The Heat Equation and the Li-Yau Harnack Inequality

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1 The Heat Eqation and the Li-Ya Harnack Ineqality Blake Hartley VIGRE Research Paper Abstract In this paper, we develop the necessary mathematics for nderstanding the Li-Ya Harnack ineqality. We begin with a derivation of the heat eqation in one dimension. A basic overview of the geometry of Riemannian manifolds is presented, followed by the generalization of the heat eqation to Riemannian manifolds. A brief discssion of the strong maximm principle and the second fndamental form is presented, followed by a proof of the Li-Ya Harnack ineqalities. 1 Introdction The flow of heat is a process whose stdy has lead to a wide variety of developments in mathematics. Indeed, Forier s analysis of an eqation describing the flow of heat in a condcting rod has lead to the development Forier series. More recently, the stdy of the heat eqation has lead to the development of the Ricci Flow, a concept which was sed by to prove the Poincaré conjectre. One of the more sefl concepts that has developed along this line of research has been the Harnack ineqalities, of which the Li-Ya Harnack ineqality is of particlar interest. The Li-Ya Harnack ineqality (LYHI) is an ineqality which applies to any soltion of the heat eqation on a Riemannian manifold with nonnegative Ricci crvatre. It provides a lower bond on the maximm possible deviation of the logarithm of a soltion to the heat eqation from being a spersoltion to Laplace s eqation; namely, if is a soltion to the heat eqation, then ln n 2t, where n is the dimension of the manifold. This ineqality holds regardless of the crvatre of the manifold and the initial conditions of the heat eqation. From this ineqality it is straightforward to derive several other important ineqalities, inclding the differential Harnack ineqality. In this paper, I attempt to give a derivation of the Li-Ya Harnack Ineqality. The reader is presmed to have some familiarity with the concept of tensors, the covariant derivate, the Riemann tensor, and the Ricci crvatre. The LYHI on a manifold with bondary reqires a basic nderstanding of the second fndamental form, of which a brief introdction is given. The proof of the LYHI for closed manifolds is completed before it is extended to manifolds with bondary, so if the reader is only interested in closed manifolds, the introdction to the second fndamental form may be skipped. 2 Derivation of the Heat Eqation in n Dimensions We begin with a simple derivation of the heat eqation, giving s a very basic intition for the heat eqation. Consider a n dimensional cbe C with side lengths of h centered at x = (x 1,..., x n ), and assme that the cbe has a niform temperatre (x,t) at time t. Now, consider the 2n adjacent 1

2 cbes C i ± centered at x = (x 1,..., x i ±h,..., x n ), and assme the cbe C i ± has a niform temperatre of (x 1,..., x i ± h,..., x n, t) at time t. The rate of heat flow from cbe C i ± to cbe C is proportional to the temperatre gradient between the two cbes and the area of contact between the two cbes: dq C ± i C dt = A (x 1,..., x i ± h,..., x n, t) (x 1,..., x n, t) h h n 1, where Q C ± i C is the amont of heat that flows from C± i into C, the first term approximates the temperatre gradient, and h n 1 is the area of the (n-1) dimensional area of contact between any two adjacent cbes. The rate at which the heat flows into C from every cbe is proportional to the rate at which the temperatre changes times the volme of the cbe: t = 1 h n [ dqc + i C + dq ] C i C dt dt [ (x1,..., x i + h,..., x n ) 2(x 1,..., x n ) + (x 1,..., x i h,..., x n ) Letting h 0 and scaling the axes so that the constant of proportionality is 1, we find that: t = h 2 ]. 2 x i 2. (1) This is the eqation that represents heat flow in flat n dimensional flat eclidean space, and is the eqation that we will generalize to arbitrary Riemannian manifolds. 3 Geometry of Riemannian Manifolds Before we may generalize eqation (1) to manifolds, we review the necessary concepts of Riemannian geometry. Henceforth, we shall make se of the Einstein smmation convention wherein an index that appears twice in the mltiplication of matrices means that the index is smmed over. Explicitly, we have: S i 1...i...i q j 1...j r T k 1...k s l 1...i...l t = S i 1...i...i q j 1...j r T k 1...k s l 1...i...l t. Let M be an n dimensional Riemannian manifold. At each point of the manifold, there is a eclidean tangent space, allowing so to form at each point a set of basis vectors and basis co-vectors. We denote these vector fields as i = and dx i, respectively. We write v i for the components of x i vectors and v i for the components of co-vectors, where this is shorthand for: v i v i i = v i x i, v i v i dx i, 2

