- Email: [email protected]
Existence of distributional solutions of a closed Dirichlet problem for an elliptic–hyperbolic equation
Nonlinear Analysis 74 (2011) 6512–6517
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Existence of distributional solutions of a closed Dirichlet problem for an elliptic–hyperbolic equation Meng Xu ∗ , Xiaoping Yang Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, PR China
article
info
Article history: Received 20 July 2010 Accepted 21 June 2011 Communicated by Enzo Mitidieri
abstract The closed Dirichlet problem for an elliptic–hyperbolic equation arising in linearizing the Monge–Ampère type equation is shown to admit a distributional solution by using an a-b-c integral method. © 2011 Elsevier Ltd. All rights reserved.
Keywords: Elliptic–hyperbolic equation Closed Dirichlet problem abc-method
1. Introduction Consider the existence of distributional solutions to the Dirichlet problem Lu ≡ (x2 − y2 )uxx + uyy + 2xux = f (x, y) u=0
in Ω on ∂ Ω
(1.1)
where Ω ⊆ {(x, y) ∈ R2 : |x| < 2, |y| < 2} is a bounded open and connected domain with piecewise C 1 boundary ∂ Ω . Eq. (1.1) arises in a modified version of linearizing the general Monge–Ampère type equation det(D2 u) = K (x, u, Du) in [1–5]. When K changes its sign, the linearized equation K (·, 0, 0)
n −1 −
aij uij + unn = f
(1.2)
i,j=1
is of mixed-type one. First, Lin [6,7] used the Tricomi-type linearized equation ywxx + wyy + ε F (ε, x, y, Dw, D2 w) = 0 for the Monge–Amperè equation, and obtained the existence of sufficiently smooth local solutions of the Darboux equation det(∇ij z ) − K det(gij )(1 − |∇g z |2 ) = 0
(1.3)
where ∇ij are covariant derivatives, and hence a sufficiently smooth isometric embedding of two-dimensional surfaces into R3 in a neighborhood of p for the following two cases: K (p) = 0 and dK (p) ̸= 0, or K ≥ 0 in a neighborhood of the point p, where K (p) is the Gaussian curvature at p of the surface.
∗
Corresponding author. Tel.: +86 25 84315877; fax: +86 25 84315875. E-mail addresses: [email protected] (M. Xu), [email protected] (X. Yang).
0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.06.034
M. Xu, X. Yang / Nonlinear Analysis 74 (2011) 6512–6517
6513
Khuri [8,9] used the linearized equation of Gallerstedt type
−y2 wxx + wyy + ε F (ε, x, y, w, Dw, D2 w) = 0 to investigate the local isometric embedding problem for two-dimensional Riemannian manifolds, and the problem of locally prescribed Gaussian curvature for surfaces in R3 . It is well known that boundary value problems in which the solution is prescribed on the entire boundary are sometimes called closed, in distinction to open problems in which the solution is prescribed on only part of the boundary. Lupo et al. [10] showed that the closed Dirichlet problem is over-determined, and then investigated the well-posedness of closed Dirichlet and mixed Dirichlet-conormal-type boundary value problems for K (y)uxx + uyy = f . Otway [11,12] considered the same problems for (x − y2 )uxx + uyy + κ ux = 0 in models of wave propagation through a linear dielectric medium (cold plasma) at frequencies lying below the geometrical optics range when κ ∈ [0, 2]. Kuz’min [13], using the method of singular perturbations and the Galerkin method, showed that there exists a unique solution u ∈ W 2,2 (G) of the mixed Dirichlet–Neumann boundary problem k(x, y)uxx + [a(x, y)uy ]y − α(x, y)ux + c (x, y)u = f (x, y) in G = (0, l) × (−1, 1) for any f ∈ W 1,2 (G) under some sufficiently smooth conditions on the coefficients and the stronger assumptions 2α ± kx ≥ δ > 0. Moreover, he obtained some results for the regularity on the solutions. Han [2,3], and Han and Khuri [4] obtained the linearized equation Lu = (aKux )x + buyy + cKux + duy = f ,
a, b > 0
(1.4)
of Darboux equation (1.3) to study the old problem of isometrically embedding a two-dimensional Riemannian manifold into Euclidean space R3 . K −1 (0) divides a small neighborhood of the origin into elliptic domains Ωκ+ on which K > 0, and hyperbolic domains Ωρ− (ρ = 1, 2) or Ωρ− (ρ = 3, 4) on which K < 0 (see [4, Lemma 2.1]). These domains are sector ones, that is, each occupies the region between two lines passing through the origin. For example, a portion of the given hyperbolic region Ωρ− ∩ Bσ (0) (ρ = 1, 2) may be written as
Ω = {(x, y)|h(x) < y < σ },
σ >0
for some Lipschitz function h(x) satisfying h(0) = 0 and ‖h(x) − |x| ‖C 1 = O(σ ). Yoshino [5] considered the linearized operator at u0 L(u0 )v ≡
d dε
det D2 (u0 + εv) = (u0 )xx vyy + (u0 )yy vxx − 2(u0 )xy vxy
of the generalized Monge–Ampère equation det(D2 u) + c (x, y)uxy = det(D2 u0 ) + c (x, y)(u0 )xy + g (x, y). When c ≡ 0 and u0 = x4 + kx2 y2 + y4 in the mixed case k > 6 in Example 2, the set {det(D2 u0 ) = 0} ⊆ R2 consists of four lines intersecting at the origin. The equation degenerates on these lines, and changes its type from elliptic to hyperbolic or vice versa when crossed one of them. This case is more complicated. In this paper, Eq. (1.1) is a special case of the linearized Eqs. (1.2) and (1.4). Eq. (1.1) degenerates on the lines y = ±x. It changes its type from elliptic to hyperbolic or vice versa, when crossed one of these lines in some neighborhood of the origin. The authors have been interested in the well-posedness for (1.1); in particular, in the regularity near the origin. We will show the existence of distributional solutions of the closed Dirichlet problem by using the a-b-c integral method in this paper. This paper is organized as follows. The main theorem (Theorem 2.2) is presented in Section 2. Two lemmas and the proof of Theorem 2.2 are given in Section 3. 2. Main theorem In order to study the regular behavior in the neighborhood of the origin, we define the space H01 (Ω ; |x2 − y2 |) to be the closure of C0∞ (Ω ) with respect to the weighted Sobolev norm
∫ ‖u‖H 1 (Ω ;|x2 −y2 |) =
Ω
1/2 2 2 2 2 2 |x − y |ux + uy + u dxdy .
