TTHE LANDIS CONJECTUREWITH SHARP RATE OF DECAY
LUCA ROSSI
Abstract.
The so called Landis conjecture states that if a solution of the equation∆ u + V ( x ) u = 0in an exterior domain decays faster than e − κ | x | , for some κ > (cid:112) sup | V | , then it mustbe identically equal to 0. This property can be viewed as a unique continuationat infinity (UCI) for solutions satisfying a suitable exponential decay. The Landisconjecture was disproved by Meshkov in the case of complex-valued functions, butit remained open in the real case. In the 2000s, several papers have addressed theissue of the UCI for linear elliptic operators with real coefficients. The results thathave been obtained require some kind of sign condition, either on the solution oron the zero order coefficient of the equation. The Landis conjecture is still opennowadays in its general form.In the present paper, we start with considering a general (real) elliptic operatorin dimension 1. We derive the UCI property with a rate of decay κ which is sharpwhen the coefficients of the operator are constant. In particular, we prove theLandis conjecture in dimension 1, and we can actually reach the threshold value κ = (cid:112) sup | V | . Next, we derive the UCI property – and then the Landis conjecture –for radial operators in arbitrary dimension. Finally, with a different approach, weprove the same result for positive supersolutions of general elliptic equations. Introduction
In [14], Kondrat (cid:48) ev and Landis asked the following question: if u is a solution ofthe equation ∆ u + V ( x ) u = 0 (1)in the exterior of a ball in R N , is it true that the condition ∃ κ > (cid:112) sup | V | , u ( x ) ≺ e − κ | x | , (2)necessarily implies u ≡ u ≺ v means u ( x ) /v ( x ) → | x | → ∞ . They also addressed the same question under thestronger requirement that u ( x ) ≺ e − κ | x | for all κ > N = 1with V constant, decaying solutions can only exist if V <
0, and they decay asexp( − (cid:112) | V || x | ). Hence, in such case, one can even take κ = (cid:112) sup | V | in condition (2).This is no longer true in higher dimension: the bounded, radial solution of ∆ u − u = 0outside a ball, which can be expressed in terms of the modified Bessel function ofsecond kind, decays like | x | − N − e −| x | . As we will see in the sequel, this discrepancybetween one and multidimensional cases holds true for general elliptic equations withvariable coefficients.The question by Kondrat (cid:48) ev and Landis received a negative answer in the paper [15]by Meshkov. There, the author exhibits two complex-valued, bounded functions CNRS, EHESS, PSL Research University, CAMS, Paris, France. a r X i v : . [ m a t h . A P ] J u l LUCA ROSSI u, V (cid:54)≡ | u ( x ) | ≤ exp( − h | x | ) for some h >
0. Onthe other hand, Meshkov shows that the power 4 / u ( x ) ≺ exp( −| x | + ε ) for some ε >
0, thennecessarily u ≡
0. These results provide a complete picture in the complex case.The conjecture has been brought back to attention in the 2000s by the worksof Bourgain and Kenig [6] and Kenig [12]. In the former, the authors improvedMeshkov’s positive answer in the case of real-valued functions, pushing the decaycondition up to u ( x ) ≤ exp( − h | x | log( | x | )). However, there is not an analogue ofMeshkov’s counterexample (nontrivial solutions with exponential decay with powerlarger than 1) in the real case. This fact led Kenig to ask in [12, Question 1] whether, in the real case , the condition u ( x ) ≺ e −| x | ε for all ε > u ≡
0. Observe that this condition is stronger than the originalrequirement (2) of [14]. However, even this weaker conjecture is still open nowadays,except for some particular situations. Kenig, Silvestre and Wang proved it in [13] indimension N = 2 and under the additional assumption that V ≤
0. The condition onthe decay is u ( x ) ≺ e − h | x | (log | x | ) for some h >
0, hence the result does not answer theoriginal question in [14]. In the case of equations set in the whole space R , the authorsare able to handle more general uniformly elliptic operators, still assuming V ≤
0, seealso [7]. As observed in [2], for equations in the whole space, this hypothesis impliesthat u ≡ u ≺
