A Pointwise Inequality for Derivatives of Solutions of the Heat Equation in Bounded Domains
aa r X i v : . [ m a t h . A P ] F e b A POINTWISE INEQUALITY FOR DERIVATIVESOF SOLUTIONS OF THE HEAT EQUATIONIN BOUNDED DOMAINS
STEFAN STEINERBERGER
Abstract.
Let u ( t, x ) be a solution of the heat equation in R n . Then, each k − th derivative also solves the heat equation and satisfies a maximum prin-ciple, the largest k − th derivative of u ( t, x ) cannot be larger than the largest k − th derivative of u (0 , x ). We prove an analogous statement for the solution ofthe heat equation on bounded domains Ω ⊂ R n with Dirichlet boundary condi-tions. As an application, we give a new and fairly elementary proof of the sharpgrowth of the second derivatives of Laplacian eigenfunction − ∆ φ k = λ k φ k with Dirichlet conditions on smooth domains Ω ⊂ R n . Introduction and Results
In Euclidean space R n , it is always possible to exchange the order of partial deriva-tives: in particular, derivatives of solutions of the heat equation also satisfy theheat equation and therefore enjoy many nice properties, in particular a maximumprinciple. Let u ( t, x ) denote a solution of the heat equation in R n . Then it isexplicitly given by u ( t, x ) = 1(4 πt ) d/ Z R n exp (cid:18) − k x − y k t (cid:19) u (0 , y ) dy. At any given point x ∈ R and any unit vector ν ∈ S n − and any k ≥
1, we can(at least formally) differentiate under the integral sign to obtain ∂ k u∂ν k ( t, x ) = 1(4 πt ) d/ Z R n exp (cid:18) − k x − y k t (cid:19) ∂ k u∂ν k (0 , x ) dy. We were interested in whether there is an analogous result on bounded domains interms of the heat kernel p t ( · , · ) of the domain Ω. Theorem 1 (Main Result) . Let Ω ⊂ R n be a domain with C − boundary and let f ∈ C k (Ω) ∩ L ∞ (Ω) , let x ∈ Ω and ν ∈ S n − . Then, for all t > , all , the solution e t ∆ f of the heat equation with Dirichlet boundary conditions u ( t, x ) = g ( x ) has X = (cid:12)(cid:12)(cid:12)(cid:12) ∂ k ∂ν k e t ∆ f ( x ) − Z Ω p t ( x , y ) ∂ k ∂ν k f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) bounded by X ≤ (cid:18) − Z Ω p t ( x , y ) dy (cid:19) max ≤ s ≤ t (cid:13)(cid:13)(cid:13)(cid:13) ∂ k ∂ν k e s ∆ f (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( ∂ Ω) . Mathematics Subject Classification.
Key words and phrases.
Heat Equation, Laplacian eigenfunction, Hessian estimate.S.S. is supported by the NSF (DMS-1763179) and the Alfred P. Sloan Foundation. Note that if t ≪ d ( x , ∂ Ω) , then the integral term in our error bound is actu-ally quite small independently of what happens on the boundary. In particular,in free space, the heat kernel always has total integral 1 and we get X ≡
0. Theresult seems to have fairly natural extensions to the Neumann Laplacian, higherderivatives and even more general parabolic equations following essentially the sametype of argument. This type of argument might have an interesting analogue onmanifolds (both with and without boundary). In the case of manifolds withoutboundary, one would expect that the underlying curvature has an additional per-turbative effect on the particles (‘stochastic parallel transport’, see e.g. Bismut[4], Elworthy & Li [8] and Thalmaier & Wang [30]). We also refer to recent de-velopments on second-order Feynman Kac formulas (see Li [16] and Thompson [31]).
An Application.
Let Ω ⊂ R n be a bounded domain with smooth boundary. Weconsider Laplacian eigenfunctions, − ∆ φ k = λ k φ k , with Dirichlet conditions on theboundary ∂ Ω. If these eigenfunctions to be normalized in L , i.e. k φ k k L = 1, thenresults of Levitan in 1952 [15], Avakumovic in 1956 [2] and H¨ormander in 1968 [12]guarantee that k φ k k L ∞ (Ω) ≤ c Ω · λ n − k , where c Ω is a constant depending only on Ω. This estimate is sharp (for an exampleon a ball, see [10, § k∇ φ k k L ∞ (Ω) ≤ c Ω · λ n +14 k and has been studied by Xu [34, 35, 36] and, subsequently, by Arnaudon, Thalmaier& Wang [1], Cheng, Thalmaier & Thompson [6], Hu, Shi & Xu [13] and Shi & Xu[21]. On compact manifolds without boundary, there are results of Xu [33] forall derivatives and by Wang & Zhou [32] for linear combinations of eigenfunctions.Recently, Frank & Seiringer [9] showed that for compact Ω ⊂ R n with C k,δ − smoothboundary, there is an estimate (cid:13)(cid:13)(cid:13)(cid:13) ∂ k φ k ∂ν k (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (Ω) . Ω ,k λ k/ k φ k k L ∞ (Ω) As an application, we give an elementary proof of this inequality for k = 2. Theorem 2.
