AN INTRODUCTION TO VISCOSITY SOLUTION THEORY. In this note, we study the general second-order fully nonlinear equations arising in various fields:

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1 AN INTRODUCTION TO VISCOSITY SOLUTION THEORY QING LIU AND XIAODAN ZHOU 1. Introduction to Fully Nonlinear Equations In this note, we study the general second-order fully nonlinear equations arising in various fields: or F(x,u(x), u(x), 2 u(x)) = (1.1) u t +F(x,u(x,t), u(x,t), 2 u(x,t)) =, (1.2) where x = (x 1,x 2,...,x n ) R n and t [, ) are independent variables, u : R n R is the unknown function, and u t, u = (u x1,u x2,...,u xn ) and 2 u respectively denote the derivative of u with respect of t, the gradient and Hessian of u with respect to x. Here F is a function defined on R n R R n S n, where S n stands for the set of all symmetric n n matrices. We are particularly interested in the class of equations with the following assumption: (A1) (Ellipticity) F(x,r,p,X) is monotone in X, i.e., F(x,r,p,X) F(x,r,p,Y) for all X,Y S n such that X Y. Regarding the notation X Y, we recall that there is a natural partial ordering in S n ; indeed, X Y means that X Y is positive semi-definite. The equation (1.2) that satisfies (A1) is called parabolic. The simplest examples of (1.1) and (1.2) with (A1) are respectively the Laplace equation n 2 u u = x = 2 i and the heat equation i=1 u t u =, where in both cases F(X) = trx. While the solutions u of the linear equations in such examples with certain boundary conditions or initial conditions can sometimes be written explicitly, solving (1.1) and (1.2) with a general nonlinear F may be difficult. Date: May 19,

2 2 QING LIU AND XIAODAN ZHOU For instance, we may discuss the elliptic equation (1.1) with a Dirichlet boundary condition: { F(x,u(x), u(x), 2 u(x)) =, for all x Ω, (1.3) (DP) u(x) = g(x), for all x Ω, (1.4) where Ω is a bounded smooth domain with boundary Ω and g is a (continuous) function on Ω. Although we might expect that for any kth order partial differential equation with boundary conditions, there exists a unique classical solution u, that is, u is of class C k and satisfies the equation everywhere in the domain, this is not always the case. A simple example is as follows. Example 1.1. Consider the following first order boundary value problem { ux = 1 in ( 1,1); (1.5) u( 1) = u(1) =. (1.6) This equationis called the Eikonalequation. It is clearthat there are no C 1 solutions. Even if we know there is a classical solution on some occasions, searching such a solution is not always straightforward, especially when the nonlinearity of F is very strong. We therefore often adopt a weak approach: (a) we choose to find a unique weak solution u of (1.1) in a function space broader than C 2 class and (b) prove the regularity of the weak solution u can reach C 2 [2]. Weak solutions to nonlinear can be mainly classified into two types: (1) Solutions in the distribution sense: We may define a weak solution via integration by parts, which is a very important approach to get a unique solution in a proper Sobolev space involving calculus of variations. (2) Solutions in the viscosity sense: This approach, which is the main topic of the note, is based on the maximum principle. The usual function space is merely the (semi-)continuous function space and the definition or estimates are usually conducted pointwise. The viscosity approach is particularly useful when F is not in the divergence form, i.e., the structure of F does not allow integration. We here list several well-known examples, in which viscosity solution theory is proved to play a significant role: (1) Hamilton-Jacobi equation: u t u 2 =. It has important applications in classical mechanics. (2) Hamilton-Jacobi-Bellman-Isaacs equation: λu+sup a A inf b B { tr(w(x,a,b) 2 u) f(x,a,b) u l(x,a,b)} =, where λ >, A,B are compact sets, W : R n A B S n is nonnegative uniformly and f and l are given functions on R n A B. This complicated equation arises from the optimal control and differential game theory.