1, as an immediate consequence of the maximumprinciple. We point out that the results of [13, 7] are deduced from a quantitativeestimate which implies that the set where a nontrivial solution is bounded from belowby e − h | x | (log | x | ) is relatively dense in R .In this paper, we deal with uniformly elliptic operators with real coefficients, whosegeneral form is L u = Tr( A ( x ) D u ) + q ( x ) · Du + V ( x ) u, defined on an exterior domain Ω ⊂ R N , i.e., a connected open set with compactcomplement. For general operators of this type, it is known since the work of Pli´s [17]that the question asked by Kondrat (cid:48) ev and Landis has a (dramatically) negativeanswer. Namely, Pli´s exibhits an operator L in R with a H¨older-continuous matrixfield A and smooth terms q , V which admits a nontrivial solution vanishing identicallyoutside a ball. This is an astonishing counterexample to the property of uniquecontinuation at infinity . Here we consider the following definition of such property. Unique Continuation at Infinity ( UCI ) . We say that a given equation satisfiesthe
UCI with a rate of decay κ , if the unique solution satisfying u ( x ) ≺ e − κ | x | is u ≡ . The
UCI implies that two distinct solutions cannot have the same behaviour atinfinity up to an additive term decaying sufficiently fast. Owing to Pli´s’ counterex-ample, the only hope to derive the
UCI is by requiring some additional hypotheseson the operator. For instance, the results by Kenig and collaborators are restrictedto dimension N = 2, whereas the counterexample is in dimension 3. Another possibleway to avoid the counterexample of [17] is by assuming a suitable regularity of the HE LANDIS CONJECTURE WITH SHARP RATE OF DECAY diffusion matrix A . It is indeed known that the pathological situation of [17] cannotarise when A is Lipschitz-continuous, see [9]. However, these restrictions do not seemto be useful in an approach based on the comparison principle and Hopf’s lemma,which is the one adopted in the present paper.In the very recent paper [2], Arapostathis, Biswas and Ganguly attack the problemusing probabilistic tools. They derive the UCI under the additional assumption that u ≥
0, or, if Ω = R N , that λ ≥
0, where λ is the generalised principal eigenvalue of the operator −L , see the definition (5) below. We point out that the condition λ ≥ V ≤ u ≥
0. The threshold for the rates ofdecay κ obtained in [2] depends on the coefficients of L and it is not optimal when q (cid:54)≡
0, see the discussion in the next subsection.1.1.
Statement of the main results.
We consider a general (real) elliptic operator L u = Tr( A ( x ) D u ) + q ( x ) · Du + V ( x ) u, in an exterior domain Ω ⊂ R N . We always assume that the matrix field A is bounded,continuous and uniformly elliptic, i.e., its smallest eigenvalue α ( x ) := min ξ ∈ R N \{ } A ( x ) ξ · ξ | ξ | satisfies inf Ω α >
0. The vector field q and the potential V belong to L ∞ (Ω). So-lutions, subsolutions and supersolutions of the equation L u = 0 are always assumedto belong to W ,Nloc and to satisfy respectively L u = 0, L u ≥ L u ≤ W ,ploc for all p < + ∞ .In general, when referred to measurable functions, the equalities or inequalities areunderstood to hold a.e., and inf, sup stand for ess inf, ess sup.Our first result concerns the case of dimension N = 1, where L is given by L u = α ( x ) u (cid:48)(cid:48) + q ( x ) u (cid:48) + V ( x ) u, defined on the half-line R + = (0 , + ∞ ). Theorem 1.1.
In the case N = 1 , any solution of L u = 0 in R + satisfies lim x → + ∞ | u ( x ) | e κx ≥ | u ( x ) | e κx , for every x > , where κ = sup | q | α + (cid:114) sup | q | α + sup | V | α . (3)As a consequence, the UCI holds when N = 1 with the rate of decay (3). Let usmake some comments about this rate of decay. If the coefficients α, q, V are constantwith q ≥ V ≤ κ in (3) is precisely the rate of decay of solutions at + ∞ .Hence, our result is sharp in that case. The fact that we are able to obtain theequality in (3) instead of the strict inequality ‘ > ’ is actually surprising for us. Asexplained before, this threshold rate of decay cannot be obtained in higher dimension.We also derive a result in the spirit of the quantitative estimate of [13, Theorem 1.2],which implies that the inequality | u ( x ) | > e − κ (cid:48) x , for any κ (cid:48) larger than κ in (3), holdsin a relatively dense set (see Proposition 2.4 below). We recall that the lower boundin [13] is e − κ | x | log( | x | ) . LUCA ROSSI
Theorem 1.1 can be readily extended to radial solutions (i.e., of the form u ( x ) = φ ( | x | )) for general elliptic equations in higher dimension. Corollary 1.2.