In terms of a Poincaré inequality with an appropriately weighted right-hand side (see Lemma 3.1), we get an equivalent norm of the form
∫ ‖u‖H 1 (Ω ;|x2 −y2 |) = 0
in
H01
(Ω ; |x2 − y2 |).
Ω
1/2 2 2 2 2 |x − y | ux + uy dxdy
6514
M. Xu, X. Yang / Nonlinear Analysis 74 (2011) 6512–6517
Next, the definition of distributional solutions is given in the following: Definition 2.1. A function u ∈ L2 (Ω ) is a distributional solution of problem (1.1) if
(u, Lϕ)L2 (Ω ) = ⟨f , ϕ⟩ for any ϕ ∈ H 1 (Ω ; |x2 − y2 |) with Lϕ ∈ L2 (Ω )
(2.1)
holds, where ⟨, ⟩ is the duality bracket associated to the H −1 -norm in the sense of Lax
‖w‖H −1 (Ω ;|x2 −y2 |) =
sup 0̸=ϕ∈C0∞ (Ω )
|⟨w, ϕ⟩| . ‖ϕ‖H 1 (Ω ;|x2 −y2 |) 0
Therefore, one has the generalized Schwarz inequality
|⟨w, ϕ⟩| ≤ ‖w‖H −1 (Ω ;|x2 −y2 |) ‖ϕ‖H 1 (Ω ;|x2 −y2 |) 0
for any w ∈ H
−1
(Ω ; |x − y |) and ϕ ∈ H01 (Ω ; |x2 − y2 |), and has a rigged triple of Hilbert spaces 2
2
H01 (Ω ; |x2 − y2 |) ⊂ L2 (Ω ) ⊂ H −1 (Ω ; |x2 − y2 |). Now, the main theorem is presented: Theorem 2.2. For every f ∈ H −1 (Ω ; |x2 − y2 |), the Dirichlet problem (1.1) admits a distributional solution u ∈ L2 (Ω ). 3. Proofs of main theorem First, we will give the appropriately weighted Poincaré inequality. Its proof is easily obtained without modification in [10,12]. Lemma 3.1. If u ∈ H 1 (Ω ; |x2 − y2 |) satisfies u = 0 on ∂ Ω , then there exists a constant C = C (Ω ) > 0 such that
‖u‖2L2 (Ω ) ≤ C
∫ Ω
2 |x − y2 |u2x + u2y dxdy.
(3.1)
The proof of Theorem 2.2 follows by classical arguments from an a priori estimate: Lemma 3.2. For any u ∈ C02 (Ω ), there exists a constant C = C (Ω ) > 0 such that
‖u‖H 1 (Ω ;|x2 −y2 |) ≤ C ‖u‖L2 (Ω ) .
(3.2)
0
Proof (a-b-c Integral Method). To obtain the estimate (3.2), we consider an arbitrary function u ∈ C02 (Ω ) and a triple (a, b, c ) of sufficiently regular functions to be determined later. Define the operator Mu = au + bux + cuy , and then estimate the quantity (Mu, Lu) from above and below. Integrating by parts, we have
(Mu, Lu) =
∫∫ − 6 Ω i =1
Ii dxdy,
(3.3)
where I1 = [a(x2 − y2 )uux ]x − ax (x2 − y2 )uux − a(x2 − y2 )u2x ; I2 = (auuy )y − ay uuy − au2y ; I3 =
1 2
1
[b(x2 − y2 )u2x ]x − bx (x2 − y2 )u2x + bxu2x ;
I4 = (bux uy )y − by ux uy −
2 1 2
1
(bu2y )x + bx u2y ; 2
I5 = [c (x2 − y2 )ux uy ]x − cx (x2 − y2 )ux uy − I6 =
1 2
1
(cu2y )y − cy u2y . 2
1 2
1
[c (x2 − y2 )u2x ]y + cy (x2 − y2 )u2x − cyu2x ; 2
(3.4)
M. Xu, X. Yang / Nonlinear Analysis 74 (2011) 6512–6517
6515
Noting that n = (n1 , n2 ) is the outer unit normal vector on ∂ Ω , s is the arc length, n1 =
=
dy ds
and n2 = − dx ds
|n|
n21 + n22 = 1 , we obtain, in terms of the Green formula,
] ∫ ∫ [ 1 1 2 2 (Mu, Lu) = − a + bx − cy (x − y ) + bx − cy u2x − [by + cx (x2 − y2 )]ux uy 2
Ω
−
1
a−
1
bx +
2
2
cy
2
− ax (x − y )uux − ay uuy dxdy
u2y
2
∫ + ∂Ω
2
∫
1
a(x − y )uux + 2
+ ∂Ω
2 1
auuy + bux uy −
2
2
b(x − y ) 2
2
u2x
c (x − y ) 2
2
u2x
−
1 2 1
+
2
bu2y
+ c (x − y )ux uy n1 ds 2
2
cu2y
n2 ds.