3 INTRODUCTION TO VISCOSITY SOLUTIONS 3 (3) Level-set mean curvature flow equation: ( ) u u t u div = u Other examples include p-laplace equations, systems involving Hamilton-Jacobi equations and so on. Hereafter, to simplify our presentation, we assume F is continuous. To conclude our introduction, let us mention several nice references for the reader. Themost elementary bookshouldbetheoneauthoredbykoike[8], whichweclosely follow in this note. Another frequently cited reference is the famous User s Guide by Crandall, Ishii and Lions [4]. The applications of viscosity solutions can be found in [1] about optimal control/game theory and in [5] about geometric motion. We refer to [2] for the regularity theory of viscosity solutions. Acknowledgment. The notes are written as reference for the introductory presentation of the first author at the University of Pittsburgh. The authors would like to thank Professor Piotr Hajlasz for invitation. 2. Definition of Viscosity Solutions We provide the definition of viscosity solutions of (1.1) in a bounded domain Ω R n. Definition 1. We say u C(Ω) is a viscosity subsolution (resp., supersolution) of (1.1) in Ω if for any test function φ C 2 (Ω) and x Ω such that u φ attains a maximum (resp., minimum) over Ω at x, we have F(x,u(x ), φ(x ), 2 φ(x )) (resp., F(x,u(x ), φ(x ), 2 φ(x )).) We call u is a viscosity solution if it is both a viscosity subsolution and a viscosity supersolution. Remark 1. It is possible to give equivalent definitions by replacing the maximum /minimum in the definition above with strict (unique) maximum/minimum or local maximum/minimum. Remark 2. We usually take C 1 test functions if F is a first order operator. In fact, one may choose to use smooth (C ) functions as tests in Definition 1 and get another equivalent definition [5, Proposition 2.2.3]. Remark 3. Itisalsopossibletodefinesubsolutionsforthespaceofallupper semicontinuous functions on Ω, denoted by USC(Ω). Similarly, the lower semicontinuous function space (LSC(Ω)) can be used to define supersolutions. We may think of the test for subsolutions in Definition 1 in a more geometric way by considering that φ touches u from above at x. If u is a viscosity sub- or supersolution, we essentially require the derivatives of the test function φ to fulfill the corresponding inequality.

4 4 QING LIU AND XIAODAN ZHOU The next result indicates that viscosity solutions are weak solutions and agree with classical solutions if their regularity is adequate. Proposition 2.1. Assume that F is elliptic as in (A1). Then u is a classical solution of (1.1) in Ω if and only if u is a viscosity solution and u C 2 (Ω). Proof. We only show the proof for the subsolution part, since the supersolution part can be proved similarly. Suppose that u is a classical subsolution. Then u C 2 (Ω) and F(x,u(x), u(x), 2 u(x)) for all x Ω. (2.1) For any φ C 2 (Ω) touching u at any x Ω from above, we have u(x ) = φ(x ) and 2 u(x ) 2 φ(x ), which, in virtue of (2.1) and (A1), yields that F(x,u(x ), φ(x ), 2 φ(x )). If u is a viscosity subsolution of class C 2, we may take u itself as its test function and get (2.1). We next revisit Example 1.1. Since it is impossible to get C 1 solutions of this problem, one may attempt to relax the definition of solutions. A seemingly natural way is to consider the problem in the space of Lipschitz continuous functions and ask a solution u to satisfy the boundary conditions and the equation u x = 1 almost everywhere in ( 1, 1), as is sometimes called a generalized solution. However such a relaxation turns out to be too loose in that we lose the uniqueness of the solutions. It is clear that u(x) = ±(1 x ) both meet the requirements. In fact any continuous function having a saw-tooth graph with slopes either 1 or -1 are qualified to be such a solution as long as the boundary condition holds. We need to choose the best solution among those described above. The viscosity solution theory proves to be a correct tool for our selection purpose. Proposition 2.2. The function u(x) = 1 x is a viscosity solution of (1.5) with boundary condition (1.6). Proof. We only need to make verification at x =, since u is smooth and satisfies (1.5) elsewhere. If there is a C 1 function φ touching u at x =, then it is clear that 1 φ x () 1; in other words, φ x () 1, which indicates that u is a subsolution. On the other hand, u is a supersolution at x = because it is not possible to touch u from below by a C 1 function. One can similarly show that v(x) = x 1 is a viscosity solution of 1 = u with the boundary condition (1.6). In general, we have the following result.