Let u be a nontrivial, radial solution of L u = 0 in an exteriordomain Ω . Then, lim | x |→ + ∞ | u ( x ) | e κ | x | = + ∞ , for all κ satisfying κ > lim | x |→ + ∞ | q | α + (cid:115) lim | x |→ + ∞ | q | α + lim | x |→ + ∞ | V | α . Next, we extend the
UCI property to radial operators in arbitrary dimension.This is achieved by applying our one-dimensional result to the spherical harmonicdecomposition of the solution. This idea of considering the harmonic decompositionof the solution is not new in the context of the unique continuation property, see [16].
Theorem 1.3.
Assume that L is of the form L u = ∆ u + q ( | x | ) x | x | · ∇ u + V ( | x | ) u. Let u be a nontrivial solution of L u = 0 in an exterior domain Ω . Then, lim | x |→ + ∞ | u ( x ) | e κ | x | = + ∞ , for all κ satisfying κ > lim r → + ∞ | q | (cid:114) lim r → + ∞ | q | r → + ∞ | V | . This theorem implies that the Landis conjecture holds for radial potentials V .We then focus on solutions with a given sign. This makes the problem much sim-pler, because the sign condition allows one to directly use some comparison argumentsin order to control the decay of the solution. One of the consequences of this is thatthe result applies to supersolutions. Theorem 1.4.
Let u be a positive supersolution of L u = 0 in an exterior domain Ω .Then, u ( x ) (cid:31) e − κ | x | , for all κ satisfying κ > lim | x |→∞ (cid:32) | q | α + (cid:114) | q | α + | V | α (cid:33) . (4)Actually, the above result is derived in Section 4 in the more general framework ofancient supersolutions of parabolic equations. We remark that Theorem 1.4 providesa lower bound on the decay of u ( x ) as | x | → ∞ , whereas our previous estimatesonly hold along some diverging sequences, which is natural because the functionsthere are allowed to change sign. An analogous result to Theorem 1.4 is derivedin [2, Corollary 4.1], using a completely different method based on the stochasticrepresentation of solutions, but with the following threshold for the rate of decay: κ > lim | x |→∞ | q | α + lim | x |→∞ (cid:114) | V | α . HE LANDIS CONJECTURE WITH SHARP RATE OF DECAY This threshold is larger than or equal to the one in (4), and it is not sharp, even foroperators with constant coefficients, if q (cid:54)≡
0. Furthermore, the threshold in (4) isexpressed in terms of the lim of the combination of q, V, α , which is in general smallerthan the combination of their lim.Finally, as in [2], we extend the result to sign-changing solutions under the assump-tion that the generalised principal eigenvalue λ is nonnegative. The latter is definedas follows: λ := sup { λ : ∃ ϕ > , ( L + λ ) ϕ ≤ } , (5)or it can be equivalently defined as the limit as r → + ∞ of the classical principaleigenvalue in Ω ∩ B r under Dirichlet boundary condition if Ω is smooth (see, e.g.,[8, 1], or the more recent paper [5]). Clearly, the hypothesis of Theorem 1.4 yields λ ≥
0, and so does condition V ≤ Theorem 1.5.
Let u be a nontrivial solution of L u = 0 in an exterior domain Ω .Assume that λ ≥ and that either Ω = R N or that lim x → ∂ Ω u ( x ) ≥ . Then, lim | x |→ + ∞ | u ( x ) | e κ | x | = + ∞ , for all κ satisfying (4) . We point out that [2] only covers the case Ω = R N (with a larger threshold for κ ).Here, in the case of an exterior domain, we do not assume any regularity of theboundary, but we need to impose a sign for the solution there. In order to deal withthe lack of regularity of the domain, we make use of the maximum principle in smalldomains derived by Berestycki, Nirenberg and Varadhan in [4], building on an ideaof Bakelman. The result of [4] actually provides a ‘refined’ maximum principle, inwhich the boundary condition is understood in a suitable weak sense. We believethis should allow one to relax the boundary condition in our Theorem 1.5 too.The following table summarises all the cases in which we derive the UCI property,with the corresponding values of the rate of decay κ . Table 1.
