((x, m), (y, j)) = inf |l((x, m), (y, j))|, l
where the inf is taken over all smooth curves connecting (x, m) and (y, j). Then d((x, m), (y, j)) is a distance. Definition 1.1. We call a function u is continuous at (x, m) in Mk if lim
d((x,m),(y,j))→0
u(y, j) = u(x, m),
and u ∈ C 0 (Mk ) if for any (x, m) ∈ Mk , u is continuous at (x, m). Similarly we can define u ∈ C α (Mk ), C 0,1 (Mk ) and C 2 (Mk ). In this paper, we study the existence of multi-valued viscosity solutions with asymptotic behavior of the Hessian equation. We shall extend the results of the Monge–Ampère equation to the Hessian equation. 2. Preliminaries To state our results, we recall some definitions; see [15,14]. Let
Γk = {λ ∈ Rn |σj (λ) > 0, j = 1, 2, . . . , k}.
∑ Γk is symmetric, that is any permutation of λ is in Γk if λ ∈ Γk . When k = 1, Γk is the half space λ ∈ Rn | λi > 0 . When k = n, Γk is the positive cone Γ + = {λ ∈ Rn |λi > 0, i = 1, . . . , n}. Definition 2.1. A function u ∈ C 2 (Mk ) is called k-convex if λ ∈ Γk , where λ = λ(D2 u(x, m)) = (λ1 , λ2 , . . . , λn ) are the eigenvalues of the Hessian matrix D2 u(x, m).
L. Dai / Nonlinear Analysis 74 (2011) 3261–3268
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Let
σkij (λ(D2 u)) =
n − ∂σk s=1
∂λs
(λ(D2 u))
∂λs 2 (D u). ∂ rij ij
1 j
If λ(x, m) ∈ Γk , (1.1) is degenerate elliptic for u at (x, m), i.e. (σk (λ(D2 u(x, m)))) ≥ 0. And σj (λ(r )), j = 1, 2, . . . , k, is concave for r with λ(r ) ∈ Γk ; see [15]. Definition 2.2. A function u ∈ C 0 (Mk ) is called a viscosity subsolution of
σk (λ(D2 u)) = 1 (x, m) ∈ Mk ,
(2.1)
if for any (y, m) ∈ Mk , ξ ∈ C 2 (Mk ), satisfying u(x, m) ≤ ξ (x, m),
(x, m) ∈ Mk and u(y, m) = ξ (y, m),
we have
σk (λ(D2 ξ (y, m))) ≥ 1. A function u ∈ C 0 (Mk ) is called a viscosity supersolution of (2.1), if for any (y, m) ∈ Mk , any k-convex function ξ ∈ C 2 (Mk ) satisfying u(x, m) ≥ ξ (x, m), (x, m) ∈ Mk
and u(y, m) = ξ (y, m),
we have
σk (λ(D2 ξ (y, m))) ≤ 1. A function u ∈ C 0 (Mk ) is called a viscosity solution of (2.1), if u is both a viscosity subsolution and a viscosity supersolution of (2.1). u is a viscosity solution of (2.1) if u is a k-convex classical solution of (2.1). Conversely, u is a k-convex classical solution of (2.1) if u is a viscosity solution of (2.1) and u is of class C 2 ; see [16]. Definition 2.3. A function u ∈ C 0 (Mk ) is called k-convex if in the viscosity sense σj (λ(D2 u(x, m))) ≥ 0 in Mk , j = 1, 2, . . . , k. u ∈ C 0 (Mk ) is 1-convex if and only if u is C 0 subharmonic; u is n-convex if and only if u is convex. The following lemmas can be found in [14]. Lemma 2.1. Let Ω be an open set in Rn and f ∈ C 0 (Rn ) be nonnegative. Assume that k-convex functions v ∈ C 0 (Ω ), u ∈ C 0 (Rn ) satisfy respectively
σk (λ(D2 v)) ≥ f (x), σk (λ(D2 u)) ≥ f (x),
x ∈ Ω, x ∈ Rn .
Moreover, u ≤ v,
x ∈ Ω,
u = v,
x ∈ ∂Ω.
(2.2)
Set
v(x), w(x) = u(x),
x ∈ Ω, x ∈ Rn \ Ω .
Then w ∈ C 0 (Rn ) is a k-convex function in Rn and satisfies in the viscosity sense
σk (λ(D2 w)) ≥ f (x),
x ∈ Rn .
Lemma 2.2. Let B be a ball in Rn and f ∈ C 0 (B) be nonnegative. Suppose u ∈ C 0 (B) satisfies in the viscosity sense σk (λ(D2 u)) ≥ f (x) in B. Then the Dirichlet problem
σk (λ(D2 u)) = f (x), u = u(x), x ∈ ∂ B
x ∈ B,
has a unique k-convex viscosity solution u ∈ C 0 (B).
(2.3) (2.4)
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L. Dai / Nonlinear Analysis 74 (2011) 3261–3268
The following lemma can be found in [12]. Lemma 2.3. Let Ω ⊂ Rn be a bounded strictly convex domain, ∂ Ω ∈ C 2 , v ∈ C 2 (Ω ). Then there exists a constant c only dependent of n, Ω , ‖v‖C 2 (Ω ) such that for any ζ ∈ ∂ Ω there exists x(ζ ) ∈ Rn satisfying
|x(ζ )| ≤ c ,
wζ < v,
x ∈ Ω \ {ζ },
where 1
wζ (x) := v(ζ ) + (|x − x(ζ )|2 − |ζ − x(ζ )|2 ), 2
x ∈ Rn .