5 INTRODUCTION TO VISCOSITY SOLUTIONS 5 Proposition 2.3. A function u is a viscosity solution of (1.1) in Ω if and only if v = u is a viscosity solution of F(x, v, v, 2 v) = in Ω. However, v(x) = x 1 is not a viscosity solution of (1.5). It does not satisfy the definition of supersolutions at x =. There is another way, so-called vanishing viscosity method, to derive the viscosity solution u(x) = 1 x. It also explains the reason why it is called a viscosity solution. Let us consider the regularized problem below. { εuxx +(u x ) 2 = 1 in ( 1,1), (2.2) u(±1) =, (2.3) which may be solved explicitly in the following manner. Suppose u ε is a smooth solution. Then the derivative v ε = u ε x satisfies v ε () = by symmetry and solves εv ε x +(vε ) 2 = 1 in ( 1,1). It is easy to see that v ε (x) = tanh(x/ε). Integrating it, we get ( ) e u ε x/ε +e x/ε (x) = εln. e 1/ε +e 1/ε Sending ε, we obtain u. In the next section we show that viscosity solutions are unique so as to rule out the other generalized solutions. 3. Uniqueness of Viscosity Solutions In this section we show the uniqueness of viscosity solutions. The key is to establish a comparison theorem. The techniques for second-order equations are a bit more complicated than those for first-order equations. We therefore only focus on the uniqueness of the solutions of the following general Hamilton-Jacobi equation { λu+h(x, u) = in Ω, (3.1) (HJ) u = g on Ω. (3.2) Here λ >, Ω R n is a bounded domain and g is a given continuous function. The function H : Ω R n R satisfies a structural assumption below: (H1) H(x,p) H(y,p) L(1 + p )( x y ) with some L > for all x,y Ω and p R n. Theorem 3.1 (Comparison Principle). Assume that Ω is bounded and λ >. Assume that (H1) holds. Suppose u C(Ω) and v C(Ω) are respectively a viscosity subsolution and a viscosity supersolution of (3.1). If u v on Ω, then u v in Ω. Proof. Suppose by contradiction that max Ω (u v) = u(x) v(x) = θ > for some x Ω. We double the variables and construct the following function: Φ ε (x,y) = u(x) v(y) x y 2 2ε

6 6 QING LIU AND XIAODAN ZHOU with ε > small. Assume that Φ ε attains a maximum over Ω at (x ε,y ε ). It is clear that Φ ε (x ε,y ε ) = u(x ε ) v(y ε ) x ε y ε 2 u(x) v(x) = θ, 2ε which is rewritten as x ε y ε 2 u(x ε ) v(y ε ) θ. (3.3) 2ε Since u and v are bounded, we obtain that x ε y ε as ε. Owing to the boundedness of Ω, we may take a subsequence of (x ε,y ε ), still indexed by ε, such that x ε,y ε x for some x Ω. By (3.3), we are led to x ε y ε 2 lim sup (3.4) ε 2ε and lim u(x ε) v(y ε )(= u(x ) v(x )) = θ >. (3.5) ε The comparison assumption u v on Ω and (3.5) enable us to get x Ω, which implies that x ε,y ε Ω when ε is sufficiently small. Now we set φ 1 (x) = v(y ε )+ x y ε 2, 2ε φ 2 (y) = u(x ε ) x ε y 2. 2ε It is easily seen that u φ 1 attains a maximum over Ω at x ε Ω, we therefore apply the definition of subsolutions to get λu(x ε )+H(x ε, φ 1 (x ε )). (3.6) Similarly, since v φ 2 attains a minimum over Ω at y ε Ω, we have Taking the difference of (3.6) and (3.7), we obtain λv(y ε )+H(y ε, φ 2 (y ε )). (3.7) λ(u(x ε ) v(y ε )) H(y ε, φ 2 (y ε )) H(x ε, φ 1 (x ε )). (3.8) Noting that φ 1 (x ε ) = φ 2 (y ε ) = (x ε y ε )/ε, we adopt (H1) to deduce that ( ) H(y ε, φ 2 (y ε )) H(x ε, φ 1 (x ε )) L x ε y ε 1+ x ε y ε ε By (3.4), we show that the right hand side of (3.8) tends to as ε. On the other hand, by (3.5), the left hand side of (3.8) is λ(u(x ε ) v(y ε )) λθ/2 > when ε > is sufficiently small. This gives a contradiction. Remark 4. The condition that u,v C(Ω) can be replaced with u USC(Ω) and v LSC(Ω) provided that one uses the semicontinous function space to define suband supersolutions, as in Remark 3.