Let Ω ⊂ R n be a compact domain with smooth boundary. There existsa constant c Ω such that for all solutions of − ∆ φ k = λ k φ k that vanish on ∂ Ω k D φ k ( x ) k L ∞ (Ω) ≤ c Ω · λ / k · k∇ φ k k L ∞ (Ω) . More generally, Theorem 1 allows us to obtain similar estimates also for higherderivatives provided there is some control on the derivatives on the boundary.2.
Proof of Theorem 1
Proof.
We describe the proof for k = 2 in detail: the more general case k ≥ k − th differential quotient). We use a probabilistic argument. Forany x ∈ Ω, let ω x ( t ) denote a Brownian motion started in x after t units of time. This Brownian motion gets ‘stuck’ once it hits the boundary (this corresponds toDirichlet boundary conditions). This gives us a way of solving the heat equationvia e t ∆ f ( x ) = E ( f ( ω x ( t ))) , where the expectation ranges over all Brownian motions ω x started in x that run for t units of time. Our goal is to control the size of second derivatives of e t ∆ f ( x ) forpoints inside the domain. Let now x ∈ Ω be fixed and let ν ∈ S n − be some fixeddirection. Calculus tells us that the second derivative of e t ∆ f at x in direction ν is given by the limit of a differential quotient (cid:18) ∂ ∂ν e t ∆ f (cid:19) ( x ) = lim ε → e t ∆ f ( x + εν ) − e t ∆ f ( x ) + e t ∆ f ( x − εν ) ε . We will, for the remainder of the proof, control exactly this differential quotient bycarefully grouping the three expectations arising from e t ∆ f ( x + εν ) − e t ∆ f ( x ) + e t ∆ f ( x − εν ) = E ( f ( ω x + εν ( t ))) − · E ( f ( ω x ( t )))+ E ( f ( ω x − εν ( t ))) . These are three independent Brownian motions started in three different points.However, these three initial points are very close to one another (ultimately, ε → A denote all Brownian motion pathsstarted in 0 ∈ R n and running for t units of time. Then, for each y ∈ Ω, we canuse translation invariance of Brownian motion in R n to write the expectations asan expectation over the set A via e t ∆ f ( y ) = E ( f ( ω y ( t ))) = E a ∈ A (cid:0) f ( a ( t ) + y ) · { a ( s )+ y ∈ Ω for all ≤ s ≤ t } (cid:1) . This has the advantage of being able to take the expectation with respect to oneuniversal set A shared by all three Brownian motions. The size of the characteristicfunction has an analytic expression which is given by the heat kernel p t ( · , · ): P (cid:0) { a ( s )+ y ∈ Ω for all ≤ s ≤ t } (cid:1) = Z Ω p t ( y, z ) dz. There is an interesting subset of A depending on x , ν and εA ε = a ∈ A : a ( s ) + x ∈ Ω for all ≤ s ≤ ta ( s ) + x + εν ∈ Ω for all ≤ s ≤ ta ( s ) + x − εν ∈ Ω for all ≤ s ≤ t .A ε ⊂ A is the set of Brownian paths that remain in Ω for all time 0 ≤ s ≤ t independently in which of the three points x , x ± εν they are started in. For pathsin A ε , the differential quotient is easy to analyze: the path ends at a certain point a ( t ) and the relative position of the three Brownian particles has been preservedsince not a single one of them has hit the boundary. Recalling that for B ⊂ Ω P ( ω x ( t ) ∈ B ) = Z B p t ( x , y ) dy, we see that X = lim ε → ε E A ε [ f ( ω x + εv ( t )) − f ( ω x ( t )) + f ( ω x − εv ( t ))] can be written as X = lim ε → E a ∈ A ε (cid:18) ∂ f∂ν ( a ( t )) (cid:19) = Z Ω p t ( x , z ) ∂ f∂ν ( z ) dz. This serves as a probabilistic proof that in R n , solving the heat kernel and differ-entiation commute since in that case A ε = A because there is no boundary. Theremainder of the argument will be concerned with A \ A ε . The cases that are in A \ A ε can be written as a disjoint union A \ A ε = [ ≤ s ≤ t A ,s ∪ A ,s ∪ A ,s , where A i,s is the event where the i − th of the three particles (having enumeratedthem in an arbitrary fashion) has hit the boundary at time s and is the first ofthe three particles to do so (in case two particles hit the boundary simultaneously,we may put this event into either of the two sets; if all three hit simultaneously, itcan be put into any of the three sets). We have a very good understanding of thesize of A \ A ε since lim ε → P ( A \ A ε ) = 1 − Z Ω p t ( x , y ) dy. It remains to understand the expected value of the differential quotient conditionalon the path being in A i,s for all 1 ≤ i ≤ ≤ s ≤ t . As it turns out, thesecases can all be analyzed in the same fashion. We illustrate the argument using theexample shown in Fig. 1. In that example, the middle particle has impacted theboundary at time s . That middle Brownian motion is stuck and E (cid:0) f ( ω x ( t )) (cid:12)(cid:12) ω x ( s ) = a ( s ) + x (cid:1) = 0 .a ( s ) + x ΩΩ c ∂ Ω a ( s ) + x + ενa ( s ) + yx − εν Figure 1.
