A Local Faber-Krahn inequality and Applications to Schrödinger's Equation
aa r X i v : . [ m a t h . A P ] N ov A LOCAL FABER-KRAHN INEQUALITY AND APPLICATIONSTO SCHR ¨ODINGER EQUATIONS
JANNA LIERL AND STEFAN STEINERBERGER
Abstract.
We prove a local Faber-Krahn inequality for solutions u to the Dirichletproblem for ∆ + V on an arbitrary domain Ω in R n . Suppose a solution u assumes aglobal maximum at some point x ∈ Ω and u ( x ) >
0. Let T ( x ) be the smallest timeat which a Brownian motion, started at x , has exited the domain Ω with probability ≥ /
2. For nice (e.g., convex) domains, T ( x ) ≍ d ( x , ∂ Ω) but we make no assumptionon the geometry of the domain. Our main result is that there exists a ball B of radius ≍ T ( x ) / such that k V k L n , (Ω ∩ B ) ≥ c n > , provided that n ≥
3. In the case n = 2, the above estimate fails and we obtain asubstitute result. The Laplacian may be replaced by a uniformly elliptic operator indivergence form. This result both unifies and strenghtens a series of earlier results. Introduction and Main Result
Faber-Krahn inequalities.
One of the earliest problems in spectral geometry is therelation between the lowest Dirichlet eigenvalue of the Laplacian and the size of the domain.More precisely, assume that a domain Ω ⊂ R n is given and that we have a nontrivial solution u of the Dirichlet problem ∆ u + λu = 0 in Ω ,u | ∂ Ω = 0 . By a solution we always mean a solution u ∈ W , (Ω) in the distributional sense. A solutionis trivial if u = 0 a.e., and nontrivial otherwise. The smallest λ for which a nontrivial solution u exists, sometimes denoted λ (Ω), can be interpreted as the base frequency of a vibratingmembrane in the shape of Ω. The Faber-Krahn inequality states that λ (Ω) ≥ πj n/ − , Γ( n/ /n | Ω | − /n , with equality if and only if Ω is a ball, where Γ denotes the Gamma function and j n/ − , denotes the smallest positive zero of a Bessel function. The Faber-Krahn inequality isa fundamental theorem in spectral geometry and one of the first results that relates thegeometry of a domain Ω to the solvability of a differential equation.More generally, a similar inequality holds for the Dirichlet problem with potential V ,∆ u + V u = 0 in Ω ,u | ∂ Ω = 0 . (1) Mathematics Subject Classification.
Key words and phrases.
Faber-Krahn inequality, Schr¨odinger operator, Feynman-Kac formula. De Carli & Hudson [8] proved that a the existence of a nontrivial solution u ∈ C (Ω) of (1)implies k V k L ∞ (Ω) ≥ πj n/ − , Γ( n/ /n | Ω | − /n , and equality is attained if Ω a metric ball and V ≡ λ (Ω) is constant. A slight refinementis given by a not very well known result of Barta [3] which implies that k V k L ∞ (Ω) ≥ λ (Ω) . This result was later put in a more general context by De Carli, Edward, Hudson & Leckband[9]. A sample result (see [9, Theorem 1.2]) is the following: if n ≥ r > n/
2, then theexistence of a nontrivial solution to (1) implies that there is a constant c n ∈ (0 , ∞ ) suchthat | Ω | n − r k V + k L r (Ω) ≥ c n > . The paper [9] also discusses the endpoint case r = n/ n = 2; this is related to the Sobolev embedding failing for n = 2. At the coreof the argument is an elegant combination of the H¨older inequality, the Sobolev inequality,Green’s identity, equation (1) and H¨older’s inequality again and bears repeating [9, (1.6)].We denote the H¨older conjugate of r > n/ q < n/ ( n − k u k L q (Ω) ≤ c n | Ω | n − r Z Ω |∇ u | dx = − c n | Ω | n − r Z Ω u ∆ u dx = c n | Ω | n − r Z Ω u V dx ≤ c n | Ω | n − r Z Ω u V + dx ≤ c n k u k L q (Ω) | Ω | n − r k V + k L r (Ω) , and from this the assertion follows after cancellation (a similar argument was already usedin [7]). These type of inequalities fit naturally into a larger family of results that relate prop-erties of an elliptic equation to an L p − norm ( p often related to the dimension of the space),we refer to the Cwikel-Lieb-Rozenblum inequality [6, 16, 23], the Alexandrov-Bakelman-Pucci estimate and various Carleman-type estimates appearing in unique continuation (seeJerison & Kenig [14] and the example in Wolff [27]).1.2. Some motivation.