3. Existence of multi-valued solutions with asymptotic behavior In this section, we establish the existence theorem of multi-valued solutions with prescribed asymptotic behavior at infinity of (2.1) under some further hypothesis on Γ . Let Ω be a bounded open strictly convex subset with smooth boundary ∂ Ω . Let Σ , diffeomorphic to an (n − 1)-disc, be the intersection of Ω and a hyperplane in Rn , and let Γ be the boundary of ∂ Σ . Let M = (Rn \ Γ ) × Z, Mk = M / ∼ k be covering spaces of Rn \ Γ as in Section 1. Σ divides Ω into two open parts, denoted as Ω + and Ω − . Fixing an x∗ ∈ Ω − , we use the convention that going through Σ from Ω − to Ω + denotes the positive direction through Σ . Our main result is the following theorem. Theorem 3.1. Let k ≥ 3. Then for any cm ∈ R, there exists a k-convex viscosity solution u ∈ C 0 (Mk ) of
σk (λ(D2 u)) = 1,
(x, m) ∈ Mk
(3.1)
satisfying
lim sup |x|k−2 u(x, m) −
c
|x|→∞
∗
2
|x|2 + cm < ∞,
(3.2)
where c∗ = (1/Cnk )1/k . Proof. We divide the proof into three steps. In the first step, we construct a viscosity subsolution of (3.1). Let Ω be a convex domain in Rn . Suppose Φ ∈ C ∞ (Ω ) is a k-convex function satisfying
σk (λ(D2 Φ )) = c0 > 1, Φ = 0, x ∈ ∂ Ω .
x ∈ Ω,
By the comparison principle, Φ ≤ 0 in Ω . And by Lemma 2.3, for each ζ ∈ ∂ Ω , there exists x(ζ ) ∈ Rn such that
wζ (x) < Φ (x),
x ∈ Ω \ {ζ },
where
wζ (x) :=
1 2
(|x − x(ζ )|2 − |ζ − x(ζ )|2 ),
x ∈ Rn ,
and supζ ∈∂ Ω |x(ζ )| < ∞. Therefore
wζ (ζ ) = 0,
wζ (x) ≤ Φ (x) ≤ 0,
σk (λ(D wζ (x))) = 2
Cnk
> 1,
x∈Ω
x∈R . n
Thus
w(x) := sup wζ (x) ζ ∈∂ Ω
satisfies
w(x) ≤ Φ (x),
x ∈ Ω.
(3.3)
From Ref. [17], we know that w satisfies
σk (λ(D2 w)) ≥ 1,
x ∈ Rn .
Define V (x) =
Φ (x), w(x),
x ∈ Ω, x ∈ Rn \ Ω .
(3.4)
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Then V ∈ C 0 (Rn ). By (3.3) and Lemma 2.1, V is a k-convex function satisfying in the viscosity sense
σk (λ(D2 V )) ≥ 1,
x ∈ Rn .
(3.5)
Fix some R1 > 0 such that
Ω ⊂⊂ BR1 .
(3.6)
Let R2 := 2R1
√
c∗ .
For a > 1, define
∫
wa (x) := inf V + BR
√ | c∗ x |
1
(sk + a) k ds,
x ∈ Rn .
2R2
1
A direct calculation gives Dij wa = (|y| + a) k
where y =
1 −1 k
[ |y|
k−1
+
a
|y|
ac∗2 xi xj
c∗ δij −
]
|y|3
,
|x| > 0,
√
c∗ x. By rotating the coordinates we may set x = (r , 0, . . . , 0)′ , therefore
D2 wa = (Rk + a)
1 −1 k
Rk−1 c∗
···
0 a
Rk−1 +
0 ···
R
c∗
···
0
···
0
, ··· a Rk−1 + c∗
···
···
0 0
R
where R = |y|. Consequently λ(D wa ) ∈ Γk for |x| > 0, and 2
a k−1 k−1 a k k σk (λ(D2 wa )) = (Rk + a)1−k Rk−1 c∗ Cnk−−11 Rk−1 + c∗ + Cnk−1 Rk−1 + c∗ R
R
1 a = k Cnk−−11 + Cnk−1 1 + k Cn R ≥
1 Cnk
(Cnk−−11 + Cnk−1 )
= 1,
0 < |x| < ∞.
Moreover,
wa (x) ≤ V (x),
|x| ≤ R1 .
(3.7)
Fix some R3 > 3R2 satisfying R3
√
c∗ > 3R2 .
We choose a1 > 1 such that for a ≥ a1 ,
wa (x) > inf V + BR
3R2
∫
1
(sk + a) k ds ≥ V (x),
|x| = R3 .
2R2
1
Then by (3.7), R3 ≥ R1 .
(3.8)
According to the definition of wa ,
wa (x) = inf V + BR
∫
√ | c∗ x |
s √ | c∗ x |
= inf V +
=
2
s
1+
a k sk
1+
BR
sk
1
− 1 ds +
∞
s 2R2
sds 2R2
a k
|x| + cm + inf V +
√ | c∗ x |
∫
− 1 ds +
1
∫
2
1
2R2
1
c∗
2R2
1
∫ BR
1+
1
a k sk
c∗ 2
|x|2 − 2R22
− 1 ds − cm −
2R22
∫ −
∞ √ | c∗ x |
s
1+
1
a k sk
− 1 ds,
x ∈ Rn .