(3.5)
Choose a = −1,
b(x, y) = Ux|x2 − y2 |,
c = Vy,
(3.6)
where U and V are two constants to be determined later. Noting that u is identically zero outside of a compact set contained in the interior of Ω (with compact support), as the arguments considered in [10] or [12], one of the coefficients in Mu fails to be continuously differentiable on all of Ω , and a cut should be introduced along the lines y = ±x since the function b is only piecewise C 1 (Ω ). The boundary integrals involving the functions a and c on either side of the lines will cancel and the boundary integrals involving b will be zero on the curve. Therefore, we have
(Mu, Lu) =
∫ ∫ [
] 1 1 −a − bx + cy (x2 − y2 ) + bx − cy u2x − [by + cx (x2 − y2 )]ux uy 2
Ω
+
2
1
1
2
2
−a + bx − cy
∫ ∫ [ =
1− Ω
1 2
[
∫∫ =
bx +
+
1+
3x2 − y2 2
+
1−
+
1+
y2 − 3x2 2
] V 1 (x2 − y2 ) + bx − Vy2 u2x − by ux uy + 1 + bx − u2y dxdy
U+ V
V
2
u2y
2
2
V
2
]
(x2 − y2 ) + Ux2 (x2 − y2 ) − Vy2 u2x + 2Uxyux uy
y2 − 3x2
U−
2
2
U−
Ω−
− ax (x − y )uux − ay uuy dxdy 2
2
2
[
∫∫
2
3x − y
Ω+
V
2
1−
u2y
dxdy V
U+
2
2
] (x2 − y2 ) + Ux2 (y2 − x2 ) − Vy2 u2x − 2Uxyux uy
u2y dxdy.
That is,
(Mu, Lu) =
[
∫∫
1−
x2 − y2 2
Ω+
+
1+
3x2 − y2 2
U−
U+ V
2
V
2
] (x2 − y2 ) − Vy2 u2x + 2Uxyux uy
u2y
∫∫
dxdy + Ω−
[ ] y2 − x2 V 2 2 2 −1 + U− (y − x ) − Vy u2x 2
2
y2 − 3x2 V u2y dxdy − 2Uxyux uy + 1 + U− 2
∫∫ ≥
[ 1−
Ω+
x2 − y2 2
U+
2
V 2
] (x2 − y2 ) − (V + U )y2 u2x + U (yux + xuy )2
6516
M. Xu, X. Yang / Nonlinear Analysis 74 (2011) 6512–6517
+
1+
√ +U
x2 − y 2 2
U−
V
y 2xux − √ uy 2
2
u2y
2
[
∫∫
dxdy +
−1 +
y2 − x2 2
Ω−
+ 1−
3x2 2
U−
V 2
U−
V
] (y2 − x2 ) − Vy2 − 2Ux2 u2x
2
u2y dxdy
i.e.
(Mu, Lu) ≥
[
∫∫
D2
1−
2
Ω+
∫∫ + Ω−
2
y
D2
1−
2
Ω+
∫∫ + Ω−
∫∫ Ω
2
2
[
≥
(x − y ) − (V + U )y
2
]
u2x
+ 1−
V
2
u2y
dxdy
2
2
2xux − √ uy 2
∫∫
≥C
U+
2
[ ] y2 − x2 V −1 + U− (y2 − x2 ) − (V + 2U )y2 + 2U (y2 − x2 ) u2x
√
+U
V
+ 1−
V
U+
2
3x2 2
U−
V
2
u2y
(x − y ) − (V + U )y 2
2
2
dxdy
]
u2x
+ 1−
V
2
u2y
dxdy
[ ] V V 3D2 2 2 2 2 −1 + 2U − (y − x ) − (V + 2U )y ux + 1 − U− u2y dxdy 2
2
2
|x2 − y2 |u2x + u2y dxdy = C ‖u‖2H 1 (Ω ;|x2 −y2 |) ,
(3.7)
0
holds provided the inequalities U ≥ 0,
V < 0,
−1 + 2U −
V 2
1−
> 0,
D2 2
U+
V 2
> 0,
V + 2U < 0,
V + U < 0, 3D2
1−
2
U−
V 2
1−
V 2
> 0,
>0
(3.8)
hold, where D = max{sup(x,y)∈Ω |x|, sup(x,y)∈Ω |y|}, the domain Ω = Ω+ ∪ Ω− , and
Ω+ = {(x, y) ∈ Ω : x2 − y2 ≥ 0}, Ω− = {(x, y) ∈ Ω : x2 − y2 ≤ 0}. That is, we must have
− V > 2U > 1 +
V 2
D2
>
2
U > 0,
1−
V 2
>
3D2 2
U > 0.