Our interest in these problems was motivated by the followingheuristic. Let Ω ⊂ R be an elongated domain as shown in Fig. 1 and let u be a nontrivialsolution of (1). The results of Faber-Krahn type discussed above show that k V k L r (Ω) (for r >
1) cannot be arbitrarily small and is bounded below by k V k L r (Ω) & r | Ω | − /r . Clearly,this lower bound on k V k L r (Ω) decays as | Ω | increases. However, basic intuition and relatedresults (e.g., [4, 12, 13, 22]) suggest that k V k L r (Ω) should not decay substantially unless theinradius (the radius of the largest ball fully contained in the domain) increases. x B x Figure 1.
Long narrow domain and its inradius
This can be seen from various points of view. The geometric perspective is that the solutionis bound to have most of its oscillation in the direction orthogonal to the direction of elonga-tion, which localizes the problem – in particular, the result should be fairly independent ofthe transversal direction. A more potential-theoretic perspective is that the far-field shouldnot act strongly in long narrow domains. Yet another perspective is to cut the long narrowdomain surgically into one that mainly contains the ball. We can then reintroduce Dirichletboundary condition by only slightly modifying the potential V and without much increasein k V k L r (Ω) . Our main result confirms this intuition.1.3. Location of the maximum.
A different question asks where a nontrivial solutionof the Dirichlet problem (1) might assume its maximum. A much simpler but nontrivialquestion is that of the inradius of the domain Ω. This question was raised in 1951 by P´olya& Szeg˝o in their classic book
Isoperimetric Inequalities in Mathematical Physics [21]. Theyraised the question whether there is a constant c > ⊂ R inradius(Ω) ≥ c · λ (Ω) − / . This inequality was first proven by Makai [19] in 1965 and, independently, by Hayman [13]in 1977. No such inequality can hold in higher dimensions because lines cutting throughthe domain affect the inradius but do not have a strong impact on the lowest eigenvalue. Acelebrated result of Lieb [17, Corollary 2] gives a complete and satisfactory answer.
Theorem (Lieb, 1983) . Let n ≥ and Ω ⊂ R n be open and non-empty. For every ε > ,there is a constant c = c ( ε, n ) > such that there exists a ball B of radius cλ (Ω) − / with | B ∩ Ω | ≥ (1 − ε ) | B | . Using an approach of Georgiev & Mukherjee [12], Rachh and the second author [22] showedthat the result also holds if the eigenvalue λ (Ω) is replaced by a Schr¨odinger potential. Inthis case, a ball B of radius | B | /n ∼ k V k − / L ∞ (Ω) , centered at a point x where the solution u of (1) assumes its maximum, has a large inter-section with Ω. Biswas [4] recently extended the result to fractional Schr¨odinger operators − ( − ∆) α/ + V . This line of reasoning was further pursued by Biswas & L˝orinczi [5]. Putdifferently, if the maximum is close to the boundary, then the potential has to be large. Itis easy to see that this is sharp: consider u ( x, y ) = sin ( nπx ) sin ( mπy ) on Ω = [0 , solving − ∆ u + ( m + n ) π u = 0 , which has global maxima and minima at points whose distance to the boundary is given by d (arg max u, ∂ Ω) ∼ m, n ) & √ m + n = k V k − / L ∞ (Ω) . We observe that this implies a form of the generalized Faber-Krahn inequality given by DeCarli & Hudson [8] (without the sharp constant), since | Ω | & inrad(Ω) n & d (arg max u, ∂ Ω) n & k V k − n/ L ∞ (Ω) , where all the implicit constants depend only on the dimension. Here and henceforth, weuse A . B to denote the existence of a universal constant such that A ≤ cB . Writing A . c ,c ,...,c n B denotes that the constant is allowed to depend on the variables in thesubscript and A ∼ B denotes that both A . B and B . A hold. Main results
Setup.
An informal summary of the types of results discussed above is the following:(i) If (1) has a solution on a small domain Ω, then k V k L p (Ω) is large (for a certainallowed range of p depending on the dimension and ‘large’ in the sense that thereexists a lower bound depending on | Ω | ).(ii) If a solution of (1) has a global maximum or minimum that is close to the boundary,then k V k L ∞ (Ω) has to be large.We will prove a result that unifies both these results for general equations of the typediv( A · ∇ u ) + V u = 0 , where we assume that A = A ( x ) is measurable and there exist constants 0 < λ < Λ < ∞ λ | ξ | ≤ h Aξ, ξ i ≤ Λ | ξ | , ∀ ξ ∈ R n . (uniform ellipticity)The uniform ellipticity constants λ and Λ impact all subsequent constants; we will suppressthis dependence for clarity of exposition. For the diffusion process ( X t ) t ≥ generated by theuniformly elliptic operator div( A · ∇ u ) on Ω with Dirichlet boundary condition, we let τ := inf { t > X t / ∈ Ω } be the first exit time of the domain Ω (or, alternatively, the first hitting time of the boundaryif no boundary condition were imposed). Definition 1.