A path in A ,s ⊂ A \ A ε : the middle point hits the boundary.It remains to understand the expected value of f ( ω x ( t )) subject to knowing thatat time s the particle is in the position a ( s ) + x + εν or a ( s ) + x − εν . At thispoint we use Markovianity: the Brownian motion does not remember its past andbehaves as if it were freshly started in that point. In particular, this shows that E (cid:0) f ( ω x ( t )) (cid:12)(cid:12) ω x ( s ) = a ( s ) + x + εν (cid:1) = E (cid:0) f ( ω a ( s )+ x + εν ( t − s )) (cid:1) . However, this is merely the formula for the solution of the heat equation and E (cid:0) f ( ω a ( s )+ x + εν ( t − s )) (cid:1) = e ( t − s )∆ f ( a ( s ) + x + εν ) . Likewise, we have that E (cid:0) f ( ω x ( t )) (cid:12)(cid:12) ω x ( s ) = a ( s ) + x − εν (cid:1) = e ( t − s )∆ f ( a ( s ) + x − εν ) . Finally, we argue that even the middle point (the one that already impacted on theboundary) can be written the same way since E (cid:0) f ( ω x ( t )) (cid:12)(cid:12) ω x ( s ) = a ( s ) + x (cid:1) = 0 = e ( t − s )∆ f ( a ( s ) + x ) . However, these three identities tell us a nice story: it tells us that evaluating the‘probabilistic’ second differential quotient amounts to, in this special case, merelyto evaluating the second differential quotient of e ( t − s )∆ f at the point a ( s ) + x in direction ν. This leads to the following conclusion: for any smooth, compact domain, the heatequation started with u (0 , x ) = f ( x ) inside Ω and constant boundary conditions u ( t, x ) = g ( x ) for x ∈ ∂ Ω has a solution u ( t, x ) = Z Ω p t ( x, y ) f ( y ) dy + Z ∂ Ω q t ( x, y ) g ( y ) dy, where q t ( x, y ) converges to the harmonic measure as t → ∞ (since Ω is smooth,regularity of the boundary does not play a role). From this we can deduce that (cid:18) ∂ e t ∆ f∂ν (cid:19) ( x ) = Z Ω p t ( x, y ) ∂ f∂ν ( y ) dy + Z t Z ∂ Ω ∂q s ∂s ( x, y ) ∂ e ( t − s )∆ f∂ν ( y ) dyds. The second integral can be easily bounded from above by (cid:12)(cid:12)(cid:12)(cid:12)Z t Z ∂ Ω ∂q s ∂s ( x, y ) ∂ e ( t − s )∆ f∂ν ( y ) dyds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z t Z ∂ Ω ∂q s ∂s ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ e ( t − s )∆ f∂ν ( y ) (cid:12)(cid:12)(cid:12)(cid:12) dyds = Z ∂ Ω q t ( x, y ) max ≤ s ≤ t (cid:12)(cid:12)(cid:12)(cid:12) ∂ e ( t − s )∆ f∂ν ( y ) (cid:12)(cid:12)(cid:12)(cid:12) dy. In particular, recalling that u ( t, x ) ≡ u (0 , x ) ≡ g ( x ) ≡
1, we have that Z Ω p t ( x, y ) dy + Z ∂ Ω q t ( x, y ) dy = 1 . This shows that the integral I = Z ∂ Ω q t ( x, y ) max ≤ s ≤ t (cid:12)(cid:12)(cid:12)(cid:12) ∂ e ( t − s )∆ f∂ν ( y ) (cid:12)(cid:12)(cid:12)(cid:12) dy can be bounded by I ≤ (cid:18) − Z Ω p t ( x , y ) dy (cid:19) max ≤ s ≤ t max y ∈ ∂ Ω (cid:12)(cid:12)(cid:12)(cid:12) ∂ e ( t − s )∆ f∂ν (cid:12)(cid:12)(cid:12)(cid:12) , where ν = ± ν always points inside the domain. This is the desired statement. (cid:3) We note that the last few steps are certainly a bit wasteful and one could obtainmore precise estimates if one were to assume additional knowledge about p t ( · , · ), q t ( · , · ) or the harmonic measure ω x . Proof of Theorem 2
The proof decouples into the following steps.(1) First, we show that second derivatives on the boundary are controlled andat most of size . ∂ Ω k∇ φ k k L ∞ . The implicit constant will only depend onthe mean curvature of the boundary ∂ Ω.(2) We then use this in combination with Theorem 1. We use time scale t = ελ − k and will show that the entire argument can be carried out with asufficiently small ε > ∂ Ω and the dimension n . If the second derivatives assume their maximumvalue in a point x , then there exists a set A in a √ t − neighborhood of x where the second derivatives are large. A itself is large in the sense of Z A p t ( x , y ) dy ≥ − ε. (3) This implies that x is not too close to the boundary: d ( x , ∂ Ω) ≥ √ t .(4) Finally, we show that if we consider a ball of radius √ t around x (and, bythe previous step, this ball is fully contained in Ω), then there exists a linesegment such that second derivatives are large on most of its length.(5) The Fundamental Theorem of Calculus then shows that the derivativeshave to grow very quickly along the line segment and this will lead to acontradiction once the second derivatives are too large: this will show that k∇ φ k k L ∞ has to be large which then contradicts known bounds.3.1. Eigenfunctions on the boundary.