(3.9)
In fact, (U , V , D) in (3.9) may have a solution, for example, one may choose D = 1, U = 3 16
, V = − 32 ; or D =
√
1 2
, V = −1; or D =
7 3, U = 24 , V = −1 or D = 72 , U = 51 , V = − 54 . Therefore, (3.7) is valid. In order to obtain an upper bound for (Mu, Lu) in terms of the L2 -norm of u, we have
(Mu, Lu) ≤
∫∫ Ω
∫∫ ≤ Ω
|u| + |U | · |x| · |x2 − y2 | · |ux | + |V | · |y| · |uy | |Lu|dxdy
∫∫ |u| · |Lu|dxdy + |U | sup |x| |x2 − y2 | |x2 − y2 ||ux | · |Lu|dxdy (x,y)∈Ω
Ω
∫∫ + |V | sup |y| (x,y)∈Ω
Ω
|uy | · |Lu|dxdy √
≤ ‖u‖L2 (Ω ) ‖Lu‖L2 (Ω ) + ∫ ∫ + |V |D Ω
2|U |D
2
∫ ∫
‖Lu‖L2 (Ω )
2
|x − y | Ω
1/2 u2y dxdy
1/2 2
u2x dxdy
‖Lu‖L2 (Ω )
√
2, U =
M. Xu, X. Yang / Nonlinear Analysis 74 (2011) 6512–6517
6517
≤ C ‖u‖H 1 (Ω ;|x2 −y2 |) ‖Lu‖L2 (Ω )
(3.10)
0
by the Schwarz inequality and Lemma 3.1, where C depends on U , V and Ω . Combining (3.10) with the lower bound (3.7), and dividing through by ‖u‖H 1 (Ω ;|x2 −y2 |) , we complete the proof of Lemma 3.2.
0
Finally, we give the proof of main theorem. It is similar to the proof of [10] or [12]. Proof of Theorem 2.2. Briefly, for any f ∈ H −1 (Ω ; |x2 − y2 |), one defines a linear functional Jf (·) by Jf (Lϕ) = ⟨f , ϕ⟩
for ϕ ∈ C0∞ (Ω ).
(3.11)
Hence, we have
|Jf (Lϕ)| ≤ ‖f ‖H −1 (Ω ;|x2 −y2 |) ‖ϕ‖H 1 (Ω ;|x2 −y2 |) ≤ C ‖f ‖H −1 (Ω ;|x2 −y2 |) ‖Lϕ‖L2 (Ω ) 0
by the generalized Schwarz inequality and Lemma 3.2. That is, this functional is bounded on a subspace Λ ≡ {Lϕ ∈ L2 (Ω ) : ϕ ∈ C0∞ (Ω )} of L2 (Ω ), using the fact that L is self-adjoint. By the standard Hahn–Banach theorem, the functional Jf extends to the closure of Λ in L2 (Ω ) in a bounded way, and then extend by zero on the orthogonal complement of Λ. Therefore, a bounded linear functional on all of L2 (Ω ) is obtained. The Riesz Representation Theorem guarantees that there exists u ∈ L2 (Ω ) satisfying (2.1). Hence, we complete the proof of Theorem 2.2. Acknowledgments The authors thank the anonymous reviewers so much for their careful reading of our manuscript and offering valuable suggestions. The authors are supported by the Natural Science Foundation of China under Grants 11001132 and 11071119. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Q. Han, Local solutions to a class of Monge–Ampère equations of mixed type, Duke Math. J. 136 (3) (2007) 421–473. Q. Han, On the isometric embedding of surfaces with Gauss curvature changing sign cleanly, Comm. Pure Appl. Math. 58 (2005) 285–295. Q. Han, Smooth local isometric embedding of surfaces with Gauss curvature changing sign cleanly, Preprint, 2010. Q. Han, M. Khuri, On the local isometric embedding in R3 of surfaces with Gaussian curvature of mixed sign, Comm. Anal. Geom. 18 (4) (2010) 649–704. M. Yoshino, Riemann–Hilbert problem and solvability of differential equations, Rend. Semin. Mat. Univ. Politec. Torino 61 (2) (2003) 183–208. C.S. Lin, The isometric embedding in R3 of a 2-dimensional Riemannian manifold with Gaussian curvature changing sign cleanly, Comm. Pure Appl. Math. 39 (1986) 867–887. C.S. Lin, The local isometric embedding in R3 of 2-dimensional Riemannian manifolds with non-negative curvature, J. Differential Geom. 21 (1985) 213–230. M. Khuri, The local isometric embedding in R3 of two-dimensional Riemannian manifolds with Gaussian curvature changing sign to finite order on a curve, J. Differential Geom. 76 (2007) 249–291. M. Khuri, Local solvability of degenerate Monge–Ampère equations and applications to geometry, Electron. J. Differential Equations 2007 (65) (2007) 1–37. D. Lupo, C.S. Morawetz, K.R. Payne, On closed boundary value problems for equations of mixed elliptic–hyperbolic type, Comm. Pure Appl. Math. 60 (9) (2007) 1319–1348. T.H. Otway, A boundary value problem for cold plasma dynamics, J. Appl. Math. 3 (1) (2003) 17–33. T.H. Otway, Energy inequalities for a model wave propagation in cold plasma, Publ. Mat. 52 (2008) 195–234. A.G. Kuz’min, Boundary-Value Problems for Transonic Flow, John Wiley & Sons Ltd., Chichester, 2002.