We define the median exit time for the diffusion starting at point x ∈ Ω as T η ( x ) := inf { t > P x ( τ ≤ t ) ≥ η } . For the purpose of Theorem 1 and Theorem 2, we make the arbitrary choice η = 1 / η . However, the implicit constants in our theorems depend on thechoice of η . Remark.
Several remarks are in order.(i) A similar quantity has already been used in [10, 15, 22, 26] under the name of‘diffusion distance’, in a setting of finite graphs.(ii) There exists a constant 0 < c < λ, Λ and the dimension suchthat | B ( x , r ) ∩ Ω | ≤ c | B ( x , r ) | implies T ( x ) ≤ r .(iii) There exists a constant c > λ, Λ and the dimension such that ∀ x ∈ Ω , T ( x ) ≥ c inf y ∈ ∂ Ω k x − y k . This constant is assumed if Ω is a ball and x is the center.(iv) If the domain has a finite capacitary width w η ′ as introduced by Aikawa in [1] thenthere is a positive constant c = c ( η, η ′ , n, λ, Λ) such that T η ( x ) ≤ c w η ′ for all x ∈ Ω.(v) If Ω ⊂ R is simply connected, then T ( x ) ∼ λ, Λ inf y ∈ ∂ Ω k x − y k . In higher dimen-sions this is true if, e.g., the domain is convex or satisfies an exterior cone condition(see [4]). In general, relating the median exit time to any kind of ‘distance to theboundary’ puts restrictions on the geometry of the domain. Main result.
Our main result is the following Faber-Krahn type inequality. It statesthat if u solves div( A · ∇ u ) + V u = 0 , and | u | has its maximum in x ∈ Ω, then T ( x ) / defines a characteristic scale such thatthe potential V has to be large somewhere inside the domain on that scale. The novelty ofthis Theorem is that it is independent of the overall shape of the domain. Theorem 1.
Let Ω ⊂ R n , n ≥ , be a bounded domain. If u is a nontrivial solution of div( A · ∇ u ) + V u = 0 in Ω ,u | ∂ Ω = 0 and | u | assumes a global maximum in x , then there exists a ball B ⊂ R n of radius T ( x ) / such that k V + k L n , (Ω ∩ B ) ≥ c n,λ, Λ , where c n,λ, Λ > depends only on the uniform ellipticity constants λ, Λ and the dimension. Remark.
Several remarks are in order.(i) L n , is the usual Lorentz space refinement of L p spaces.(ii) The proof yields a slightly stronger result: we can apply this result whenever u assumes global maximum in x and u ( x ) > u assumes a globalminimum in x and u ( x ) <
0. Since u is nontrivial, the global maximum of | u | falls into one of these categories.(iii) We note the following immediate corollary: if − ∆ u = λ (Ω) u , then1 . k V k L n , (Ω ∩ B ( x ,r )) = k λ (Ω) k L n (Ω ∩ B ( x ,r )) ∼ (cid:0) λ (Ω) n r n (cid:1) n ∼ λ (Ω) T ( x ) . This implies the existence of a ball of radius ∼ T ( x ) / & λ − / that has largeparts inside Ω (cf. the properties of the median exit time in Remark 2.1). Thisresult was first established by Lieb [17] and refined by Georgiev & Mukherjee [12].2.3. The case n = 2 . The case n = 2 is slightly different: Theorem 1 stated in n = 2dimensions fails. We illustrate this with an example on the unit disk D ⊂ R given by DeCarli, Edward, Hudson & Leckband [9]: define the radial function u ε ( r ) by u ( r ) = ( − log ε − ε − r if 0 ≤ r ≤ ε − log r if ε ≤ r ≤ . Both u ε and its derivative u ′ ε are continuous. We observe that ∆ u ε ∼ ε − {| x |≤ ε } and k u k L ∞ ∼ − log ε . Put differently, we have ∆ u ε = V ε u ε for a potential V ε satisfying V ε ( x ) ∼ ε − ( − log ε ) − {| x |≤ ε } . Obviously, k V ε k L ( D ) ∼ ( − log ε ) − is not bounded from below. Hence there cannot be alower bound on k V ε k L (Ω ∩ B ) for a ball B ⊂ D . Theorem 2 will show that this type oflogarithmic behavior is actually the worst possible case: note that the convolution | V ε | ∗ (cid:12)(cid:12) (log | x | )1 {| x |≤ } (cid:12)(cid:12) (0) = Z {| y |≤ ε } | log ( | y | ) | log (cid:0) ε (cid:1) ε dy ∼ ε , | V ε | ∗ (cid:12)(cid:12) (log | x | )1 {| x |≤ } (cid:12)(cid:12) (0) & . We will prove that this holds in general.