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L. Dai / Nonlinear Analysis 74 (2011) 3261–3268
Let
µ(m, a) = inf V +
∫
BR
∞
1+
s 2R2
1
1
a k sk
− 1 ds − cm − 2R22 .
Then µ(m, a) is continuous and monotonic increasing for a and when a → ∞, µ(m, a) → ∞, 1 ≤ m ≤ k. Moreover,
wa (x) =
c∗ 2
|x|2 + cm + µ(m, a) − O(|x|2−k ),
when |x| → ∞.
(3.9)
Define, for a ≥ a1 and 1 ≤ m ≤ k, um,a (x) =
max{V (x), wa (x)} − µ(m, a), wa (x) − µ(m, a),
|x| ≤ R3 , |x| ≥ R3 .
Then by (3.9), for 1 ≤ m ≤ k, um,a (x) =
c∗ 2
|x|2 + cm − O(|x|2−k ),
when |x| → ∞,
and by the definition of V and (3.6)–(3.8), um,a (x) = −µ(m, a),
x ∈ Γ.
Choose a2 ≥ a1 sufficiently large such that when a ≥ a2 , V (x) − µ(m, a) = V (x) − inf V − BR
≤ cm ≤
c∗ 2
1
∞
∫
s
1+
2R2
|x|2 + cm ,
1
a k
sk
− 1 ds + cm + 2R22
|x| ≤ R3 .
Therefore, um,a (x) ≤
c∗ 2
|x|2 + cm ,
a ≥ a2 , x ∈ R n .
By Lemma 2.1, um,a ∈ C 0 (Rn ) is k-convex and satisfies in the viscosity sense
σk (λ(D2 um,a )) ≥ 1,
x ∈ Rn .
It is easy to see that there exists a continuous function a(m) (a), 2 ≤ m ≤ k, satisfying lim a(m) (a) = ∞,
a→∞
and, for 2 ≤ m ≤ k,
µ(m, a(m) (a)) = µ(1, a). So there exists a3 ≥ a2 such that when a ≥ a3 , a(m) (a) > a2 , 2 ≤ m ≤ k. Let a(1) (a) = a, and define ua (x, m) = um,a(m) (a) (x),
(x, m) ∈ Mk .
Then by the definition of um,a , when a ≥ a3 , ua ∈ C 0 (Mk ) is a locally k-convex function satisfying ua (x, m) = ua (x, m) ≤
c∗ 2 c∗ 2
|x|2 + cm − O(|x|2−k ), |x|2 + cm ,
lim ua (x, m) = −µ(1, a),
x→x
when |x| → ∞,
x ∈ Rn , 1 ≤ m ≤ k, x ∈ Γ , 1 ≤ m ≤ k,
and in the viscosity sense,
σk (λ(D2 ua )) ≥ 1,
(x, m) ∈ Mk .
In the second step, we define the Perron solution of (3.1). For a ≥ a3 , let Sa denote the set of k-convex functions v ∈ C 0 (Mk ) which can be extended to Γ and satisfies
σk (λ(D2 v)) ≥ 1, (x, m) ∈ Mk , lim v(x, m) ≤ −µ(1, a), x ∈ Γ , x→x
v(x, m) ≤
c∗ 2
|x|2 + cm ,
x ∈ Rn , 1 ≤ m ≤ k.
(3.10)
L. Dai / Nonlinear Analysis 74 (2011) 3261–3268
3267
Clearly, ua ∈ Sa . Hence Sa ̸= ∅. Define ua (x, m) := sup{v(x, m)|v ∈ Sa },
(x, m) ∈ Mk .
In the third step, we prove that ua is a viscosity solution of (3.1). From the definition of ua , ua is a viscosity subsolution of (3.1) and satisfies ua (x, m) ≤
c∗ 2
|x|2 + cm ,
x ∈ Rn .
We only need to prove that ua is a viscosity supersolution of (3.1) satisfying (3.2). For any x0 ∈ Rn \ Γ , fix ε > 0 such that B = Bε (x0 ) ⊂ Rn \ Γ . Then the lifting of B into Mk is the k disjoint balls denoted as {B(i) }ki=1 . For any (x, m) ∈ B(i) , by Lemma 2.2, Dirichlet problem
σk (λ(D2 u˜ )) = 1, u˜ = ua ,
(x, m) ∈ B(i) ,
(3.11)
(x, m) ∈ ∂ B(i)
has a k-convex viscosity solution u˜ ∈ C 0 (B(i) ). From the comparison principle, ua ≤ u˜ ,
(x, m) ∈ B(i) .
Define
ψ(x, m) =
u˜ (x, m), ua (x, m),
(x, m) ∈ B(i) , (x, m) ∈ Mk \ {B(i) }ki=1 .
By Lemma 2.1,
σk (λ(D2 ψ(x, m))) ≥ 1,
x ∈ Rn .
Because
σk (λ(D2 u˜ )) = 1 = σk (λ(D2 g )), u˜ = ua ≤ g , (x, m) ∈ ∂ B(i) , where g (x, m) = u˜ ≤ g ,
c∗ 2
(x, m) ∈ B(i) ,
|x|2 + cm , from the comparison principle,
(x, m) ∈ B(i) .