3 respectively. We define the rank of a tensor to be (N, M), where N and M are the nmber of raised and lowered indexes, respectively. We define the metric on M to be a positive-definite tensor g ij which allows s to define a notion of distance on the manifold. The metric carries with it all of the information abot the manifold that we are interested in. We define the inverse metric g ij by the eqation: δ i j = g ik g kj, where δ i j is the Kronecker delta, defined by: δ i j = { 1, if i = j 0, if i j The metric gives a notion of distance as follows: if ξ i (t), t [0, T ] is a parametrized, continosly differentiable path in M, then we define the length of ξ by: L(ξ) = T 0 g ij dξ i dt dξ j dt dt. The metric allows s to define a notion of an inner prodct between two tensors. vectors ξ = ξ i, η = η i, we set: ξ, η = g ij ξ i η j. For two This notion easily generalizes to arbitrary tensors with the same nmber of vector/co-vector indexes. If ξ = ξ i 1...i r j 1...j s, η = η i 1...i r j 1...j s, then: ξ, η = g i1 k 1 g irk r g j 1l1 g jsls ξ i 1...i r j 1...j s η k 1...k r l 1...l s. We may se the metric and its inverse to raise and lower the indexes of arbitrary tensors. Given an arbitrary Vector T = T i i or co-vector S = S i dx i, we have T i = g ij T j, S i = g ij S j, respectively. We define Levi-Cività connection on M to be a matrix Γ k ij by: at each point of M Γ k ij = 1 2 gkl ( i g jl + j g li l g ij ). We henceforth refer to this as the connection on M. Given an arbitrary vector field v = v k k, the connection allows s to define the covariant derivative of v: i v k i v k + Γ k ijv j. 3

4 For a scalar, on the other hand, the covariant derivative is simply the partial derivative: i = i. Finally, the covariant derivative obeys the prodct rle: if X and Y are arbitrary tensor fields, then: XY = ( X)Y + X( Y ). These rles allow for the straightforward generalization of the covariant derivative to arbitrary tensors. Indeed, the covariant derivative of a co-vector follows from the prodct rle applied to the contraction of a vector with a co-vector, forming a scalar, and higher rank tensors may be contracted with vectors and co-vectors to form scalars, at which point the prodct rle may again be applied. With this generalization, we find that the covariant derivative is a map with (N, M) tensors to (N, M + 1) tensors. We may define covariant derivatives in arbitrary directions as follows: if X = X i i, we define the covariant derivative in the X direction to be: X = X i i. While it is possible to define other connections on M, the Levi-Cività connection is niqely defined by two important properties: Γ k ij is torsion free: Γk ij = Γk ji. Γ k ij is compatible with the metric: kg ij = 0. Now, if X = X i i, Y = Y i i, and Z = Z i i are vector fields on M, then we may define the Riemann crvatre tensor by: Rm(X, Y )Z = X Y Z Y X Z [X,Y ] Z, where [X, Y ] is the Lie bracket of X and Y, and will often be zero. It is a straightforward, albeit tedios, matter to show that, nder the Levi-Civit connection, the Riemann crvatre tensor has the form: R k lij = iγ k lj lγ k ij + Γ k miγ m lj Γk mjγ m li. By lowering the pper index of the Riemann tensor, we obtain the commonly sed form of the Riemann tensor: R klij = g km R m lij. By contracting the first and third indexes of the Riemann tensor, we obtain the Ricci tensor, also known as the Ricci crvatre: R ij = R k ikj. 4