Theorem 2 (Local Faber-Krahn inequality, n = 2) . Let Ω ⊂ R be a bounded domain. Let u is a nontrivial ( u = 0) solution of div( A · ∇ u ) + V u = 0 in Ω ,u | ∂ Ω = 0 . There exists a constant c > depending only on λ, Λ such that if | u | assumes a globalmaximum in x , then there exists a ball B = B ( x, ( cT ( x )) / ) ⊂ R such that Z B | V + ( y ) | log (cid:18) cT ( x ) | x − y | (cid:19) dy ≥ c λ, Λ . Lieb-type inequalities.
One particularly interesting consequence are Lieb-type in-equalities that follow from the same approach. Avoiding any notion of ‘distance to theboundary’ of the domain, Lieb’s theorem relates the norm of the potential to the (Eu-clidean) geometry of the domain.
Theorem 3 (Lieb-type inequality) . Let n ≥ and let Ω ⊂ R n be a bounded domain. Let u be a nontrivial solution of the Dirichlet problem div( A · ∇ u ) + V u = 0 in Ω ,u | ∂ Ω = 0 . Suppose | u | assumes its maximum in x ∈ Ω . Then there is a constant C = C ( n, λ, Λ) > such that the following holds: If the potential V is so small that, for some η ∈ (0 , , k V + k L n , ( B ) < C − η for all balls B ⊂ R n of radius at most T η ( x ) / , then the point x must be so close to the boundary that | B ( x , T η ( x ) / ) ∩ Ω | ≥ − η − η | B ( x , T η ( x ) / ) | . Proof of Theorem 1
This section is devoted to the proof of Theorem 1. The argument decouples nicely intoseveral different parts. First, we derive the fundamental inequality (2) that was alreadyused in [22, 26]. The other subsections develop different types of tools that will allow us toextract the desired information from inequality (2). As before, we let ( X t ) t ≥ be the diffusionprocess on Ω, generated by the uniformly elliptic operator div( A · ∇ ), with absorption atthe boundary. We introduce a cemetery state ∆ and set V (∆) = 0 and u (∆) = 0.3.1. A preliminary lower bound.
We consider the solution of (1) as a steady-state so-lution of the parabolic equation ∂ t u − (div( A · ∇ u ) + V u ) = 0 . By the Feynman-Kac formula, ∀ t ≥ , u ( x ) = E x (cid:18) u ( X t ) exp (cid:18)Z t V ( X s ) ds (cid:19)(cid:19) . We may assume that u assumes a global maximum in x and u ( x ) > − u and note that − u also solves (1)). Recall that τ is the first exit time from Ω. Then, u ( x ) = E x (cid:18) u ( X t )1 { τ>t } exp (cid:18)Z t V ( X s ) ds (cid:19)(cid:19) ≤ u ( x ) E x (cid:18) { τ>t } exp (cid:18)Z t V + ( X s ) ds (cid:19)(cid:19) . Since u ( x ) >
0, this simplifies to E x (cid:18) { τ>t } exp (cid:18)Z t V + ( X s ) ds (cid:19)(cid:19) ≥ . (2)3.2. Khasminskii’s Lemma.Lemma 4 (Khasminskii’s lemma) . Let V ≥ be a measurable function and ( X s ) s ≥ be aMarkov process on R n with the property that for some t > and α < , sup x ∈ R n E x (cid:20)Z t V ( X s ) ds (cid:21) = α. Then sup x ∈ R n E x (cid:20) exp (cid:18)Z t V ( X s ) ds (cid:19)(cid:21) ≤ − α . Khasminskii’s lemma is a classical tool in connection with the Feynman-Kac formula. Forthe convenience of the reader, we repeat the proof given in [18, 25].
Proof.
The argument proceeds by showing the stronger result E x " m ! (cid:18)Z t V ( B s ) ds (cid:19) m ≤ α m for all non-negative integers m . From this, the desired result then follows by summation.Expanding the power allows us to rewrite the statement as E x (cid:20) m ! Z t Z t · · · Z t V ( B s ) V ( B s ) . . . V ( B s m ) ds . . . ds m (cid:21) ≤ α m . There are n ! ways of ordering an n − tuple of point; therefore, defining∆ m = { ( s , . . . , s n ) : 0 ≤ s ≤ s · · · ≤ s m ≤ t } , we have the equivalent statement E x (cid:20)Z ∆ m V ( B s ) V ( B s ) . . . V ( B s m ) ds . . . ds m (cid:21) ≤ α m . This is where the Markovian property enters: for any fixed s ≤ s ≤ · · · ≤ s m − V ( B s ) V ( B s ) . . . V ( B s m − ) E x "Z ts n − V ( B ( s m )) ds m ≤ V ( B s ) V ( B s ) . . . V ( B s m − ) sup y ∈ R n E y (cid:20)Z t V ( B ( s m )) ds m (cid:21) ≤ V ( B s ) V ( B s ) . . . V ( B s m − ) α, where we have used that V ≥
0. The lemma now follows by induction. (cid:3)
A technical estimate.