Hence ψ ∈ Sa . By the definition of ua , ua ≥ ψ in Mk . Consequently u˜ ≤ ua in B(i) . As a result, u˜ = ua ,
(x, m) ∈ B(i) .
Since x0 is arbitrary, we know that ua is a k-convex viscosity solution of (3.1). By the definition of ua , ua ≤ ua ≤ g ,
(x, m) ∈ Mk ,
so ua satisfies (3.2). Theorem 3.1 is proved.
Acknowledgements The author would like to thank Professor Jiguang Bao for his guidance and Professor Yanyan Li and Professor Luis Caffarelli for their suggestions. The author is also indebted to the anonymous referees for the deep and helpful comments. References [1] F. Almgren, F. Almgren’s Big Regularity Paper. Q -Valued Functions Minimizing Dirichlet’s Integral and the Regularity of Area-Minimizing Rectifiable Currents up to Codimension 2, in: World Scientific Monograph Series in Mathematics, vol. 1, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. [2] G.C. Evans, A necessary and sufficient condition of Wiener, Amer. Math. Monthly 54 (1947) 151–155. [3] G.C. Evans, Surfaces of minimal capacity, Proc. Natl. Acad. Sci. USA 26 (1940) 489–491. [4] G.C. Evans, Lectures on Multiple Valued Harmonic Functions in Space, in: Univ. Calif. Publ. Math. (NS), vol. 1, 1951, pp. 281–340. [5] G. Levi, Generalization of a spatial angle theorem, Uspehi Mat. Nauk. 26 (1971) 199–204. (in Russian) Translated from the English by Ju. V. Egorov. [6] L. Caffarelli, Certain multiple valued harmonic functions, Proc. Amer. Math. Soc. 54 (1976) 90–92. [7] L. Caffarelli, On the hölder continuity of multiple valued harmonic functions, Indiana Univ. Math. J. 25 (1976) 79–84. [8] L. Gosse, S. Jin, X. Li, Two moment systems for computing multiphase semiclassical limits of the Schrödinger equation, Math. Models Methods Appl. Sci. 13 (2003) 1689–1723. [9] S. Izumiya, G.T. Kossioris, G.N. Makrakis, Multivalued solutions to the eikonal equation in stratified media, Quart. Appl. Math. 59 (2001) 365–390. [10] S. Jin, S. Osher, A level set method for the computation of multivalued solutions to quasi-linear hyperbolic PDEs and Hamilton–Jacobi equations, Commun. Math. Sci. 1 (2003) 575–591.
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[11] S. Jin, H. Liu, S. Osher, Y. Tsai, Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation, J. Comput. Phys. 205 (2005) 222–241. [12] L. Caffarelli, Y.Y. Li, Some multi-valued solutions to Monge–Ampère equations, Comm. Anal. Geom. 14 (2006) 411–441. [13] L. Ferrer, A. Martínez, F. Milán, An extension of a theorem by K. Jörgens and a maximum principle at infinity for parabolic affine spheres, Math. Z. 230 (1999) 471–486. [14] L.M. Dai, J.G. Bao, Multi-valued solutions to Hessian equations, Nonlinear Differential Equations Appl. (in press). [15] L. Caffarelli, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985) 261–301. [16] J.I.E. Urbas, On the existence of nonclassical solutions for two class of fully nonlinear elliptic equations, Indiana Univ. Math. J. 39 (2) (1990) 355–382. [17] H. Ishii, P.L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations 83 (1) (1990) 26–78.
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Existence of multi-valued solutions with asymptotic behavior of Hessian equations Limei Dai School of Mathematics and Information Science, Weifang University, Shandong 261061, People’s Republic of China
article
abstract
info
Article history: Received 27 November 2009 Accepted 5 February 2011
In this paper, we use the Perron method to prove the existence of multi-valued solutions with asymptotic behavior at infinity of Hessian equations. © 2011 Elsevier Ltd. All rights reserved.
Keywords: Hessian equations Multi-valued solutions Asymptotic behavior
1. Introduction In this paper, we study the multi-valued solutions of the Hessian equation
σk (λ(D2 u)) = 1,
(1.1)
where σk (λ) denotes the kth elementary symmetric function of λ = (λ1 , . . . , λn ), which is defined by
σk (λ) =
−
λi1 · · · λik ,
k = 1, . . . , n,
i1 <···
λ = λ(D2 u) are the eigenvalues of the Hessian matrix D2 u. For k = 1 (1.1) is the Poisson equation 1u = 1, and for k = n (1.1) is the Monge–Ampère equation det(D2 u) = 1. From the theory of analytic functions, we know that the typical two dimensional examples of multi-valued harmonic functions are
1
u1 (z ) = Re z k
u2 (z ) = Arg(z ),
,
z ∈ C \ {0}, z ∈ C \ {0},
and
u3 (z ) = Re( (z − 1)(z + 1)),
z ∈ C \ {±1}.
By 1970s, Almgren [1] had realized that a minimal variety near a multiplicity-k disc could be well approximated by the graph of a multi-valued function minimizing a suitable analog of the ordinary Dirichlet integral. Many facts about harmonic functions are also true for these Dirichlet minimizing multi-valued functions. Evans [2–4], Levi [5] and Caffarelli [6,7] studied the multi-valued harmonic functions. Evans [3] proved that the conductor potential of a surface with minimal capacity was a double-valued harmonic function. In [7], Caffarelli proved the Hölder continuity of the multi-valued harmonic functions.