7 INTRODUCTION TO VISCOSITY SOLUTIONS 7 Corollary 3.2 (Uniqueness). Assume that Ω is bounded and λ >. Assume that (H1) holds. Then the viscosity solution of (HJ) with g continuous is unique. Proof. Suppose there are two viscosity solutions u 1 and u 2 of (3.1) satisfying u 1 = u 2 = g on Ω. Then by Theorem 3.1, we get u 1 u 2. We can also have u 2 u 1 for the same reason. Hence, we conclude that u 1 = u 2. Remark 5. It is easy to see in the proof of Theorem 3.1 that one may take a continuous function λ(x) with inf Ω λ > in place of the constant λ and the comparison result and therefore the uniqueness still hold. We next aim to apply the theorem to the uniqueness problem for (1.5)-(1.6). We prove a more general theorem. Theorem 3.3. Let f > be a given continuous function and Ω be a bounded domain. Assume min Ω f >. Then the viscosity solutions of with boundary condition are unique. u = f(x) in Ω, (3.9) u = on Ω (3.1) Proof. In view of Proposition 2.3, it is equivalent to show that the solutions to the equation f(x) u = in Ω (3.11) with (3.1) are unique. Let us take w = e u. Then it can be proved, as in Proposition 3.4 below, that u is a viscosity solution of (3.11) if and only if w solves f(x)w w = in Ω (3.12) in the viscosity sense. Since the viscosity solution w to (3.12) with w = 1 on Ω due to Corollary 3.2, the viscosity solutions to (3.11) are unique as well. Proposition 3.4. Assume h : R R is a strictly increasing function of class C 2. The inverse function of h is denoted by h 1. Let u be a viscosity solution (resp., subsolution, supersolution) of (1.1) in Ω. Then v = h(u) is a viscosity solution (resp., subsolution, supersolution) of F in Ω. ( x,h 1 (v)(x), v(x) h (v(x)), 1 h (v(x)) 2 v(x) h (v(x)) (h (v(x))) 2 v(x) v(x) ) = We remark that there is another simpler proof of the comparison theorem for(3.9) by Ishii [7]

8 8 QING LIU AND XIAODAN ZHOU 4. Existence of Viscosity Solutions There are various approaches to the existence theorem of viscosity solutions to a certain equation. We list several well-known methods. (1) Vanishing viscosity method: We may add an artificial viscosity in the equation to get a smoother solution and pass to the limit as the viscosity approaches. (2) Optimal control/differential game: We set up an optimal control or game problem, whose value function represents or approximates the viscosity solution. (3) Perron s method: The supremum of all subsolutions or the infimum of all supersolutions with boundary data constraint gives a viscosity solution. The vanishing viscosity method is the earliest one to find solutions to fully nonlinear equations. One may refer to the early work [3] for more details. Perron s method is also classical but was first adopted in the theory of viscosity solutions by Ishii [6]. We here elaborate the second method above, which has much attractive recent development. We again first take the Eikonal equation in Example 1.1 to simplify our exhibition. It is not difficult to see that the equation is intimately related to the following optimal control problem: Suppose a particle moving from any x ( 1,1) at speed less than or equal to 1 is controlled to arrive the endpoints ±1 as soon as possible. It is obvious that the optimal arrival time under control is the viscosity solution u(x) = 1 x of (1.5) with (1.6). We next consider a general version of this control problem; namely, suppose Ω is a bounded domain in R n and one controls the motion of a particle in Ω started from x Ω with speed no more than 1 to reach the boundary Ω as quickly as possible. A more mathematical description is as follows. Let A denote the set of all possible control processes A = {α is Lebesgue measurable [, ) R n : α(t) B 1 () for all t }, where B r (a) denotes the open ball centered at a with radius r. We then use y(t;x,α) to denote the position of the particle after time t, started from x and controlled by α A. It is certainly the unique (Lipschitz continuous) solution of the ODE { y (s) = α(s), It follows immediately that y() = x. y(t;x 1,α) y(t;x 2,α) = x 1 x 2 for all t [, ), x 1,x 2 Ω and α A. (4.1)