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Existence of distributional solutions of a closed Dirichlet problem for an elliptic–hyperbolic equation Meng Xu ∗ , Xiaoping Yang Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, PR China
article
info
Article history: Received 20 July 2010 Accepted 21 June 2011 Communicated by Enzo Mitidieri
abstract The closed Dirichlet problem for an elliptic–hyperbolic equation arising in linearizing the Monge–Ampère type equation is shown to admit a distributional solution by using an a-b-c integral method. © 2011 Elsevier Ltd. All rights reserved.
Keywords: Elliptic–hyperbolic equation Closed Dirichlet problem abc-method
1. Introduction Consider the existence of distributional solutions to the Dirichlet problem Lu ≡ (x2 − y2 )uxx + uyy + 2xux = f (x, y) u=0
in Ω on ∂ Ω
(1.1)
where Ω ⊆ {(x, y) ∈ R2 : |x| < 2, |y| < 2} is a bounded open and connected domain with piecewise C 1 boundary ∂ Ω . Eq. (1.1) arises in a modified version of linearizing the general Monge–Ampère type equation det(D2 u) = K (x, u, Du) in [1–5]. When K changes its sign, the linearized equation K (·, 0, 0)
n −1 −
aij uij + unn = f
(1.2)
i,j=1
is of mixed-type one. First, Lin [6,7] used the Tricomi-type linearized equation ywxx + wyy + ε F (ε, x, y, Dw, D2 w) = 0 for the Monge–Amperè equation, and obtained the existence of sufficiently smooth local solutions of the Darboux equation det(∇ij z ) − K det(gij )(1 − |∇g z |2 ) = 0
(1.3)
where ∇ij are covariant derivatives, and hence a sufficiently smooth isometric embedding of two-dimensional surfaces into R3 in a neighborhood of p for the following two cases: K (p) = 0 and dK (p) ̸= 0, or K ≥ 0 in a neighborhood of the point p, where K (p) is the Gaussian curvature at p of the surface.
∗
Corresponding author. Tel.: +86 25 84315877; fax: +86 25 84315875. E-mail addresses: [email protected] (M. Xu), [email protected] (X. Yang).
0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.06.034
M. Xu, X. Yang / Nonlinear Analysis 74 (2011) 6512–6517
6513
Khuri [8,9] used the linearized equation of Gallerstedt type
−y2 wxx + wyy + ε F (ε, x, y, w, Dw, D2 w) = 0 to investigate the local isometric embedding problem for two-dimensional Riemannian manifolds, and the problem of locally prescribed Gaussian curvature for surfaces in R3 . It is well known that boundary value problems in which the solution is prescribed on the entire boundary are sometimes called closed, in distinction to open problems in which the solution is prescribed on only part of the boundary. Lupo et al. [10] showed that the closed Dirichlet problem is over-determined, and then investigated the well-posedness of closed Dirichlet and mixed Dirichlet-conormal-type boundary value problems for K (y)uxx + uyy = f . Otway [11,12] considered the same problems for (x − y2 )uxx + uyy + κ ux = 0 in models of wave propagation through a linear dielectric medium (cold plasma) at frequencies lying below the geometrical optics range when κ ∈ [0, 2]. Kuz’min [13], using the method of singular perturbations and the Galerkin method, showed that there exists a unique solution u ∈ W 2,2 (G) of the mixed Dirichlet–Neumann boundary problem k(x, y)uxx + [a(x, y)uy ]y − α(x, y)ux + c (x, y)u = f (x, y) in G = (0, l) × (−1, 1) for any f ∈ W 1,2 (G) under some sufficiently smooth conditions on the coefficients and the stronger assumptions 2α ± kx ≥ δ > 0. Moreover, he obtained some results for the regularity on the solutions. Han [2,3], and Han and Khuri [4] obtained the linearized equation Lu = (aKux )x + buyy + cKux + duy = f ,
a, b > 0
(1.4)
of Darboux equation (1.3) to study the old problem of isometrically embedding a two-dimensional Riemannian manifold into Euclidean space R3 . K −1 (0) divides a small neighborhood of the origin into elliptic domains Ωκ+ on which K > 0, and hyperbolic domains Ωρ− (ρ = 1, 2) or Ωρ− (ρ = 3, 4) on which K < 0 (see [4, Lemma 2.1]). These domains are sector ones, that is, each occupies the region between two lines passing through the origin. For example, a portion of the given hyperbolic region Ωρ− ∩ Bσ (0) (ρ = 1, 2) may be written as
Ω = {(x, y)|h(x) < y < σ },
σ >0
for some Lipschitz function h(x) satisfying h(0) = 0 and ‖h(x) − |x| ‖C 1 = O(σ ). Yoshino [5] considered the linearized operator at u0 L(u0 )v ≡
d dε
det D2 (u0 + εv) = (u0 )xx vyy + (u0 )yy vxx − 2(u0 )xy vxy
of the generalized Monge–Ampère equation det(D2 u) + c (x, y)uxy = det(D2 u0 ) + c (x, y)(u0 )xy + g (x, y). When c ≡ 0 and u0 = x4 + kx2 y2 + y4 in the mixed case k > 6 in Example 2, the set {det(D2 u0 ) = 0} ⊆ R2 consists of four lines intersecting at the origin. The equation degenerates on these lines, and changes its type from elliptic to hyperbolic or vice versa when crossed one of them. This case is more complicated. In this paper, Eq. (1.1) is a special case of the linearized Eqs. (1.2) and (1.4). Eq. (1.1) degenerates on the lines y = ±x. It changes its type from elliptic to hyperbolic or vice versa, when crossed one of these lines in some neighborhood of the origin. The authors have been interested in the well-posedness for (1.1); in particular, in the regularity near the origin. We will show the existence of distributional solutions of the closed Dirichlet problem by using the a-b-c integral method in this paper. This paper is organized as follows. The main theorem (Theorem 2.2) is presented in Section 2. Two lemmas and the proof of Theorem 2.2 are given in Section 3. 2. Main theorem In order to study the regular behavior in the neighborhood of the origin, we define the space H01 (Ω ; |x2 − y2 |) to be the closure of C0∞ (Ω ) with respect to the weighted Sobolev norm
∫ ‖u‖H 1 (Ω ;|x2 −y2 |) =
Ω
1/2 2 2 2 2 2 |x − y |ux + uy + u dxdy .