E-mail address: [email protected]. 0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.02.004
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At the beginning of this century, the multi-valued solutions of the eikonal equation were considered in [8,9], respectively. Later, Jin et al. provided a level set method for the computation of multi-valued geometric solutions to general quasilinear PDEs and multi-valued physical observables to the semiclassical limit of the Schrödinger equations; see [10,11]. Recently, Caffarelli and Li have investigated the multi-valued solutions of the Monge–Ampère equation in [12] where they first introduced the geometric situation of the multi-valued solutions and then obtained the existence, boundedness, regularity and the asymptotic behavior at infinity of the multi-valued viscosity solutions. The multi-valued solutions for the Dirichlet problem of the Monge–Ampère equation on exterior planar domains are discussed by Ferrer, Martínez and Milán in [13] using complex variable methods. In [14], the author and Bao studied the multi-valued solutions of Hessian equation where we obtained the existence and boundedness of the multi-valued solutions. The geometric situation of the multi-valued functions is given in [12]. Let n ≥ 2, D ⊂ Rn be a bounded domain with smooth boundary ∂ D, and let Σ ⊂ D be homeomorphic in Rn to an n − 1 dimensional closed disc, i.e., there exists a homeomorphism ψ : Rn → Rn such that ψ(Σ ) is an n − 1 dimensional closed disc. Let Γ = ∂ Σ , the boundary of Σ . Thus Γ is homeomorphic to an n − 2 dimensional sphere for n ≥ 3. Let Z be the set of integers and M = (D \ Γ ) × Z denote a covering of D \ Γ with the following standard parameterization: fixing an x∗ ∈ D \ Γ , and connecting x∗ by a smooth curve in D \ Γ to a point x in D \ Γ . If the curve goes through Σ m ≥ 0 times in the positive direction (fixing such a direction), then we arrive at (x, m) in M. If the curve goes through Σ m ≥ 0 times in the negative direction, then we arrive at (x, −m) in M. For k = 2, 3, . . . , we introduce an equivalence relation ‘‘∼ k’’ on M as follows: (x, m) and (y, j) in M are ‘‘∼ k’’ equivalent if x = y and m − j is an integer multiple of k. We let Mk := M / ∼ k denote the k-sheet cover of D \ Γ , and let
∂ ′ Mk :=
k
(∂ D × {m}).
m =1
For n = 2, we can understand the covering space Mk more clearly from the above example u3 . In this example, Γ = {1, −1} and Σ is the interval (−1, 1). Each time the point z goes around −1 or 1, it crosses the interval (−1, 1) one time. We define a distance in Mk as follows: for any (x, m), (y, j) ∈ Mk , let l((x, m), (y, j)) denote a smooth curve in Mk which connects (x, m) and (y, j), and let |l((x, m), (y, j))| denote its length. Define d((x, m), (y, j)) = inf |l((x, m), (y, j))|, l
where the inf is taken over all smooth curves connecting (x, m) and (y, j). Then d((x, m), (y, j)) is a distance. Definition 1.1. We call a function u is continuous at (x, m) in Mk if lim
d((x,m),(y,j))→0
u(y, j) = u(x, m),
and u ∈ C 0 (Mk ) if for any (x, m) ∈ Mk , u is continuous at (x, m). Similarly we can define u ∈ C α (Mk ), C 0,1 (Mk ) and C 2 (Mk ). In this paper, we study the existence of multi-valued viscosity solutions with asymptotic behavior of the Hessian equation. We shall extend the results of the Monge–Ampère equation to the Hessian equation. 2. Preliminaries To state our results, we recall some definitions; see [15,14]. Let
Γk = {λ ∈ Rn |σj (λ) > 0, j = 1, 2, . . . , k}.
∑ Γk is symmetric, that is any permutation of λ is in Γk if λ ∈ Γk . When k = 1, Γk is the half space λ ∈ Rn | λi > 0 . When k = n, Γk is the positive cone Γ + = {λ ∈ Rn |λi > 0, i = 1, . . . , n}. Definition 2.1. A function u ∈ C 2 (Mk ) is called k-convex if λ ∈ Γk , where λ = λ(D2 u(x, m)) = (λ1 , λ2 , . . . , λn ) are the eigenvalues of the Hessian matrix D2 u(x, m).