The purpose of this subsection is to provide a self-containedproof of the following lemma for the convenience of the reader. Stronger results could beobtained by appealing to the literature centered around special functions (especially resultsdealing with incomplete Gamma functions) but are not needed here.
Lemma 5.
Let n ∈ N and d > . Let x ∈ R n . If n = 2 , then Z d c s exp (cid:18) − | x | c s (cid:19) ds . c ,c (cid:18) (cid:26) , − log (cid:18) | x | c d (cid:19)(cid:27)(cid:19) exp (cid:18) − | x | c d (cid:19) . If n ∈ { , } , then Z d c s n/ exp (cid:18) − | x | c s (cid:19) ds . c ,c ,n | x | − n exp (cid:18) − | x | c d (cid:19) . If n ≥ , then there exists a polynomial q ( · ) of degree at most ( n − / such that Z d c s n/ exp (cid:18) − | x | c s (cid:19) ds . c ,c ,n | x | − n q (cid:18) | x | c d (cid:19) exp (cid:18) − | x | c d (cid:19) . Proof.
The substitution z = s/ | x | shows Z d c s n/ exp (cid:18) − | x | c s (cid:19) ds . c | x | − n Z d/ | x | exp( − / ( c z )) z n/ dz. Another substitution ( y = 1 / ( c z )) yields Z d/ | x | exp( − / ( c z )) z n/ dz . c ,n Z ∞| x | / ( c d ) y n − e − y dy. We first consider the case n = 2. If | x | / ( c d ) ≤ Z ∞| x | / ( c d ) y − e − y dy . Z | x | / ( c d ) y − e − y dy . Z | x | / ( c d ) y − dy . − log (cid:18) | x | c d (cid:19) , and if | x | / ( c d ) ≥ Z ∞| x | / ( c d ) y − e − y dy ≤ c d | x | Z ∞| x | / ( c d ) e − y dy = c d | x | exp (cid:18) − | x | c d (cid:19) ≤ exp (cid:18) − | x | c d (cid:19) . Summarizing, this establishes Z ∞| x | / ( c d ) y e − y dy . (cid:18) (cid:26) , − log (cid:18) | x | c d (cid:19)(cid:27)(cid:19) exp (cid:18) − | x | c d (cid:19) , which is the desired statement for n = 2. The cases n ∈ { , } are even simpler: If | x | / ( c d ) ≤ ∼ (cid:18) − | x | c d (cid:19) ∼ ∼ Z ∞ y n − e − y dy ∼ Z ∞| x | / ( c d ) y n − e − y dy. If | x | / ( c d ) ≥ y n − ≤ y ≥ Z ∞| x | / ( c d ) y n − e − y dy ≤ Z ∞| x | / ( c d ) e − y dy = exp (cid:18) − | x | c d (cid:19) . If n ≥
5, then we can bound y n − . y k , where k is the smallest integer bigger or equalthan ( n − /
2. It then suffices to remark that integration by parts implies, for k ∈ N , Z ∞| x | / ( c d ) y k e − y dy = (cid:18) | x | c d (cid:19) k exp (cid:18) − | x | c d (cid:19) + Z ∞| x | / ( c d ) y k − e − y dy and an iterative application implies the assertion. (cid:3) An upper bound on a convolution.
It is well known (see the work of Aronson [2])that the uniformly elliptic operator div( A · ∇ ) admits a heat kernel p t ( x, y ) satisfying theGaussian upper bound p s ( x, y ) ≤ c s n/ exp (cid:18) − | x − y | c s (cid:19) , ∀ s > , x, y ∈ R n , for some c , c > λ, Λ , n . The last ingredient for the proof of Theorem1 is the following estimate. Lemma 6.