Validity of the
UCI N = 1 κ = sup | q | α + (cid:114) sup | q | α + sup | V | αu is radial, κ > lim | x |→∞ | q | α + (cid:115) lim | x |→∞ | q | α + lim | x |→∞ | V | α or L is radial u ≥ κ > lim | x |→∞ (cid:32) | q | α + (cid:114) | q | α + | V | α (cid:33) or V ≤ R N and λ ≥ x → ∂ Ω u ( x ) ≥ λ ≥ LUCA ROSSI The one-dimensional case
In this section, N = 1 and the operator L is defined in the half-line R + = (0 , + ∞ ) by L u = α ( x ) u (cid:48)(cid:48) + q ( x ) u (cid:48) + V ( x ) u. We assume that α, q, V ∈ L ∞ ( R + ) and that inf α >
0. We let β, γ denote thefollowing quantities: β := sup R + | q | α , γ := sup R + | V | α . The strategy we employ to prove the
UCI property relies on the comparison withsuitable solutions for the following nonlinear operators with constant coefficients: L ∗ u := u (cid:48)(cid:48) − β | u (cid:48) | − γ | u | , L ∗ u := u (cid:48)(cid:48) + β | u (cid:48) | + γ | u | . These are the “extremal” operators associated with L , in the sense that L ∗ ≤ L ≤ L ∗ , that is, solutions for L are supersolutions for L ∗ and subsolutions for L ∗ . As a matterof fact, we will actually deal with functions u satisfying more generally L ∗ u ≤ ≤ L ∗ u rather than L u = 0. Concerning the regularity, we have that if u ∈ W , loc ( R + )solves L u = 0, then u (cid:48) ∈ C ( R + ) and therefore, using the equation, we find that u ∈ W , ∞ loc ( R + ). Thus, we work in this regularity framework.Positive, decreasing solutions of L ∗ = 0 and negative, increasing solutions of L ∗ = 0satisfy u (cid:48)(cid:48) + βu (cid:48) − γu = 0. They decay at + ∞ as e − κx , with κ given by (3), i.e., κ = β (cid:114) β γ. (6)Our aim is to show that the same κ provides a lower bound for the exponential rateof decay (along some sequence) for sign-changing functions satisfying L ∗ u ≤ ≤ L ∗ u .Throughout this section, κ denotes the above quantity.The comparison principle between sub and supersolutions for the extremal oper-ators requires the positivity of the supersolution. This condition implies that thegeneralised principal eigenvalue of the nonlinear operator has the sign that ensuresthe validity of the maximum principle. We further require that the derivatives of thefunctions do not vanish simultaneously, in order to reduce to the linear case. Proposition 2.1.
Let ( a, b ) be a bounded interval and let u , u ∈ W , ∞ (( a, b )) satisfy max [ a,b ] u > , min [ a,b ] u > , | u (cid:48) | + | u (cid:48) | (cid:54) = 0 in ( a, b ) , and either L ∗ u ≤ ≤ L ∗ u or L ∗ u ≤ ≤ L ∗ u in ( a, b ) . Then max [ a,b ] u u = max (cid:26) u ( a ) u ( a ) , u ( b ) u ( b ) (cid:27) . HE LANDIS CONJECTURE WITH SHARP RATE OF DECAY Moreover, unless u /u is constant, the above maximum cannot be attained at someinterior point and in addition if it is attained at y (resp. z ), there holds u (cid:48) ( a ) u ( a ) < u (cid:48) ( a ) u ( a ) (cid:18) resp. u (cid:48) ( b ) u ( b ) > u (cid:48) ( b ) u ( b ) (cid:19) . Proof.