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Let
σkij (λ(D2 u)) =
n − ∂σk s=1
∂λs
(λ(D2 u))
∂λs 2 (D u). ∂ rij ij
1 j
If λ(x, m) ∈ Γk , (1.1) is degenerate elliptic for u at (x, m), i.e. (σk (λ(D2 u(x, m)))) ≥ 0. And σj (λ(r )), j = 1, 2, . . . , k, is concave for r with λ(r ) ∈ Γk ; see [15]. Definition 2.2. A function u ∈ C 0 (Mk ) is called a viscosity subsolution of
σk (λ(D2 u)) = 1 (x, m) ∈ Mk ,
(2.1)
if for any (y, m) ∈ Mk , ξ ∈ C 2 (Mk ), satisfying u(x, m) ≤ ξ (x, m),
(x, m) ∈ Mk and u(y, m) = ξ (y, m),
we have
σk (λ(D2 ξ (y, m))) ≥ 1. A function u ∈ C 0 (Mk ) is called a viscosity supersolution of (2.1), if for any (y, m) ∈ Mk , any k-convex function ξ ∈ C 2 (Mk ) satisfying u(x, m) ≥ ξ (x, m), (x, m) ∈ Mk
and u(y, m) = ξ (y, m),
we have
σk (λ(D2 ξ (y, m))) ≤ 1. A function u ∈ C 0 (Mk ) is called a viscosity solution of (2.1), if u is both a viscosity subsolution and a viscosity supersolution of (2.1). u is a viscosity solution of (2.1) if u is a k-convex classical solution of (2.1). Conversely, u is a k-convex classical solution of (2.1) if u is a viscosity solution of (2.1) and u is of class C 2 ; see [16]. Definition 2.3. A function u ∈ C 0 (Mk ) is called k-convex if in the viscosity sense σj (λ(D2 u(x, m))) ≥ 0 in Mk , j = 1, 2, . . . , k. u ∈ C 0 (Mk ) is 1-convex if and only if u is C 0 subharmonic; u is n-convex if and only if u is convex. The following lemmas can be found in [14]. Lemma 2.1. Let Ω be an open set in Rn and f ∈ C 0 (Rn ) be nonnegative. Assume that k-convex functions v ∈ C 0 (Ω ), u ∈ C 0 (Rn ) satisfy respectively
σk (λ(D2 v)) ≥ f (x), σk (λ(D2 u)) ≥ f (x),
x ∈ Ω, x ∈ Rn .
Moreover, u ≤ v,
x ∈ Ω,
u = v,
x ∈ ∂Ω.
(2.2)
Set
v(x), w(x) = u(x),
x ∈ Ω, x ∈ Rn \ Ω .
Then w ∈ C 0 (Rn ) is a k-convex function in Rn and satisfies in the viscosity sense
σk (λ(D2 w)) ≥ f (x),
x ∈ Rn .
Lemma 2.2. Let B be a ball in Rn and f ∈ C 0 (B) be nonnegative. Suppose u ∈ C 0 (B) satisfies in the viscosity sense σk (λ(D2 u)) ≥ f (x) in B. Then the Dirichlet problem
σk (λ(D2 u)) = f (x), u = u(x), x ∈ ∂ B
x ∈ B,
has a unique k-convex viscosity solution u ∈ C 0 (B).
(2.3) (2.4)
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The following lemma can be found in [12]. Lemma 2.3. Let Ω ⊂ Rn be a bounded strictly convex domain, ∂ Ω ∈ C 2 , v ∈ C 2 (Ω ). Then there exists a constant c only dependent of n, Ω , ‖v‖C 2 (Ω ) such that for any ζ ∈ ∂ Ω there exists x(ζ ) ∈ Rn satisfying
|x(ζ )| ≤ c ,
wζ < v,
x ∈ Ω \ {ζ },
where 1
wζ (x) := v(ζ ) + (|x − x(ζ )|2 − |ζ − x(ζ )|2 ), 2
x ∈ Rn .
3. Existence of multi-valued solutions with asymptotic behavior In this section, we establish the existence theorem of multi-valued solutions with prescribed asymptotic behavior at infinity of (2.1) under some further hypothesis on Γ . Let Ω be a bounded open strictly convex subset with smooth boundary ∂ Ω . Let Σ , diffeomorphic to an (n − 1)-disc, be the intersection of Ω and a hyperplane in Rn , and let Γ be the boundary of ∂ Σ . Let M = (Rn \ Γ ) × Z, Mk = M / ∼ k be covering spaces of Rn \ Γ as in Section 1. Σ divides Ω into two open parts, denoted as Ω + and Ω − . Fixing an x∗ ∈ Ω − , we use the convention that going through Σ from Ω − to Ω + denotes the positive direction through Σ . Our main result is the following theorem. Theorem 3.1. Let k ≥ 3. Then for any cm ∈ R, there exists a k-convex viscosity solution u ∈ C 0 (Mk ) of
σk (λ(D2 u)) = 1,
(x, m) ∈ Mk
(3.1)
satisfying
lim sup |x|k−2 u(x, m) −
c
|x|→∞
∗
2
|x|2 + cm < ∞,
(3.2)
where c∗ = (1/Cnk )1/k . Proof. We divide the proof into three steps. In the first step, we construct a viscosity subsolution of (3.1). Let Ω be a convex domain in Rn . Suppose Φ ∈ C ∞ (Ω ) is a k-convex function satisfying
σk (λ(D2 Φ )) = c0 > 1, Φ = 0, x ∈ ∂ Ω .
x ∈ Ω,
By the comparison principle, Φ ≤ 0 in Ω . And by Lemma 2.3, for each ζ ∈ ∂ Ω , there exists x(ζ ) ∈ Rn such that
wζ (x) < Φ (x),
x ∈ Ω \ {ζ },
where
wζ (x) :=
1 2
(|x − x(ζ )|2 − |ζ − x(ζ )|2 ),
x ∈ Rn ,
and supζ ∈∂ Ω |x(ζ )| < ∞. Therefore
wζ (ζ ) = 0,
wζ (x) ≤ Φ (x) ≤ 0,
σk (λ(D wζ (x))) = 2
Cnk
> 1,
x∈Ω
x∈R . n
Thus
w(x) := sup wζ (x) ζ ∈∂ Ω
satisfies
w(x) ≤ Φ (x),
x ∈ Ω.