We start the argument by noting thateigenfunctions − ∆ φ k = λ k φ k in smooth domains with vanishing Dirichlet bound-ary conditions on ∂ Ω cannot have a particularly large second derivative on theboundary. We will prove this by expressing the Laplacian in local coordinates atthe boundary: more precisely, let ∂ Ω be a hypersurface in R n and let ν be a unitnormal vector to ∂ Ω. Using ∆ ∂ Ω to denote the Laplacian in the induced metric on ∂ Ω and ∆ to denote the classical Laplacian in R n , we have the identity∆ u = ∆ ∂ Ω u + ( n − H ∂u∂n + ∂ u∂n , where H is the mean curvature of the boundary ∂ Ω in that point. This identity isderived, for example, in the book by Sperb [29, Eq. 4.68]. x ν ∂ Ω Figure 2.
A point on the boundary and a normal direction.This equation is particularly useful in our case because φ k vanishes identically onthe boundary and thus ∆ ∂ Ω φ k = 0. The identity is also frequently used on levelsets of u (since then the Laplacian ∆ ∂ Ω u vanishes for the same reason), see for example Kawohl & Horak [14]. We refer to Reilly [20] for a friendly introductionto this identity in low dimensions. Since φ k vanishes on the boundary, we have ∂ φ k ∂ν + ( n − H ∂φ k ∂ν = 0 . However, the mean curvature H is bounded depending only on ∂ Ω because thedomain is smooth. Therefore, using the gradient estimate of Hu, Shi & Xu [13] oncompact Riemannian manifolds with boundary, we get (cid:12)(cid:12)(cid:12)(cid:12) ∂ φ k ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ≤ n k H k L ∞ ( ∂ Ω) (cid:12)(cid:12)(cid:12)(cid:12) ∂φ k ∂ν (cid:12)(cid:12)(cid:12)(cid:12) . Ω k∇ φ k k L ∞ . Ω λ / k φ k k L ∞ . λ n +14 k . This shows that second derivatives in the normal direction cannot be much largerthan first derivatives, since they are generated as a combination of first derivativesand the curvature. As for second derivatives in directions orthogonal to the normaldirection, we note that due to the smoothness of Ω, points at distance ε are ∼ ε away from the boundary, where the implicit constant depends on the curvatureof ∂ Ω. This allows us to bound the second differential quotient by ≤ c Ω k∇ f k L ∞ , where c Ω depends only on the local curvature of ∂ Ω.3.2.
Applying Theorem 1.