The argument is classical, see e.g. [18, Theorem 2.10], even if here we dealwith nonlinear operators. We define w := u /u . Assume that M := max [ a,b ] w > x ∈ ( a, b ). It follows that u ( x ) >
0. Moreover, w (cid:48) ( x ) = 0, that is, u (cid:48) ( x ) u ( x ) = u ( x ) u (cid:48) ( x ), which, because | u (cid:48) | + | u (cid:48) | (cid:54) = 0,implies that u (cid:48) and u (cid:48) have the same strict sign. This means that L ∗ u j = ˜ L u j or L ∗ u j = ˜ L u j for j = 1 , J of x , where ˜ L is a linear operatorof the type ˜ L u = u (cid:48)(cid:48) + ˜ βu (cid:48) + ˜ γu . We then compute, in J ,0 ≤ ˜ L u = ˜ L ( u w )= u (cid:18) w (cid:48)(cid:48) + (cid:16) u (cid:48) u + ˜ β (cid:17) w (cid:48) (cid:19) + (cid:0) u (cid:48)(cid:48) + ˜ βu (cid:48) + ˜ γu (cid:1) w = u (cid:18) w (cid:48)(cid:48) + (cid:16) u (cid:48) u + ˜ β (cid:17) w (cid:48) + ˜ L u u w (cid:19) . This means that w is a subsolution in J of an equation with nonpositive zero orderterm. We can therefore apply the strong maximum principle and infer that w ≡ M in J . We have thereby shown that the set where w attains its maximum is both openand closed in ( a, b ), i.e., it is either empty or it coincides with the whole ( a, b ).It remains to prove the last statement of the proposition. Suppose that w is notconstant and that its maximum is attained at a (the other case is analogous). Thenthe Hopf lemma (see, e.g., [10]) implies that w (cid:48) ( a ) <
0, that is,0 > u (cid:48) ( a ) u ( a ) − u (cid:48) ( a ) u ( a ) u ( a ) , from which the desired inequality follows because u ( a ) , u ( a ) > (cid:3) Proposition 2.1 will be used to compare sub and supersolutions of Cauchy problems.Let us anticipate how this will be done, since, contrary to the usual application, wewill use subsolutions to get upper bounds and supersolutions to get lower bounds.Namely, let u, v be respectively a subsolution and a positive, monotone supersolutionof an extremal operator such that u ( a ) = v ( a ) and u (cid:48) ( a ) > v (cid:48) ( a ). Then v is smallerthan u in a right neighbourhood of y . If they cross at some point b > a , then we wouldget a contradiction with the maximum principle of Proposition 2.1. This means that u > v to the right of y , as long as v, v (cid:48) do not vanish.The idea of the proof of Theorem 1.1 consists in distinguishing the region where u is less steep than the exponential e − κx from the points where it is steeper. We recallthat κ is given by (6). The steepness refers to the ratio − u (cid:48) ( x ) /u ( x ). On one hand, inthe first case u decays at most as e − κx . On the other, we will show that if u is steeperthan e − κx at a point ¯ x then | u | hits the x -axis at some ˜ x > ¯ x with a certain slope andthen it eventually crosses back the exponential function | u (¯ x ) | e − κx at a later point.This ‘bouncing property’, depicted in Figure 1, is the object of the next lemma. LUCA ROSSI
Figure 1.
The ‘bounce’ of | u | when steeper than e − κx . Lemma 2.2.