(3.3)
From Ref. [17], we know that w satisfies
σk (λ(D2 w)) ≥ 1,
x ∈ Rn .
Define V (x) =
Φ (x), w(x),
x ∈ Ω, x ∈ Rn \ Ω .
(3.4)
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Then V ∈ C 0 (Rn ). By (3.3) and Lemma 2.1, V is a k-convex function satisfying in the viscosity sense
σk (λ(D2 V )) ≥ 1,
x ∈ Rn .
(3.5)
Fix some R1 > 0 such that
Ω ⊂⊂ BR1 .
(3.6)
Let R2 := 2R1
√
c∗ .
For a > 1, define
∫
wa (x) := inf V + BR
√ | c∗ x |
1
(sk + a) k ds,
x ∈ Rn .
2R2
1
A direct calculation gives Dij wa = (|y| + a) k
where y =
1 −1 k
[ |y|
k−1
+
a
|y|
ac∗2 xi xj
c∗ δij −
]
|y|3
,
|x| > 0,
√
c∗ x. By rotating the coordinates we may set x = (r , 0, . . . , 0)′ , therefore
D2 wa = (Rk + a)
1 −1 k
Rk−1 c∗
···
0 a
Rk−1 +
0 ···
R
c∗
···
0
···
0
, ··· a Rk−1 + c∗
···
···
0 0
R
where R = |y|. Consequently λ(D wa ) ∈ Γk for |x| > 0, and 2
a k−1 k−1 a k k σk (λ(D2 wa )) = (Rk + a)1−k Rk−1 c∗ Cnk−−11 Rk−1 + c∗ + Cnk−1 Rk−1 + c∗ R
R
1 a = k Cnk−−11 + Cnk−1 1 + k Cn R ≥
1 Cnk
(Cnk−−11 + Cnk−1 )
= 1,
0 < |x| < ∞.
Moreover,
wa (x) ≤ V (x),
|x| ≤ R1 .
(3.7)
Fix some R3 > 3R2 satisfying R3
√
c∗ > 3R2 .
We choose a1 > 1 such that for a ≥ a1 ,
wa (x) > inf V + BR
3R2
∫
1
(sk + a) k ds ≥ V (x),
|x| = R3 .
2R2
1
Then by (3.7), R3 ≥ R1 .
(3.8)
According to the definition of wa ,
wa (x) = inf V + BR
∫
√ | c∗ x |
s √ | c∗ x |
= inf V +
=
2
s
1+
a k sk
1+
BR
sk
1
− 1 ds +
∞
s 2R2
sds 2R2
a k
|x| + cm + inf V +
√ | c∗ x |
∫
− 1 ds +
1
∫
2
1
2R2
1
c∗
2R2
1
∫ BR
1+
1
a k sk
c∗ 2
|x|2 − 2R22
− 1 ds − cm −
2R22
∫ −
∞ √ | c∗ x |
s
1+
1
a k sk
− 1 ds,
x ∈ Rn .
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Let
µ(m, a) = inf V +
∫
BR
∞
1+
s 2R2
1
1
a k sk
− 1 ds − cm − 2R22 .
Then µ(m, a) is continuous and monotonic increasing for a and when a → ∞, µ(m, a) → ∞, 1 ≤ m ≤ k. Moreover,
wa (x) =
c∗ 2
|x|2 + cm + µ(m, a) − O(|x|2−k ),
when |x| → ∞.
(3.9)
Define, for a ≥ a1 and 1 ≤ m ≤ k, um,a (x) =
max{V (x), wa (x)} − µ(m, a), wa (x) − µ(m, a),
|x| ≤ R3 , |x| ≥ R3 .
Then by (3.9), for 1 ≤ m ≤ k, um,a (x) =
c∗ 2
|x|2 + cm − O(|x|2−k ),
when |x| → ∞,
and by the definition of V and (3.6)–(3.8), um,a (x) = −µ(m, a),
x ∈ Γ.
Choose a2 ≥ a1 sufficiently large such that when a ≥ a2 , V (x) − µ(m, a) = V (x) − inf V − BR
≤ cm ≤
c∗ 2
1
∞
∫
s
1+
2R2
|x|2 + cm ,
1
a k
sk
− 1 ds + cm + 2R22
|x| ≤ R3 .
Therefore, um,a (x) ≤
c∗ 2
|x|2 + cm ,
a ≥ a2 , x ∈ R n .
By Lemma 2.1, um,a ∈ C 0 (Rn ) is k-convex and satisfies in the viscosity sense
σk (λ(D2 um,a )) ≥ 1,
x ∈ Rn .
It is easy to see that there exists a continuous function a(m) (a), 2 ≤ m ≤ k, satisfying lim a(m) (a) = ∞,
a→∞
and, for 2 ≤ m ≤ k,
µ(m, a(m) (a)) = µ(1, a). So there exists a3 ≥ a2 such that when a ≥ a3 , a(m) (a) > a2 , 2 ≤ m ≤ k. Let a(1) (a) = a, and define ua (x, m) = um,a(m) (a) (x),
(x, m) ∈ Mk .