Let us fix x ∈ Ω and ν ∈ S n − so that the secondderivative in x in direction ν is among the largest that can occur, i.e. ∂ φ k ∂ν ( x ) = max x ∈ Ω ,ν ∈ S n − (cid:12)(cid:12)(cid:12)(cid:12) ∂ φ k ∂ν ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = c λ n +34 k . We assume that this largest second directional derivative is positive (for ease ofexposition): if it is negative, we consider without loss of generality − φ k instead. Ourgoal is to now deduce a contradiction once c is sufficiently large. Let ε be a smallparameter (our goal will be to show that there is a sufficiently small but positiveparameter ε > ε >
0: we use Theorem 2 for time t = ελ − k incombination with e t ∆ φ k ( x ) = e − λ k t φ k ( x )to obtain e − ε ∂ ∂ν φ k ( x ) ≤ Z Ω p t ( x , y ) ∂ ∂ν φ k ( y ) dy + c λ n +14 k , where the existence of an absolute constant c depending only on Ω follows from § ν are‘large’ (positive and at least half the value of the maximum) A = (cid:26) x ∈ Ω : ∂ φ k ∂ν ( x ) ≥ ∂ φ k ∂ν ( x ) (cid:27) . This allows us to bound the inequality above by replacing the partial derivatives in y by the maximum partial derivative e − ε ∂ ∂ν φ k ( x ) ≤ Z Ω p t ( x , y ) ∂ ∂ν φ k ( y ) dy + c λ n +14 k ≤ Z Ω \ A p t ( x , y ) (cid:18) ∂ ∂ν φ k ( x ) (cid:19) dy + Z A p t ( x , y ) (cid:18) ∂ ∂ν φ k ( x ) (cid:19) dy + c λ n +14 k . Dividing by the largest derivative, we get1 − ε ≤ e − ε ≤ Z Ω \ A p t ( x , y ) dy + Z A p t ( x , y ) dy + c c √ λ k . Recall that Z Ω \ A p t ( x , y ) dy + Z A p t ( x , y ) dy = Z Ω p t ( x, y ) dy ≤ . This shows that once λ k is sufficiently large, say, so large that c c √ λ k ≤ ε we have 1 − ε ≤ Z Ω \ A p t ( x , y ) dy + Z A p t ( x , y ) dy. However, if 1 − ε ≤ a/ b and a + b ≤
1, then a ≤ ε and thus Z A p t ( x , y ) dy ≥ − ε. This means that if we start a Brownian motion in x and let it run for t = ελ − k units of time, then the likelihood of never hitting the boundary and ending up inthe set A is actually quite large. We note that this argument can be used for each ε > ε in the manner outlined above).3.3. x is far away from the boundary. This section uses the inequality Z A p t ( x , y ) dy ≥ − ε to prove that x is at least ≥ c · √ t away from the boundary. Indeed, we obtain aslightly stronger result and show that we could choose c = 1 for ε sufficiently small(depending only on Ω). Since the boundary is smooth and compact, there existsan length scale δ such that at the scale of δ (or below) the boundary behavesroughly like a hyperplane. Suppose d ( x , ∂ Ω) ≤ . √ t ≪ δ . Then the geometrylooks a bit as in Fig. 3. x √ t∂ Ω d ( x , ∂ Ω) Figure 3.
A maximal point being closer than √ t to the boundary. Since ∂ Ω is smooth and we operate at small length scales (relative to ∂ Ω), we knowthat d ( x , ∂ Ω) ≤ . √ t implies that B √ t ( x ) intersects the boundary ∂ Ω in alarge segment. This also shows that a typical Brownian motion run for t units oftime will hit the boundary with nontrivial probability bounded away from 0 whichcontradicts our lower bound on the survival probability of Brownian motion up totime t − ε ≤ Z A p t ( x , y ) dy ≤ Z Ω p t ( x , y ) dy. It remains to make this more quantitative. There is a simple way of boundingthe survival probability of a Brownian motion as follows: let us assume that x is in the origin, the boundary ∂ Ω is oriented so that it is given by the hyper-plane { x ∈ R n : x = d } at distance d from the origin. The boundary ∂ Ω is not ahyperplane but for sufficiently small length scales behaves effectively as one, thecurvature acts as a lower order term (which we will account for in the subsequentargument). The main question is then whether, within t units of time, the firstcomponent of the Brownian motion is ever larger than d . The first component ofan n − dimensional Brownian motion is a one-dimensional Brownian motion and wecan apply the reflection principle to conclude P (cid:18) max ≤ s ≤ t B ( s ) ≥ d (cid:19) = P ( | B ( t ) | ≥ d ) . This refines our estimate to1 − ε ≤ Z A p t ( x , y ) dy ≤ Z Ω p t ( x , y ) dy ≤ − P [ | B ( t ) | ≥ · d ( x , ∂ Ω)] , where the factor of 2 compensates for the higher order term coming from the cur-vature of ∂ Ω (and thus valid for t sufficiently small depending only on ∂ Ω). Thus P [ | B ( t ) | ≥ · d ( x , ∂ Ω)] ≤ ε.B ( t ) is distributed like the Gaussian N (0 , t ) and thus, by rescaling, P [ | B ( t ) | ≥ · d ( x , ∂ Ω)] = P (cid:20) | B (1) | ≥ √ t · d ( x , ∂ Ω) (cid:21) ≤ ε. Knowing that this quantity is less than 4 ε will lead to a lower bound on d ( x , ∂ Ω).The standard tail bound, valid for all z > P ( B (0 , ≥ z ) ≥ √ π (cid:18) z − z (cid:19) e − z / implies, for z ≥