Let n ≥ , let f : R n → [0 , ∞ ) . For every d > , sup x ∈ R n Z R n f ( y ) Z d p s ( x, y ) ds dy . c ,c ,n sup | B |≤ d n k f k L n , ( B ) , where the supremum ranges over all balls B ⊂ R n with volume | B | ≤ d n .Proof. We fix x ∈ R n and let B = B ( x, d ). We choose countably many balls B i = B ( x i , d ), i ≥
2, in such a way that every point in R n is contained in at most N of these balls, where N depends only on the dimension n . Then Z R n f ( y ) Z d p s ( x, y ) ds dy ≤ ∞ X i =1 Z B i f ( y ) Z d p s ( x, y ) ds dy. By Lemma 5, Z B i f ( y ) Z d p s ( x, y ) ds dy . c ,c Z B i f ( y ) | x − y | n − q (cid:18) | x − y | c d (cid:19) exp (cid:18) − | x − y | c d (cid:19) dy We note that there is one term that becomes singular on the diagonal x = y while the otherterms all exhibit decay. The refined H¨older inequality due to O’Neil [20], k f g k L ( R n ) . n k f k L n , ( R n ) k g k L nn − , ∞ ( R n ) , together with the fact that 1 | x − y | n − ∈ L nn − , ∞ ( R n , dy ) , implies that Z B i f ( y ) | x − y | n − dy . k f k L n , ( B i ) for each i ≥
1. We apply the above estimate to each ball B i that is at most distance d/ x . The number of such B i ’s can be bounded in terms on n only.It remains to consider those balls B i whose distance to x is at least d/
2. A simple countingestimate shows that we have roughly ∼ ℓ n − balls at distance ∼ ℓ · d from x . If B i is sucha ball then 1 | x − y | n − q (cid:18) | x − y | c d (cid:19) exp (cid:18) − | x − y | c d (cid:19) . ℓ · deg q d n − ℓ n − exp (cid:18) − ℓ c (cid:19) . Using this asymptotic, we will bound the integral I = Z B i f ( y ) | x − y | n − q (cid:18) | x − y | c d (cid:19) exp (cid:18) − | x − y | c d (cid:19) dy. Applying the usual H¨older inequality with p = n/ p ′ = n/ ( n − I . ℓ · deg q − n +2 exp (cid:18) − ℓ c (cid:19) sup | B |≤ d n k f k L n ( B ) . Altogether, we obtainsup x ∈ R n Z R n f ( y ) Z d p s ( x, y ) ds dy . sup | B |≤ d n k f k L n , ( B ) + ∞ X ℓ =1 ℓ · deg q − n +2 ℓ n − exp (cid:18) − ℓ c (cid:19)! sup | B |≤ d n k f k L n ( B ) . sup | B |≤ d n k f k L n , ( B ) . (cid:3) Proof of Theorem 1.
Given a point x at which the solution attains its maximum,estimate (2) together with the Cauchy-Schwarz inequality imply that1 ≤ E x (cid:18) { τ>t } exp (cid:18)Z t V + ( X s ) ds (cid:19)(cid:19) ≤ P ( τ > t ) / (cid:18) E x exp (cid:18)Z t V + ( X s ) ds (cid:19)(cid:19) / for all t >
0. We choose t = T ( x ) to be the median exit time T ( x ) that we introduced inDefinition 1. Then P x ( τ > T ( x )) / ≤ √ , and therefore E x " exp Z T ( x )0 V + ( X s ) ds ! ≥ . Khasminskii’s Lemma, discussed in Section 3.2, now impliessup x ∈ R n E x "Z T ( x )0 V + ( X s ) ds ≥ . Finally, by Lemma 6,12 ≤ sup x ∈ R n E x "Z T ( x )0 V + ( X s ) ds . n,λ, Λ sup | B |≤ T ( x ) n/ (cid:13)(cid:13) V + (cid:13)(cid:13) L n , ( B ) . Recall that V ≡ Proofs of Theorem 2 and 3
Proof of Theorem 2.
The proof follows the same line of thought as the proof ofTheorem 1, the only difference is one technical lemma which we provide here. Lemma 7.
Let d > and assume g : R → (0 , ∞ ) satisfies g ( x ) . (cid:18) d | x | {| x |≤ d } (cid:19) exp (cid:18) − | x | d (cid:19) . Then sup x ∈ R Z R f ( x − y ) g ( y ) dy . sup x ∈ R Z {| y |≤ d } | f ( x − y ) | log (cid:18) d | y | (cid:19) dy. Proof.