In terms of a Poincaré inequality with an appropriately weighted right-hand side (see Lemma 3.1), we get an equivalent norm of the form
∫ ‖u‖H 1 (Ω ;|x2 −y2 |) = 0
in
H01
(Ω ; |x2 − y2 |).
Ω
1/2 2 2 2 2 |x − y | ux + uy dxdy
6514
M. Xu, X. Yang / Nonlinear Analysis 74 (2011) 6512–6517
Next, the definition of distributional solutions is given in the following: Definition 2.1. A function u ∈ L2 (Ω ) is a distributional solution of problem (1.1) if
(u, Lϕ)L2 (Ω ) = ⟨f , ϕ⟩ for any ϕ ∈ H 1 (Ω ; |x2 − y2 |) with Lϕ ∈ L2 (Ω )
(2.1)
holds, where ⟨, ⟩ is the duality bracket associated to the H −1 -norm in the sense of Lax
‖w‖H −1 (Ω ;|x2 −y2 |) =
sup 0̸=ϕ∈C0∞ (Ω )
|⟨w, ϕ⟩| . ‖ϕ‖H 1 (Ω ;|x2 −y2 |) 0
Therefore, one has the generalized Schwarz inequality
|⟨w, ϕ⟩| ≤ ‖w‖H −1 (Ω ;|x2 −y2 |) ‖ϕ‖H 1 (Ω ;|x2 −y2 |) 0
for any w ∈ H
−1
(Ω ; |x − y |) and ϕ ∈ H01 (Ω ; |x2 − y2 |), and has a rigged triple of Hilbert spaces 2
2
H01 (Ω ; |x2 − y2 |) ⊂ L2 (Ω ) ⊂ H −1 (Ω ; |x2 − y2 |). Now, the main theorem is presented: Theorem 2.2. For every f ∈ H −1 (Ω ; |x2 − y2 |), the Dirichlet problem (1.1) admits a distributional solution u ∈ L2 (Ω ). 3. Proofs of main theorem First, we will give the appropriately weighted Poincaré inequality. Its proof is easily obtained without modification in [10,12]. Lemma 3.1. If u ∈ H 1 (Ω ; |x2 − y2 |) satisfies u = 0 on ∂ Ω , then there exists a constant C = C (Ω ) > 0 such that
‖u‖2L2 (Ω ) ≤ C
∫ Ω
2 |x − y2 |u2x + u2y dxdy.
(3.1)
The proof of Theorem 2.2 follows by classical arguments from an a priori estimate: Lemma 3.2. For any u ∈ C02 (Ω ), there exists a constant C = C (Ω ) > 0 such that
‖u‖H 1 (Ω ;|x2 −y2 |) ≤ C ‖u‖L2 (Ω ) .
(3.2)
0
Proof (a-b-c Integral Method). To obtain the estimate (3.2), we consider an arbitrary function u ∈ C02 (Ω ) and a triple (a, b, c ) of sufficiently regular functions to be determined later. Define the operator Mu = au + bux + cuy , and then estimate the quantity (Mu, Lu) from above and below. Integrating by parts, we have
(Mu, Lu) =
∫∫ − 6 Ω i =1
Ii dxdy,
(3.3)
where I1 = [a(x2 − y2 )uux ]x − ax (x2 − y2 )uux − a(x2 − y2 )u2x ; I2 = (auuy )y − ay uuy − au2y ; I3 =
1 2
1
[b(x2 − y2 )u2x ]x − bx (x2 − y2 )u2x + bxu2x ;
I4 = (bux uy )y − by ux uy −
2 1 2
1
(bu2y )x + bx u2y ; 2
I5 = [c (x2 − y2 )ux uy ]x − cx (x2 − y2 )ux uy − I6 =
1 2
1
(cu2y )y − cy u2y . 2
1 2
1
[c (x2 − y2 )u2x ]y + cy (x2 − y2 )u2x − cyu2x ; 2
(3.4)
M. Xu, X. Yang / Nonlinear Analysis 74 (2011) 6512–6517
6515
Noting that n = (n1 , n2 ) is the outer unit normal vector on ∂ Ω , s is the arc length, n1 =
=
dy ds
and n2 = − dx ds
|n|
n21 + n22 = 1 , we obtain, in terms of the Green formula,
] ∫ ∫ [ 1 1 2 2 (Mu, Lu) = − a + bx − cy (x − y ) + bx − cy u2x − [by + cx (x2 − y2 )]ux uy 2
Ω
−
1
a−
1
bx +
2
2
cy
2
− ax (x − y )uux − ay uuy dxdy
u2y
2
∫ + ∂Ω
2
∫
1
a(x − y )uux + 2
+ ∂Ω
2 1
auuy + bux uy −
2
2
b(x − y ) 2
2
u2x
c (x − y ) 2
2
u2x
−
1 2 1
+
2
bu2y
+ c (x − y )ux uy n1 ds 2
2
cu2y
n2 ds.