2, 2 ε ≥ P ( B (0 , ≥ z ) ≥ πz e − z / . We want to argue that this forces z ≥ z = r log 1 ε because plugging in z leads to12 πz e − z / = 12 π q log ε e − z / = 12 π √ ε q log ε which is larger than 2 ε for all 0 ≤ ε ≤ . ε sufficiently small (whichhere is an absolute constant), we have the desired inequality. This forces z ≥ r log 1 ε and thus d ( x , ∂ Ω) ≥ √ t · r log 1 ε . Recalling that t = ελ − k , we have d ( x , ∂ Ω) ≥ · √ ε · r log 1 ε · λ − / k . For ε ≤ e − , it is at least ε / λ − k = √ t away from the boundary. We willnot, strictly speaking, need this and could absorb any constant that arises here inthe final step. Similar arguments have already been used in other settings in theliterature, often to prove bounds on the location of the maximum of the solution[7, 17, 19] and also in the context of Hermite-Hadamard inequalities [18].3.4. Parts of A are close to x . The next step is to argue that Z A p t ( x , y ) dy ≥ − ε implies that A has a large intersection with a ball B √ t ( x ). We know from the pre-vious section that for ε sufficiently small the entire ball is contained in Ω. Domainmonotonicity of the heat kernel allows to compare the heat kernel p t ( x , · ) to theheat kernel in R n which is strictly larger. This implies1 − ε ≤ Z A p t ( x , y ) dy ≤ Z A πt ) n/ exp (cid:18) − k x − y k t (cid:19) dy ≤ . Hence, since the Euclidean heat kernel has total integral 1, Z A c πt ) n/ exp (cid:18) − k x − y k t (cid:19) dy ≤ ε. Let us now consider B √ t ( x ) ∩ A c . We clearly have Z B √ t ( x ) ∩ A c πt ) n/ exp (cid:18) − k x − y k t (cid:19) ≥ πt ) n/ e − / · | B √ t ( x ) ∩ A c | . From this we deduce | B √ t ( x ) ∩ A c | ≤ e / ε · (4 πt ) n/ ≤ ε (4 πt ) n/ . We recall that t = ελ − / k and that B √ t ( x ) = ω n t n/ , where ω n is the volume of the unit ball in n dimensions. Therefore | B √ t ( x ) ∩ A | = | B √ t ( x ) | − | B √ t ( x ) ∩ A c |≥ (cid:18) − · (4 π ) n/ ω n ε (cid:19) | B √ t ( x ) | . Hence there exists a ε sufficiently small (depending only on the dimension) suchthat 99% of the volume of B √ t ( x ) is contained in A . Piercing Rays.
We can now conclude the argument as follows (see Fig. 4).The previous section implies that | B √ t ( x ) ∩ A | ≥ (cid:18) − · (4 π ) n/ ω n ε (cid:19) | B √ t ( x ) | .x ν Figure 4.
A ball with a small percentage of its mass, the set A c ,removed (in practice A c may look a lot more complicated). Thereexists a long line that intersects the ball and A along a large set.Thus, for ε sufficiently small, we have that 99% of the volume of the ball B √ t ( x ) isactually contained in A . We now use the pigeonhole principle (or Fubini’s Theorem)to argue that there exists a line in R n that points in direction ν ℓ = (cid:8) z + t · ν : z ∈ B √ t ( x ) ∧ t ∈ R (cid:9) such that the line intersects the ball B √ t ( x ) along a long segment | ℓ ∩ B √ t ( x ) | ≥ c √ t, where c > | ℓ ∩ B √ t ( x ) ∩ A || ℓ ∩ B √ t ( x ) | ≥ .ν Figure 5.
Cutting off the ‘edges’. This can be achieved as follows: we choose a small c and remove all those partsof a sphere that have intersection less than c √ t in direction ν (see Fig. 5). Bymaking c sufficiently small (depending only on the dimension), we can ensure thatno more than 1% of the total volume of the sphere has been lost. We know thatat least 99% of the measure of the ball is in A : in the worst case, the ‘edges’ thathave been removed are in A which ensures that at least a 0 . / . ≥ .
98 portionof the remaining mass is in A . If there were no line going through the remainderof the ball for which | ℓ ∩ B √ t ( x ) ∩ A || ℓ ∩ B √ t ( x ) | ≥ , then we can integrate over all the fibers and conclude that at most 97% of theremaining mass is contained in A which contradicts our inequality.3.6. Fundamental Theorem of Calculus.