We may assume that the supremum is assumed in the origin (after possibly trans-lating the function). The desired inequality then reads Z R f ( y ) g ( y ) dy . sup x ∈ R Z {| y |≤ d } | f ( x − y ) | log (cid:18) d | y | (cid:19) dy. As in the proof of Lemma 6, we cover R by disks D i = B ( x i , d ), i ≥
1, of radius d in sucha way that x = 0, each point in R is at most distance d/
10 away from some x i , and eachpoint in R is contained in at most c of these disks (a simple lattice construction shows thisto be possible; one could ask for constructions that minimize c and questions of these typeshave been studied independently, see F¨uredi & Loeb [11] - this is, of course, not requiredhere). We can now bound Z R f ( y ) g ( y ) dy . Z R f ( y ) (cid:18) d | y | {| y |≤ d } (cid:19) exp (cid:18) − | y | d (cid:19) dy . Z {| y |≤ d } | f ( y ) | log (cid:18) d | y | (cid:19) dy + Z R | f ( y ) | exp (cid:18) − | y | d (cid:19) dy. The first term is easy to bound since, trivially, Z {| y |≤ d } | f ( y ) | log (cid:18) d | y | (cid:19) dy ≤ sup x ∈ R Z {| y |≤ d } | f ( x − y ) | log (cid:18) d | y | (cid:19) dy. We now deal with the second term: Clearly, Z R | f ( y ) | exp (cid:18) − | y | d (cid:19) dy ≤ ∞ X i =1 Z D i | f ( y ) | exp (cid:18) − | y | d (cid:19) dy. We may assume that the disks i are ordered in increasing distance from the origin so that d ( D i ,
0) = inf x ∈ D i | x | ≥ d √ c i for some c > i . This implies Z D i | f ( y ) | exp (cid:18) − | y | d (cid:19) dy . exp ( − c · i ) Z D i | f ( y ) | dy. This leads to summable decay, it now suffices to show the uniform estimatesup i ∈ N Z D i | f ( y ) | dy . sup x ∈ R Z {| y |≤ d } | f ( x − y ) | log (cid:18) d | y | (cid:19) dy. This seems a bit tricky at first (because the logarithm vanishes at | y | = d ) but is easilycompensated by the fact that for every point in x ∈ R , there exists a disk D j whose center x j is at most distance d/
10 away. We fix an arbitrary i and let A = { j ∈ N : | x j − x i | ≤ d } . This set is finite and its cardinality only depends on c (this could be made explicit by fixinga sufficiently fine lattice but this is not required). We now claim that for all y ∈ D i | f ( y ) | ≤ X a ∈ A | f ( x j − ( x j − y )) | log (cid:18) d | x j − y | (cid:19) . This is easy to see: for every y there exists x j with | x j − y | ≤ d/
10 which ensures thatat least one logarithmic factor is bigger than log (100) ≥
1. Therefore, with a change ofvariables, Z D i | f ( y ) | dy ≤ X j ∈ A Z D a | f ( z ) | log (cid:18) d | z − x j | (cid:19) dz = X j ∈ A Z {| z |≤ d } | f ( z + x j ) | log (cid:18) d | z | (cid:19) dz ≤ ( A ) sup x ∈ R Z {| z |≤ d } | f ( x − z ) | log (cid:18) d | z | (cid:19) dz . c sup x ∈ R Z {| z |≤ d } | f ( x − z ) | log (cid:18) d | z | (cid:19) dz. (cid:3) Proof of Theorem 2.
Arguing exactly as in the proof of Theorem 1, we arrive at12 ≤ sup x ∈ R E x "Z T ( x )0 V + ( X s ) ds . Interchanging the order of integration and using Lemma 5 yields E x "Z T ( x )0 V + ( X s ) ds . Z R V + ( y − x ) (cid:18) (cid:26) , log (cid:18) c T ( x ) | y | (cid:19)(cid:27)(cid:19) exp (cid:18) − | y | c T ( x ) (cid:19) dy. The argument concludes by using Lemma 7 to bound the supremum viasup x ∈ R E x "Z T ( x )0 V + ( X s ) ds . sup x ∈ R Z {| y |≤ ( c T ( x )) / } | V + ( x − y ) | log (cid:18) c T ( x ) | y | (cid:19) dy, which is the desired statement. (cid:3) Proof of Theorem 3.
Proof.