Then by the definition of um,a , when a ≥ a3 , ua ∈ C 0 (Mk ) is a locally k-convex function satisfying ua (x, m) = ua (x, m) ≤
c∗ 2 c∗ 2
|x|2 + cm − O(|x|2−k ), |x|2 + cm ,
lim ua (x, m) = −µ(1, a),
x→x
when |x| → ∞,
x ∈ Rn , 1 ≤ m ≤ k, x ∈ Γ , 1 ≤ m ≤ k,
and in the viscosity sense,
σk (λ(D2 ua )) ≥ 1,
(x, m) ∈ Mk .
In the second step, we define the Perron solution of (3.1). For a ≥ a3 , let Sa denote the set of k-convex functions v ∈ C 0 (Mk ) which can be extended to Γ and satisfies
σk (λ(D2 v)) ≥ 1, (x, m) ∈ Mk , lim v(x, m) ≤ −µ(1, a), x ∈ Γ , x→x
v(x, m) ≤
c∗ 2
|x|2 + cm ,
x ∈ Rn , 1 ≤ m ≤ k.
(3.10)
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Clearly, ua ∈ Sa . Hence Sa ̸= ∅. Define ua (x, m) := sup{v(x, m)|v ∈ Sa },
(x, m) ∈ Mk .
In the third step, we prove that ua is a viscosity solution of (3.1). From the definition of ua , ua is a viscosity subsolution of (3.1) and satisfies ua (x, m) ≤
c∗ 2
|x|2 + cm ,
x ∈ Rn .
We only need to prove that ua is a viscosity supersolution of (3.1) satisfying (3.2). For any x0 ∈ Rn \ Γ , fix ε > 0 such that B = Bε (x0 ) ⊂ Rn \ Γ . Then the lifting of B into Mk is the k disjoint balls denoted as {B(i) }ki=1 . For any (x, m) ∈ B(i) , by Lemma 2.2, Dirichlet problem
σk (λ(D2 u˜ )) = 1, u˜ = ua ,
(x, m) ∈ B(i) ,
(3.11)
(x, m) ∈ ∂ B(i)
has a k-convex viscosity solution u˜ ∈ C 0 (B(i) ). From the comparison principle, ua ≤ u˜ ,
(x, m) ∈ B(i) .
Define
ψ(x, m) =
u˜ (x, m), ua (x, m),
(x, m) ∈ B(i) , (x, m) ∈ Mk \ {B(i) }ki=1 .
By Lemma 2.1,
σk (λ(D2 ψ(x, m))) ≥ 1,
x ∈ Rn .
Because
σk (λ(D2 u˜ )) = 1 = σk (λ(D2 g )), u˜ = ua ≤ g , (x, m) ∈ ∂ B(i) , where g (x, m) = u˜ ≤ g ,
c∗ 2
(x, m) ∈ B(i) ,
|x|2 + cm , from the comparison principle,
(x, m) ∈ B(i) .
Hence ψ ∈ Sa . By the definition of ua , ua ≥ ψ in Mk . Consequently u˜ ≤ ua in B(i) . As a result, u˜ = ua ,
(x, m) ∈ B(i) .
Since x0 is arbitrary, we know that ua is a k-convex viscosity solution of (3.1). By the definition of ua , ua ≤ ua ≤ g ,
(x, m) ∈ Mk ,
so ua satisfies (3.2). Theorem 3.1 is proved.
Acknowledgements The author would like to thank Professor Jiguang Bao for his guidance and Professor Yanyan Li and Professor Luis Caffarelli for their suggestions. The author is also indebted to the anonymous referees for the deep and helpful comments. References [1] F. Almgren, F. Almgren’s Big Regularity Paper. Q -Valued Functions Minimizing Dirichlet’s Integral and the Regularity of Area-Minimizing Rectifiable Currents up to Codimension 2, in: World Scientific Monograph Series in Mathematics, vol. 1, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. [2] G.C. Evans, A necessary and sufficient condition of Wiener, Amer. Math. Monthly 54 (1947) 151–155. [3] G.C. Evans, Surfaces of minimal capacity, Proc. Natl. Acad. Sci. USA 26 (1940) 489–491. [4] G.C. Evans, Lectures on Multiple Valued Harmonic Functions in Space, in: Univ. Calif. Publ. Math. (NS), vol. 1, 1951, pp. 281–340. [5] G. Levi, Generalization of a spatial angle theorem, Uspehi Mat. Nauk. 26 (1971) 199–204. (in Russian) Translated from the English by Ju. V. Egorov. [6] L. Caffarelli, Certain multiple valued harmonic functions, Proc. Amer. Math. Soc. 54 (1976) 90–92. [7] L. Caffarelli, On the hölder continuity of multiple valued harmonic functions, Indiana Univ. Math. J. 25 (1976) 79–84. [8] L. Gosse, S. Jin, X. Li, Two moment systems for computing multiphase semiclassical limits of the Schrödinger equation, Math. Models Methods Appl. Sci. 13 (2003) 1689–1723. [9] S. Izumiya, G.T. Kossioris, G.N. Makrakis, Multivalued solutions to the eikonal equation in stratified media, Quart. Appl. Math. 59 (2001) 365–390. [10] S. Jin, S. Osher, A level set method for the computation of multivalued solutions to quasi-linear hyperbolic PDEs and Hamilton–Jacobi equations, Commun. Math. Sci. 1 (2003) 575–591.
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