5 By contracting the Ricci tensor with the inverse metric on both indexes, we obtain the Ricci scalar, also known as the Ricci crvatre: The Second Fndamental Form R = g ij R ij. The proof of the LYHI on manifolds with bondary reqires a generalization of the concept of a convex srface. In order to state this definition, we mst first define the second fndamental form. Sppose we have an n dimensional manifold M, and the sbmanifold M with co-dimension 1 (i.e. M is (n-1) dimensional). We define a nit normal vector field ν as follows: ν(p) = 1 and ν(p) T p N {µ T p M µ, v = 0 for all v T p N}, where T p R is the tangent space of the manifold R at the point p. Now, ν can be either an otward pointing on inward pointing vector field; we choose to define it as a the otward pointing vector field. With an explicit ν in hand, we are now able to define the second fndamental form: II p (v, w) v ν, w. (2) The second fndamental form gives s the following definition of convexity: Definition 1. A manifold M is convex iff, for all p M and v T p M, II p (v, v) 0. 4 Generalization of the Heat Eqation to Riemannian Manifolds Before we may generalize the heat eqation to arbitrary Riemannian manifolds, we need to first write the heat eqation as a tensor eqation. This may be easily accomplished by noting that, in eclidean space with the standard strctre: g ij = g ij = δ i j, Γ k ij 0, and i i. Now, we rewrite the derivatives in the heat eqation as covariant derivatives: t = 2 t 2 = i i = g ij i j, t =, (3) where is the generalization of the Laplacian operator to crved manifolds. Both sides of this eqation are scalars, so that we have generalized the heat eqation to a tensor eqation. 5

6 This definition of the Laplacian reqires a minor clarification. When we take the covariant derivative of a scalar, a (0,0) tensor, we obtain a co-vector, a (0,1) tensor. Applying the covariant derivative again, we obtain a (0,2) tensor. In order to obtain the Laplacian, a scalar, we contract this tensor with both indexes of the inverse metric, giving s a (0,0) tensor (a scalar): = g ij ( i ( j )). (4) With the flly general eqation in or hands, it now seems natral for s to consider qestions of the existence and niqeness of soltions to this eqation. While it is tre that the existence of soltions to this eqation may be proven nder appropriate assmptions of the initial conditions and bondary conditions (i.g. Nemann bondary conditions and smooth initial conditions), the proof of this wold take s astray from or goal of proving the LYHI. The proof of niqeness also reqires appropriate initial conditions and bondary conditions assmptions, bt follows as a natral conseqence of the maximm principle, one of the necessary tools in or proof of the LYHI. We ths opt to prove niqeness as an application of the maximm principle before we move on to the proof of the LYHI. 5 The Strong Maximm Principle Let M be as above. Let U = M\ M, and let U T = U (0, T ]. We define the heat operator as: ( t ). The heat eqation may then be stated as: = ( t ) = 0 (5) Strong Maximm Principle. Assme C (U T ) and q > 0 in U T, U connected. Then: i) If is a sbsoltion of (4), i.e. 0 in U T, and if attains a non-negative absolte maximm over U T at an interior point (x 0, t 0 ) U T, then is constant on U t0. ii) If is a spersoltion of (4), i.e. 0 in U T, and if attains a non-positive absolte minimm over U T at an interior point (x 0, t 0 ) U T, then is constant on U t0. Proof. See [2] We also amend an important lemma, known as the Hopf bondary point lemma: Hopf Bondary Point Lemma. Sppose 0 in U T, and that is not constant. By the strong maximm principle, attains a maximm at a point p on the bondary. Then: (p) > 0, n Where n is an otwardly oriented normal coordinate. Proof. See [3]. Before moving on to the proof of the niqeness of soltions to the heat eqation, we first prove a sefl corollary: 6