We conclude with an application ofthe Fundamental Theorem. Let ℓ be the line constructed in the preceding sectionand let ℓ = ℓ ∩ B √ t ( x ) denote the line segment that lies fully in the ball B √ t ( x ).By construction ℓ has length at least | ℓ | ≥ c √ t . Denoting the beginning and theendpoint of ℓ by a, b , respectively, we have ∂φ k ∂ν ( b ) − ∂φ k ∂ν ( a ) = Z ℓ ∂ φ k ∂ν ( x ) dx. However, 97% of the line segment is actually in A and, recalling the definition of A = (cid:26) x ∈ Ω : ∂ ∂ν φ k ( y ) ≥ ∂ φ k ∂ν ( x ) (cid:27) , we have Z ℓ ∂ φ k ∂ν ( x ) dx ≥ . · | ℓ | ∂ φ k ∂ν ( x ) + Z ℓ ∩ A c ∂ φ k ∂ν ( x ) dx. However, we can bound the remaining integral by the supremum norm, meaning ∂ φ k ∂ν ( x ) ≥ − ∂ φ k ∂ν ( x )and get that Z ℓ ∩ A c ∂ φ k ∂ν ( x ) dx ≥ − . · | ℓ | ∂ φ k ∂ν ( x ) . Therefore, recalling that | ℓ | ≥ c √ t , ∂φ k ∂ν ( b ) − ∂φ k ∂ν ( a ) ≥ . · | ℓ | ∂ φ k ∂ν ( x ) ≥ . · c · √ ε · λ − / k ∂ φ k ∂ν ( x ) . We combine this with the upper bound ∂φ k ∂ν ( b ) − ∂φ k ∂ν ( a ) ≤ · k∇ φ k k L ∞ ≤ c · λ n +14 k and, since ε is an absolute constant, this is the desired result. Concluding Remarks.
It seems that certain variations of this argument arefeasible and some may prove to be useful in other settings. We quickly highlightone such variation. In the first half of the proof, we establish that if there is a largesecond derivative, then there are large second derivatives in a large subset of the ∼ √ t neighborhood around the point. If the Hessian is very large in a point, thenit is also large in a large set in the neighborhood. Bochner’s formula states that12 ∆ |∇ u | = h∇ ∆ u, ∇ u i + k D u k . In the case of a Laplacian eigenfunction, the identity simplifies to12 ∆ |∇ φ k | = − λ k k∇ φ k k + k D φ k k . If the maximum size of the Hessian exceeds √ λ k k ∆ φ k k L ∞ by a large constant in apoint x ∈ Ω, then ∆ |∇ φ k | ∼ k D φ k k ∼ λ n +32 k ( ⋄ )This, however, shows that |∇ φ k | is undergoing growth. We remark that if anarbitrary f : R n → R is nonnegative f ≥ f ≥ C in B R (0) , then k f k L ∞ ( B R (0)) & n C · R . This can be seen by comparing f with the function gg ( x ) = C k x k n . We have ∆( f − g ) ≥ C/ f − g assumes its maximum on the boundary.We also know that f (0) − g (0) = 0 and thus the maximum is nonnegative andthus at least of size ∼ CR . Applying this to our inequality ( ⋄ ) in a B √ ελ − / k ( x )neighborhood, we obtain k∇ φ k k L ∞ (cid:18) B √ ελ − / k ( x ) (cid:19) & ε,n λ n +32 k λ − k = λ n +12 k . For sufficiently large C , this then contradicts the gradient estimate. A difficultywith this argument is that ( ⋄ ) is not a priori satisfied in the entire ball but merelyin a very large subset: however, in the complement of that set, we always have12 ∆ |∇ φ k | = − λ k k∇ φ k k + k D φ k k ≥ − λ k k∇ φ k k which cannot be arbitrarily negative and is a constant factor smaller than theHessian. One would expect that a more refined maximum principle could then beapplied in that case. This type of argument may be simpler to apply on a manifoldsince it does not rely on arguments along sub-manifolds. References [1] M. Arnaudon, A. Thalmaier, F.-Y. Wang, Gradient Estimates on Dirichlet and NeumannEigenfunctions, International Mathematics Research Notices 20 (2020), p. 7279–7305.[2] G. Avakumovic, ¨Uber die Eigenfunktionen auf geschlossenen Riemannschen Mannig-faltigkeiten, Math. Z. 65 (1956), p. 327–344.[3] P. Berard. On the wave equation on a compact Riemannian manifold without conjugatepoints. Math. Z., 155 (1977): p. 249–276.[4] J. M. Bismut, Large Deviations and the Malliavin Calculus, Progr. Math., Vol. 45, BirkhauserBoston, Cambridge, MA, 1984 [5] M. Blair and C. Sogge, Logarithmic improvements in L p bounds for eigenfunctions at thecritical exponent in the presence of nonpositive curvature Inventiones mathematicae 217(2019), p. 703–748.[6] L.-J. Cheng, A. Thalmaier and J. Thompson, Uniform gradient estimates on manifolds witha boundary and applications Analysis and Mathematical Physics 8 (2018), p. 571–588.[7] B. Georgiev and M. Mukherjee, Nodal geometry, heat diffusion and Brownian motion Analysis& PDE 11 (2017), p. 133–148.[8] K.D. Elworthy and X.