9 INTRODUCTION TO VISCOSITY SOLUTIONS 9 We define the first exit time of Ω from x with control α to be { if y(t;x,α) Ω for all t > ; τ(x,α) = inf{t : y(t;x,α) / Ω} otherwise. We consider the following cost functional J(x;α) = τ(x,α) f(y(t;x,α))dt, where f : Ω R is a given Lipschitz continuous function, as known as the running cost function, and define the value function ( ) τ(x,α) v(x) = inf J(x;α) = inf f(y(t;x,α))dt. (4.2) α A α A We assume min Ω f >. It is not difficult to see that in this case v(x) is bounded uniformly in x Ω provided that Ω is bounded. Theorem 4.1. Let Ω be a bounded domain in R n. Assume f is a Lipschitz continuous function on Ω and min Ω f >. Then the value function v(x) defined in (4.2) is the unique viscosity solution of the Eikonal equation { u = f(x) in Ω, (4.3) u = on Ω. (4.4) Note that the uniqueness is guaranteed by Corollary 3.2 and Remark 5. When f 1, it is not hard to observe that the optimal control is to move the particle straight toward the closest point on Ω to x. Hence, the value function T of the optimal control problem must be the distance function to Ω, and thus is justified with ease to be the viscosity solution of the Eikonal equation. However, in general, we are not able to do such straightforward calculations. We instead need to establish the so-called dynamic programming principle for our problem { } v(x) = inf v(y(t;x,α))+ f(y(s;x,α))ds. (DPP) α A Theorem 4.2. Let Ω be a bounded domain in R n. Then for any x Ω, (DPP) holds when t > is sufficiently small. Proof. Let α A such that v(x) = We assume that t < τ(x,α ), then v(x) = τ(x,α ) f(y(s;x,α ))ds+ f(y(t;x,α ))dt. τ(x,α ) f(y(s;x,α ))ds.

10 1 QING LIU AND XIAODAN ZHOU Letting x t = y(t;x,α ), we have which implies τ(xt,α) v(x t ) = inf ( f(y(s;x t,α))ds) α A It easily follows that v(y(t;x,α )) τ(x,α ) t τ(xt,α ) f(y(s;x,α ))ds. inf {v(y(t;x,α))+ f(y(s,x,α))ds} v(x). α A On the other hand, let α 1,α 2 A, such that and Let inf {v(y(t;x,α))+ f(y(s,x,α))ds} = v(y(t;x,α 1 ))+ α A v(y(t;x,α 1 )) = ˆα(s) = τ(y(t;x,α1 ),α 2 ) f(y(s;x t,α ))ds, f(y(s;y(t;x,α 1 ),α 2 )ds. { α 1 (s), if s t; α 2 (s t), if t < s t+τ(y(t;x,α 1 ),α 2 ). f(y(s,x,α 1 ))ds Then ˆα A and we have τ(x,ˆα) The equality therefore follows. f(y(t;x, ˆα))dt v(x). With (DPP), we are able to show that v is continuous in Ω and then prove Theorem 4.1. Proposition 4.3. Let Ω be a bounded domain in R n. Assume f is a Lipschitz continuous function on Ω and min Ω f >. Then the value function v defined in (4.2) is continuous in Ω. Proof. We first notice that for any δ > small and x Ω with dist(x, Ω) δ, there exists α δ A such that τ(x;α δ ) δ. Since f is bounded in Ω by a constant M > from above, we get J(x,α δ ) Mδ, which yields v(x) Mδ and therefore the continuity of v on the boundary. In general, for any x 1,x 2 Ω with x 1 x 2 δ, we may take α 1 A such that v(x 1 ) J(x 1,α 1 ) ε. It is clear that τ 1 := τ(x 1,α 1 ) (v(x 1 )+ε)/min Ω f C for some C >. Now we adopt the same control α 1 for the dynamics starting from x 2 and If τ(x 2,α 1 ) < τ 1, we have y(t;x 2,α 1 ) y(t;x 1,α 1 ) δ for all t. J(x 2,α 1 ) J(x 1,α 1 ).