7 Corollary 1. Sppose is sch that = 0 on U T, and if attains an absolte extremm over U T at an interior point (x 0, t 0 ). Then is constant on U t0. Proof. We begin by noting that is both a spersoltion and a sbsoltion of (4). Sppose attains an absolte maximm over U T at an interior point (x 0, t 0 ). We may assme that the maximm is non-negative, since ( + C) = 0 for C constant. Ths, we apply (i) from the strong maximm principle, and find that is constant. Similarly, sppose attains an absolte minimm over U T at an interior point (x 0, t 0 ). We may assme that the minimm is non-positive, since ( + C) = 0 for C constant. Ths, applying (ii), is constant on U t Uniqeness of Soltions to the Heat Eqation Theorem 1. Sppose solves the heat eqation ( = 0) with the following bondary conditions: 1. Nemann bondary conditions on M [0, T ]; v is thermally inslated from the otside world = 0; physically, this states that the manifold 2. Initial conditions on M [0]: (x, 0) 0 (x), with 0 (x) is smooth. is niqe. Proof. Sppose there exist two soltions, satisfying bondary conditions 1 and 2. Then w = satisfies the bondary condition w(x, 0) = 0 as well as the Nemann bondary condition. Now, sppose w attains an absolte extremm at an interior point (x 0, t 0 ) U T. Then w is constant by corollary 1. However, w(x, 0) = 0 w 0. Ths, w attains its maximm and minimm on the bondary. If this isn t the case, then w attains its maximm minimm on the bondary. By the Hopf bondary point lemma, (p) > 0, n which contradicts the Nemann bondary condition. Ths, =, proving the niqeness. 6 The Li-Ya Harnack Ineqality We begin by stating the LYHI: Li-Ya Harnack Ineqality. Let M be an n dimensional, compact, Riemannian manifold with non-negative Ricci crvatre and a convex bondary M if M. Let (x, t) satisfy the heat eqation = 0 with Nemann bondary conditions ( n = 0) on M (0, T ], and assme that (x, t) > 0 (x, t) M [0, T ]. Then: H = t n 0 (x, t) M (0, T ]. (6) 2t Before beginning the proof, let s see how eqation (6) is eqivalent to the ineqality presented in the introdction: 7

8 ( ) ( ( )) ln = g ij i ( j ln ) = g ij j i = g ij i j 1 + ( j ) i = i j gij 2 = t 2 2. Sbstitting this into eqation (6), we find that: H = ln + n 2t 0, ln n 2t. We may now begin the proof of the LYHI. We follow the proof given in the original paper by Li and Ya, [1]. We first prove a technical reslt which may seem arbitrary at this point, bt is pivotal to the proof of the LYHI. Lemma 1. (Li-Ya, Lemma 1.1 of [1]) Let f(x, t) be a smooth fnction on M [0, T ] sch that t f = f + f 2, and set F := t( f 2 t f) = t f. If F has nonnegative Ricci crvatre, then F satisfies the ineqality Proof. We begin with three straightforward calclations: ( t )F 2 F, f 1 t F + 2 nt F 2. (7) f 2 = g kl k l (g ij i f j f) = g ij g kl k ( l i f j f + i f l j f) = g ij g kl ( k l i f j f + l i f k j f + k i f l j f + i f k l j f) = 2 f, f f, 2 f = 2 f, f + 2 Hess(f) 2, (8) and f, f = g ij g kl k l i f j f = g ij g kl k i l f j f = g ij g kl ( i k l f j f + R p lki pf j f) = f, f + g ij g kl g pq R qlki p f j f = f, f + g ij g kl g pq R kilq p f j f = f, f + g ij g pq R iq p f j f = f, f + Ric( f, f), (9) ( f) 2 = g ij i j f 2 g ij 2 i j f 2 = n Hess(f) 2. (10) Where the ineqality is simply the Cachy-Schwartz ineqality, and g ij 2 = n is a property of the metric. We may now pt (8), (9), and (10) together to find that: 8