(3.5)
Choose a = −1,
b(x, y) = Ux|x2 − y2 |,
c = Vy,
(3.6)
where U and V are two constants to be determined later. Noting that u is identically zero outside of a compact set contained in the interior of Ω (with compact support), as the arguments considered in [10] or [12], one of the coefficients in Mu fails to be continuously differentiable on all of Ω , and a cut should be introduced along the lines y = ±x since the function b is only piecewise C 1 (Ω ). The boundary integrals involving the functions a and c on either side of the lines will cancel and the boundary integrals involving b will be zero on the curve. Therefore, we have
(Mu, Lu) =
∫ ∫ [
] 1 1 −a − bx + cy (x2 − y2 ) + bx − cy u2x − [by + cx (x2 − y2 )]ux uy 2
Ω
+
2
1
1
2
2
−a + bx − cy
∫ ∫ [ =
1− Ω
1 2
[
∫∫ =
bx +
+
1+
3x2 − y2 2
+
1−
+
1+
y2 − 3x2 2
] V 1 (x2 − y2 ) + bx − Vy2 u2x − by ux uy + 1 + bx − u2y dxdy
U+ V
V
2
u2y
2
2
V
2
]
(x2 − y2 ) + Ux2 (x2 − y2 ) − Vy2 u2x + 2Uxyux uy
y2 − 3x2
U−
2
2
U−
Ω−
− ax (x − y )uux − ay uuy dxdy 2
2
2
[
∫∫
2
3x − y
Ω+
V
2
1−
u2y
dxdy V
U+
2
2
] (x2 − y2 ) + Ux2 (y2 − x2 ) − Vy2 u2x − 2Uxyux uy
u2y dxdy.
That is,
(Mu, Lu) =
[
∫∫
1−
x2 − y2 2
Ω+
+
1+
3x2 − y2 2
U−
U+ V
2
V
2
] (x2 − y2 ) − Vy2 u2x + 2Uxyux uy
u2y
∫∫
dxdy + Ω−
[ ] y2 − x2 V 2 2 2 −1 + U− (y − x ) − Vy u2x 2
2
y2 − 3x2 V u2y dxdy − 2Uxyux uy + 1 + U− 2
∫∫ ≥
[ 1−
Ω+
x2 − y2 2
U+
2
V 2
] (x2 − y2 ) − (V + U )y2 u2x + U (yux + xuy )2
6516
M. Xu, X. Yang / Nonlinear Analysis 74 (2011) 6512–6517
+
1+
√ +U
x2 − y 2 2
U−
V
y 2xux − √ uy 2
2
u2y
2
[
∫∫
dxdy +
−1 +
y2 − x2 2
Ω−
+ 1−
3x2 2
U−
V 2
U−
V
] (y2 − x2 ) − Vy2 − 2Ux2 u2x
2
u2y dxdy
i.e.
(Mu, Lu) ≥
[
∫∫
D2
1−
2
Ω+
∫∫ + Ω−
2
y
D2
1−
2
Ω+
∫∫ + Ω−
∫∫ Ω
2
2
[
≥
(x − y ) − (V + U )y
2
]
u2x
+ 1−
V
2
u2y
dxdy
2
2
2xux − √ uy 2
∫∫
≥C
U+
2
[ ] y2 − x2 V −1 + U− (y2 − x2 ) − (V + 2U )y2 + 2U (y2 − x2 ) u2x
√
+U
V
+ 1−
V
U+
2
3x2 2
U−
V
2
u2y
(x − y ) − (V + U )y 2
2
2
dxdy
]
u2x
+ 1−
V
2
u2y
dxdy
[ ] V V 3D2 2 2 2 2 −1 + 2U − (y − x ) − (V + 2U )y ux + 1 − U− u2y dxdy 2
2
2
|x2 − y2 |u2x + u2y dxdy = C ‖u‖2H 1 (Ω ;|x2 −y2 |) ,
(3.7)
0
holds provided the inequalities U ≥ 0,
V < 0,
−1 + 2U −
V 2
1−
> 0,
D2 2
U+
V 2
> 0,
V + 2U < 0,
V + U < 0, 3D2
1−
2
U−
V 2
1−
V 2
> 0,
>0
(3.8)
hold, where D = max{sup(x,y)∈Ω |x|, sup(x,y)∈Ω |y|}, the domain Ω = Ω+ ∪ Ω− , and
Ω+ = {(x, y) ∈ Ω : x2 − y2 ≥ 0}, Ω− = {(x, y) ∈ Ω : x2 − y2 ≤ 0}. That is, we must have
− V > 2U > 1 +
V 2
D2
>
2
U > 0,
1−
V 2
>
3D2 2
U > 0.