11 INTRODUCTION TO VISCOSITY SOLUTIONS 11 Otherwise, y 1 := y(τ 1 ;x 2,α 1 ) Ω but dist(y 1, Ω) δ. In this case, we may then turn to the control α δ above to guarantee J(y 1,α δ ) Mδ. Connecting the controls α 1 and α δ, i.e., letting { α 1 (t) if t τ 1, ˆα(t) := α δ (t τ(x 1,α 1 )) if t > τ 1, we end up with an estimate J(x 2, ˆα) J(x 1,α 1 )+ τ1 In either case, we have (4.5). It follows that Letting ε, we are led to f(y(t;x 2,α 1 )) f(y(t;x 1,α 1 ))dt+mδ. (4.5) v(x 2 ) v(x 1 )+L f Cδ +Mδ +ε. v(x 2 ) v(x 1 ) (M +L f C)δ = (M +L f C) x 2 x 1. We conclude the proof by interchanging the roles of x 1 and x 2 in the argument above. Proof of Theorem 4.1. The boundary condition is easily verified. We next show that v is a subsolution of (4.3). Assume that there exists φ C 1 (Ω) and x Ω such that max(v φ) = v(x ) φ(x ) = Ω By (DPP), we have φ(x ) inf α A {φ(y(t;x,α))+ We take α a for any fixed a B 1 () and get Then we have φ(x ) φ(y(t;x,a))+ φ(x ) φ(y(t;x,a)) f(y(s;x,α))ds} f(y(s;x,a))ds We may estimate the right hand side in the following way: f(y(s;x,a))ds = f(y(s;x,a))ds. (4.6) f(y(s;x,a)) f(x )ds+ f(x )ds f(y(s;x,a)) f(x ) ds+f(x )t 1 2 L ft 2 +f(x )t. On the other hand, Taylor expansion for the left hand side of (4.6) gives (4.7) φ(x ) (φ(x )+ φ(x ) at+o(t)) = φ(x ) at+o(t) (4.8)

12 12 QING LIU AND XIAODAN ZHOU Connecting (4.7) and (4.8), we get a φ(x )t f(x )t+o(t) Taking the supremum for the left hand side over a B 1 (), we have φ(x ) t f(x )t+o(t), which yields that φ(x ) f(x ) by dividing t and then sending t. Wenextshowthatv isasupersolutionof(4.3). Assumethatthereexistφ C 1 (Ω) and x Ω such that By (DPP), we have φ(x ) inf α A min(v φ) = v(x ) φ(x ) =. Ω { φ(y(t;x,α))+ We may take α A such that { φ(x ) φ(y(t;x,α ))+ Noticing that } f(y(s;x,α))ds. } f(y(s;x,α ))ds tε. y(s;x,α ) x s and thus f(y(s;x,α )) f(x) L f s, the inequality above yields that { } φ(x ) α ds+f(x)t +o(t) tε. The bracket on the right hand side can be bounded from below by ( φ(x ) + f(x))t. We obtain ( φ(x ) f(x ))t tε+o(t), which implies that φ(x ) f(x ). References [1] M. Bardi and I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi- Bellman equations, With appendices by Maurizio Falcone and Pierpaolo Soravia. Birkhuser Boston, Inc., Boston, MA, [2] L. Caffarelli and X. Cabre, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, 43, American Mathematical Society, Providence, RI, [3] M. G. Crandall, L. C. Evans and P. L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), [4] M. G. Crandall, H. Ishii and P. L. Lions, User s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), [5] Y. Giga, Surface evolution equations, a level set approach, Monographs in Mathematics 99, Birkhäuser Verlag, Basel, 26.

13 INTRODUCTION TO VISCOSITY SOLUTIONS 13 [6] H. Ishii, Perron s method for Hamilton-Jacobi equations, Duke Math. J. 55 (1987), [7] H. Ishii, A simple, direct proof of uniqueness for solutions of the Hamilton-Jacobi equations of eikonal type, Proc. Amer. Math. Soc., 1 (1987), [8] S. Koike, A Beginner s Guide to the Theory of Viscosity Solutions, MSJ Memoirs, 13. Mathematical Society of Japan, Tokyo, 24. Qing Liu, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 1526, USA, qingliu@pitt.edu Xiaodan Zhou, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 1526, USA, xiz78@pitt.edu