9 f 2 = 2 f, f + 2 Hess(f) 2 = 2 f, f + 2Ric( f, f) + 2 Hess(f) 2 2 f, f + 2 n ( f)2. (11) Finally, we may find or reslt by noting that: as was to be shown. ( t )F = t( f 2 ( t f)) t F (12) t(2 f, f + 2 n ( f)2 + t f) t F ( ) F = 2t, f + 2t ( ) F 2 ( F t t t n t t ) t F = 2 F, f 1 t F + 2 nt F 2, (13) We are now ready to proceed with the proof of the LYHI. Proof. Set f = ln(). We begin by noting that, from above: ln = t 2 = t ln() ln() 2 = f, (14) so that f satisfies the assmption of Lemma 1. We ths define F as in Lemma 1. Comparing ineqality (6) to eqation (14), we find that the LYHI is eqivalent to: or f n 2t, F n 2. (15) We may prove this by means of the reslt of Lemma 1, namely: ( t )F 2 F, f + 2 ( nt F F n ). (16) 2 If (15) were not tre, then there wold be a maximm point of F on M [0, T ] sch that F (x 0, t 0 ) > n/2. Since F (x, 0) = 0 by definition, t 0 > 0. We first consider the case in which M is a closed manifold. Since M lacks a bondary, x 0 is necessarily an interior point. Since (x 0, t 0 ) is a maximm for F, it is clear that: F (x 0, t 0 ) = 0, F (x 0, t 0 ) 0, t F (x 0, t 0 ) 0. Sbstitting this into (16), it we find that: 9

10 0 ( t )F ( F (x 0, t 0 ) F (x 0, t 0 ) n ) > 0, nt 0 2 Which is clearly a contradiction. Ths, the theorem is has been demonstrated for manifolds withot bondary. Now, sppose M.The above argment shows that (x 0, t 0 ) M (0, T ]. Now, by applying the Hopf bondary point lemma, we see that F n > 0, where n is the normally oriented, otward pointing coordinate. Now, let s set p a basis {x i }for the tangent space of M at (x 0, t 0 ) sch that the coordinate x n is normally oriented and otward pointing. Ths, in this coordinate system, we may express the Nemann bondary condition as n f = 0. Now, we have: F n (x 0, t 0 ) = t n ( f 2 t f) = 2t = 2t n i f i f t ( f) n i f i f = n f, f = 2t 0 II x0 ( f, f) < 0, since the bondary is convex and t 0 > 0. This contradicts the Hopf bondary point lemma, and proves or theorem. Having proven the LYHI, we conclde the paper with a demonstration that the ineqality is not strict; it may be satrated by some soltions to the heat eqation. Consider the following soltion to the heat eqation in n dimensional Eclidean space with the sal strctre: (x, t) = 1 t e x 2 4t, ln() = 1 2 Sbstitting this into the LYHI, we find that: ln(t) x 2 4t. so that 2 ln() x 2 i = 1 2t, ln() = 2 ln() x 2 i Which both satisfies and satrates the LYHI. = n 2t, Acknowledgments I am immensely thankfl for the help, spport, criticism, and gidance of Professor Artem Plemotov. I also received several sefl pointers from Professor WIlliam Dnbar of Bard College at Simon s Rock. Finally, I am deeply indebted to Doctor Reto Müller and his enlightening thesis. [4] 10

11 Bibliography [1] Chow, Bennett. The Ricci Flow: Techniqes and Applications. Providence, RI: American Mathematical Society, Print. [2] L.C. Evans, Partial Differential Eqations. Gradate Stdies in Mathematics 19, Amer. Math. Soc., Providence, RI, 1998 [3] P. Li and S.-T. Ya, On the Parabolic Kernel of the Schrödinger Operator. Acta Math. 156 (1986), [4] Mller, Reto. Differential Harnack Ineqalities and the Ricci Flow. Zrich, Switzerland: Eropean Mathematical Society, Print. 11