Let η ∈ (0 , x ∈ Ω be a point in which | u | assumes its maximum. Suppose | B ( x , T η ( x ) / ) ∩ Ω | < − η − η | B ( x , T η ( x ) / ) | . By Definition 1 of the median exit time T η ( x ), we have P x ( τ > T η ( x )) < − η. This, estimate (2), and the Cauchy-Schwarz inequality imply that1 ≤ E x " { τ>T η ( x ) } exp Z T η ( x )0 V + ( X s ) ds ! ≤ P ( τ > T η ( x )) / E x " exp Z T η ( x )0 V + ( X s ) ds ! / ≤ (1 − η ) / E x " exp Z T η ( x )0 V + ( X s ) ds ! / . Thus, E x " exp Z T η ( x )0 V + ( X s ) ds ! ≥ − η . By Khasminskii’s Lemma, sup x ∈ R n E x Z T η ( x )0 V + ( X s ) ds ≥ η. Lemma 6 then implies that there is a constant C = C ( n, λ, Λ) such that η ≤ sup x ∈ R n E x Z T η ( x )0 V + ( X s ) ds ≤ C sup | B |≤ T η ( x ) n/ k V k L n , ( B ) , where the supremum ranges over all balls B of volume at most T η ( x ) n/ . If, however, k V + k L n , ( B ) < C − η for all balls B of volume at most T η ( x ) n/ , then we have a contra-diction, and therefore | B ( x , T η ( x ) / ) ∩ Ω | ≥ − η − η | B ( x , T η ( x ) / ) | . (cid:3) References [1] H. Aikawa, Norm estimate of Green operator, perturbation of Green function and integrability ofsuperharmonic functions, Math. Ann. 312 no. 2 (1998), 289318.[2] D. Aronson, Non-negative solutions of linear parabolic equations, Ann. Sci. Norm. Sup. 22 (1968),607–694.[3] J. Barta, Sur la vibration fundamentale d’une membrane, C. R. Acad. Sci. Paris 204 (1937), 472–473.[4] A. Biswas, Location of maximizers of eigenfunctions of fractional Schroedingers equation, MathematicalPhysics, Analysis and Geometry 20, no. 4, pages 14, 2017.[5] A. Biswas and J L˝orinczi, Universal Constraints on the Location of Extrema of Eigenfunctions ofNon-Local Schr¨odinger Operators, arXiv:1710.11596[6] M. Cwikel, Weak type estimates for singular values and the number of bound states of Schrodingeroperators, Ann. Math., (2) 106 (1977), 93–100.[7] L. De Carli, S. Hudson, Geometric remarks on the level curves of harmonic functions, Bull. LondonMath. Soc. 42 no. 1 (2010), 83–95.[8] L. De Carli, S. Hudson, A Faber-Krahn inequality for solutions of Schrdinger’s equation. Adv. Math.230 (2012), no. 4-6, 2416–2427.[9] L. De Carli, J. Edward, S. Hudson, M. Leckband, Minimal support results for Schrdinger equations.Forum Math. 27 (2015), no. 1, 343–371.[10] X. Cheng, M. Rachh and S. Steinerberger, On the Diffusion Geometry of Graph Laplacians and Appli-cations, arXiv:1611.03033[11] Z. F¨uredi and P. Loeb, On the best constant for the Besicovitch covering theorem. Proc. Amer. Math.Soc. 121 (1994), no. 4, 1063–1073.[12] B. Georgiev and M. Mukherjee, Nodal Geometry, Heat Diffusion and Brownian Motion,arXiv:1602.07110[13] W.K. Hayman, Some bounds for principal frequency, Applicable Anal. 7, no. 3, 247–254, 1977/78.[14] D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrdinger oper-ators. With an appendix by E. M. Stein. Ann. of Math. (2) 121 (1985), no. 3, 463–494. [15] J. Lu and S. Steinerberger, A Variation on the Donsker-Varadhan Inequality for the Principal Eigen-value, Proc. Royal Soc. A, accepted.[16] E. Lieb, Bounds on the eigenvalues of the Laplace and Schr¨odinger operators. Bull. Amer. Math. Soc.,82 (1976), no. 5, 751–753.[17] E. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math.74 (1983), 441–448.[18] J. Lorinczi, F. Hiroshima and V. Betz, Feynman-Kac-type theorems and Gibbs measures on path space.With applications to rigorous quantum field theory. De Gruyter Studies in Mathematics, 34. Walter deGruyter & Co., Berlin, 2011.[19] E. Makai, A lower estimation of the principal frequencies of simply connected membranes. Acta Math.Acad. Sci. Hungar. 16 (1965), 319–323.[20] R. O’Neil, Convolution operators and L ( p, q ) spaces. Duke Math. J. 30 1963 129–142.[21] G. P´olya and G. Szeg˝o. Isoperimetric Inequalities in Mathematical Physics. Annals of MathematicsStudies, no. 27, Princeton University Press, Princeton, N. J., 1951.[22] M. Rachh and S. Steinerberger, On the location of maxima of solutions of Schroedinger’s equation, toappear in Comm. Pure Appl. Math[23] G. Rozenblum, Distribution of the discrete spectrum of singular differential operators, Dokl. Acad.Nauk SSSR, 202 (1972), 1012-1015; translation in Soviet Math. Dokl., 13 (1972), 245–249.[24] I. Seo On minimal support properties of solutions of Schrdinger equations. J. Math. Anal. Appl. 414(2014), no. 1, 21–28.[25] B. Simon, Schr¨odinger semigroups. Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526.[26] S. Steinerberger, Lower bounds on nodal sets of eigenfunctions via the heat flow, Communications inPartial Differential Equations, 39, 2014, 2240–2261.[27] T. Wolff, Note on counterexamples in strong unique continuation problems. Proc. Amer. Math. Soc.114 (1992), no. 2, 351–356. Department of Mathematics, University of Connecticut, 341 Mansfield Road, Storrs, CT 06268
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