(3.9)
In fact, (U , V , D) in (3.9) may have a solution, for example, one may choose D = 1, U = 3 16
, V = − 32 ; or D =
√
1 2
, V = −1; or D =
7 3, U = 24 , V = −1 or D = 72 , U = 51 , V = − 54 . Therefore, (3.7) is valid. In order to obtain an upper bound for (Mu, Lu) in terms of the L2 -norm of u, we have
(Mu, Lu) ≤
∫∫ Ω
∫∫ ≤ Ω
|u| + |U | · |x| · |x2 − y2 | · |ux | + |V | · |y| · |uy | |Lu|dxdy
∫∫ |u| · |Lu|dxdy + |U | sup |x| |x2 − y2 | |x2 − y2 ||ux | · |Lu|dxdy (x,y)∈Ω
Ω
∫∫ + |V | sup |y| (x,y)∈Ω
Ω
|uy | · |Lu|dxdy √
≤ ‖u‖L2 (Ω ) ‖Lu‖L2 (Ω ) + ∫ ∫ + |V |D Ω
2|U |D
2
∫ ∫
‖Lu‖L2 (Ω )
2
|x − y | Ω
1/2 u2y dxdy
1/2 2
u2x dxdy
‖Lu‖L2 (Ω )
√
2, U =
M. Xu, X. Yang / Nonlinear Analysis 74 (2011) 6512–6517
6517
≤ C ‖u‖H 1 (Ω ;|x2 −y2 |) ‖Lu‖L2 (Ω )
(3.10)
0
by the Schwarz inequality and Lemma 3.1, where C depends on U , V and Ω . Combining (3.10) with the lower bound (3.7), and dividing through by ‖u‖H 1 (Ω ;|x2 −y2 |) , we complete the proof of Lemma 3.2.
0
Finally, we give the proof of main theorem. It is similar to the proof of [10] or [12]. Proof of Theorem 2.2. Briefly, for any f ∈ H −1 (Ω ; |x2 − y2 |), one defines a linear functional Jf (·) by Jf (Lϕ) = ⟨f , ϕ⟩
for ϕ ∈ C0∞ (Ω ).
(3.11)
Hence, we have
|Jf (Lϕ)| ≤ ‖f ‖H −1 (Ω ;|x2 −y2 |) ‖ϕ‖H 1 (Ω ;|x2 −y2 |) ≤ C ‖f ‖H −1 (Ω ;|x2 −y2 |) ‖Lϕ‖L2 (Ω ) 0
by the generalized Schwarz inequality and Lemma 3.2. That is, this functional is bounded on a subspace Λ ≡ {Lϕ ∈ L2 (Ω ) : ϕ ∈ C0∞ (Ω )} of L2 (Ω ), using the fact that L is self-adjoint. By the standard Hahn–Banach theorem, the functional Jf extends to the closure of Λ in L2 (Ω ) in a bounded way, and then extend by zero on the orthogonal complement of Λ. Therefore, a bounded linear functional on all of L2 (Ω ) is obtained. The Riesz Representation Theorem guarantees that there exists u ∈ L2 (Ω ) satisfying (2.1). Hence, we complete the proof of Theorem 2.2. Acknowledgments The authors thank the anonymous reviewers so much for their careful reading of our manuscript and offering valuable suggestions. The authors are supported by the Natural Science Foundation of China under Grants 11001132 and 11071119. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Q. Han, Local solutions to a class of Monge–Ampère equations of mixed type, Duke Math. J. 136 (3) (2007) 421–473. Q. Han, On the isometric embedding of surfaces with Gauss curvature changing sign cleanly, Comm. Pure Appl. Math. 58 (2005) 285–295. Q. Han, Smooth local isometric embedding of surfaces with Gauss curvature changing sign cleanly, Preprint, 2010. Q. Han, M. Khuri, On the local isometric embedding in R3 of surfaces with Gaussian curvature of mixed sign, Comm. Anal. Geom. 18 (4) (2010) 649–704. M. Yoshino, Riemann–Hilbert problem and solvability of differential equations, Rend. Semin. Mat. Univ. Politec. Torino 61 (2) (2003) 183–208. C.S. Lin, The isometric embedding in R3 of a 2-dimensional Riemannian manifold with Gaussian curvature changing sign cleanly, Comm. Pure Appl. Math. 39 (1986) 867–887. C.S. Lin, The local isometric embedding in R3 of 2-dimensional Riemannian manifolds with non-negative curvature, J. Differential Geom. 21 (1985) 213–230. M. Khuri, The local isometric embedding in R3 of two-dimensional Riemannian manifolds with Gaussian curvature changing sign to finite order on a curve, J. Differential Geom. 76 (2007) 249–291. M. Khuri, Local solvability of degenerate Monge–Ampère equations and applications to geometry, Electron. J. Differential Equations 2007 (65) (2007) 1–37. D. Lupo, C.S. Morawetz, K.R. Payne, On closed boundary value problems for equations of mixed elliptic–hyperbolic type, Comm. Pure Appl. Math. 60 (9) (2007) 1319–1348. T.H. Otway, A boundary value problem for cold plasma dynamics, J. Appl. Math. 3 (1) (2003) 17–33. T.H. Otway, Energy inequalities for a model wave propagation in cold plasma, Publ. Mat. 52 (2008) 195–234. A.G. Kuz’min, Boundary-Value Problems for Transonic Flow, John Wiley & Sons Ltd., Chichester, 2002.