-M. Li, Formulae for the derivatives of heat semigroups, J. Funct. Anal.125 (1994) p. 252–286.[9] R. Frank and R. Seiringer, Quantum Corrections to the Pekar Asymptotics of a StronglyCoupled Polaron, Comm. Pure Appl. Math, to appear[10] D. Grieser, Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary.Commun. Partial Differential Equ. 27 (2002), p. 1283–1299.[11] A. Hassell and M. Tacy, Improvement of eigenfunction estimates on manifolds of nonpositivecurvature. Forum Mathematicum 27 (2015), pp. 1435–1451.[12] L. H¨ormander, The spectral function of an elliptic operator. Acta Math. 88 (1968), p. 341–370.[13] J. Hu, Y. Shi and B. Xu, The gradient estimate of a Neumann eigenfunction on a compactmanifold with boundary Chinese Annals of Mathematics B 36 (2015), p. 991–1000[14] B. Kawohl and J. Horak, On the Geometry of the p − Laplacian operator, Discrete and Con-tinuous Dynamical Systems 10 (2017), p. 799–813.[15] B. Levitan, On the asymptotic behavior of the spectral function of a self-adjoint differentialequation of second order. Isv. Akad. Nauk SSSR Ser. Mat. 16 (1952), p. 325–352.[16] X.-M. Li, Doubly Damped Stochastic Parallel Translations and Hessian Formulas, in: Sto-chastic Partial Differential Equations and Related Fields (SPDERF 2016), Springer Proceed-ings in Mathematics & Statistics 229, Springer[17] J. Lierl and S. Steinerberger, A Local Faber-Krahn inequality and Applications toSchrodinger’s Equation, Comm. in PDE 43 (2018), p. 66–81.[18] J. Lu and S. Steinerberger, A dimension-free Hermite-Hadamard inequality via gradient es-timates for the torsion function, Proc. Amer. Math. Soc. 148 (2020), p. 673–679.[19] M. Rachh and S. Steinerberger, On the location of maxima of solutions of Schroedinger’sequation, Comm. Pure. Appl. Math. 71 (2018), p. 1109–1122[20] R. Reilly, Mean Curvature, The Laplacian, and Soap Bubbles, The American MathematicalMonthly 89 (1982), p. 180–198[21] Y. Shi and B. Xu, Gradient estimate of an eigenfunction on a compact Riemannian manifoldwithout boundary, Annals of Global Analysis and Geometry 38 (2010), p. 21–26.[22] H. Smith, Sharp L → L q bounds on the spectral projectors for low regularity metrics, Math.Res. Lett., 13 (2006), p. 967–974[23] H. Smith and C. Sogge, On the Lp norm of spectral clusters for compact manifolds withboundary. Acta Mathematica 198 (2007), p. 107–153.[24] C. Sogge, Concerning the Lp norm of spectral clusters for second-order elliptic operators oncompact manifolds. J. Funct. Anal. 77 (1988), p. 123–138[25] C. Sogge, Eigenfunction and Bochner-Riesz estimates on manifolds with boundary. Math.Res. Lett. 9 (2002), p. 205–216.[26] C. Sogge, Riemannian manifolds with maximal eigenfunction growth. Duke Math. J. 114(2002), p. 387–437.[27] C. Sogge, J. Toth, and S. Zelditch. About the blowup of quasimodes on Riemannian manifolds.J. Geom. Anal., 21 (2011): p. 150–173[28] C. Sogge and S. Zelditch, Riemannian manifolds with maximal eigenfunction growth. DukeMath. J. 114 (2002), p. 387–437[29] R. Sperb, Maximum principles and their applications, Mathematics in Science and Engineer-ing, vol. 157, Academic Press, New York, 1981.[30] A. Thalmaier, F.-Y. Wang, Gradient estimates for harmonic functions on regular domains inRiemannian manifolds, J. Funct. Anal. 155 (1998), p. 109–124.[31] J. Thompson, Derivatives of Feynman–Kac Semigroups, Journal of Theoretical Probability32 (2019), p. 950–973.[32] J. Wang and L. Zhou, Gradient Estimate for Eigenforms of Hodge Laplacian, Math. Res.Lett. 19 (2012), p. 575–588. [33] B. Xu, Derivatives of the spectral function and Sobolev norms of eigenfunctions on a closedRiemannian manifold. Ann. Glob. Anal. Geom. 26 (2004), p. 231–252.[34] X. Xu, Eigenfunction estimates on compact manifolds with boundary and H¨ormander multi-plier theorem. Ph. D. Thesis, Johns Hopkins University (2004)[35] X. Xu, New proof of the H¨ormander multiplier theorem on compact manifolds without bound-ary. Proc. Am. Math. Soc. 135 (2007), p. 1585–1595.[36] X. Xu, Gradient estimates for eigenfunctions of compact manifolds with boundary and theH¨ormander multiplier theorem. Forum Math. 21 (2009), 455–476. Department of Mathematics, University of Washington, Seattle
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