Let u ∈ W , ∞ loc ( R + ) satisfy L ∗ u ≤ ≤ L ∗ u in R + and assume that thereexists ¯ x > for which the following occur: u (¯ x ) (cid:54) = 0 , − u (cid:48) (¯ x ) u (¯ x ) > κ. Then there exists h > such that − u (¯ x + h ) u (¯ x ) > e − κh . Proof. If κ = 0, i.e. q ≡ V ≡
0, the result trivially holds. Suppose that κ (cid:54) = 0. Up toreplacing the function u with u (¯ x + · ) /u (¯ x ), we can assume without loss of generalitythat ¯ x = 0 and that u (0) = 1, u (cid:48) (0) < − κ . With this change, the proof amounts toshowing that − u ( h ) > e − κh for some h > v of the equation v (cid:48)(cid:48) + βv (cid:48) − γv = 0, that is, v ( x ) = Ae − κx + Be λx , for some A, B ∈ R and λ = − β (cid:114) β γ ≥ . Imposing v (0) = 1 and v (cid:48) (0) = − κ (cid:48) , where κ (cid:48) is a fixed number satisfying κ < κ (cid:48) < − u (cid:48) (0), reduces to the system (cid:40) A + B = 1 − Aκ + Bλ = − κ (cid:48) . It follows that B ( κ + λ ) = κ − κ (cid:48) <
0, whence
B <
A >
1. As aconsequence, v (cid:48) is negative and v vanishes at some point ξ >
0, which means that L ∗ v = 0 in (0 , ξ ). Applying Proposition 2.1 with u = v , u = u we deduce that, forevery b > u > , b ], there holdsmax [0 ,b ] vu = max (cid:26) , v ( b ) u ( b ) (cid:27) . The above left-hand side is larger than 1 because v (cid:48) (0) > u (cid:48) (0). It follows that v ( b ) > u ( b ). This means that the inequality u < v holds as long as u remains HE LANDIS CONJECTURE WITH SHARP RATE OF DECAY positive, and therefore u must vanish somewhere in (0 , ξ ]. Let ˜ x denote the first zeroof u in (0 , ξ ].We claim that u (cid:48) < v (cid:48) in (0 , ˜ x ). Assume that this is not the case. Let ζ be thesmallest point in (0 , ˜ x ) where u (cid:48) = v (cid:48) . In the interval (0 , ζ ) we have that u (cid:48) < v (cid:48) < < u < v , from which we obtain( u − v ) (cid:48)(cid:48) ≤ − β ( u − v ) (cid:48) + γ ( u − v ) ≤ − β ( u − v ) (cid:48) . This inequality can be rewritten as (log( v (cid:48) − u (cid:48) )) (cid:48) ≥ − β . Therefore, log( v (cid:48) − u (cid:48) ) isbounded from below in [0 , ζ ), contradicting u (cid:48) ( ζ ) = v (cid:48) ( ζ ).Let us call ˜ u := − u . The following properties hold at the point ˜ x :˜ u (˜ x ) = 0 , ˜ u (cid:48) (˜ x ) ≥ − v (cid:48) (˜ x ) = Aκe − κ ˜ x − Bλe λ ˜ x > κe − κ ˜ x . We will derive the desired lower bound for ˜ u by comparison with a solution associatedwith the extremal operator L ∗ , for which the following holds. Lemma 2.3.
Assume that κ > . Let w be the solution of the Cauchy problem L ∗ w = 0 in R + w (0) = 0 w (cid:48) (0) = 1 . (7) Then there exists ˆ x > such that w (cid:48) > in (0 , ˆ x ) and moreover w (ˆ x ) > κ e − κ ˆ x . Let us postpone the proof of Lemma 2.3 until the end of the current one. Considerthe function w and the number ˆ x provided by the lemma. Then, for ε ∈ (0 , ˆ x ), set u ( x ) := κe − κ ˜ x w ( x − ˜ x + ε ) . The functions ˜ u, u satisfy L ∗ u = 0 ≤ L ∗ ˜ u in [˜ x, ˜ x + ˆ x − ε ]. We can therefore applyProposition 2.1 and derivemax [˜ x, ˜ x +ˆ x − ε ] ˜ uu = max (cid:26) ˜ u (˜ x ) u (˜ x ) , ˜ u (˜ x + ˆ x − ε ) u (˜ x + ˆ x − ε ) (cid:27) = max (cid:26) , ˜ u (˜ x + ˆ x − ε ) u (˜ x + ˆ x − ε ) (cid:27) . (8)In order to get a lower bound for the above left-hand side, we compute˜ u (˜ x + √ ε ) − u (˜ x + √ ε ) = ˜ u (cid:48) (˜ x ) √ ε − u (cid:48) (˜ x − ε )( √ ε + ε ) + o ( √ ε )= √ ε (cid:0) ˜ u (cid:48) (˜ x ) − κe − κ ˜ x (cid:1) + o ( √ ε ) . Recalling that ˜ u (cid:48) (˜ x ) > κe − κ ˜ x , we deduce that ˜ u (˜ x + √ ε ) > u (˜ x + √ ε ) for ε smallenough. It then follows from (8) that ˜ u (˜ x + ˆ x − ε ) > u (˜ x + ˆ x − ε ) for ε sufficientlysmall, that is, − u (˜ x + ˆ x − ε ) > κe − κ ˜ x w (ˆ x ). Letting ε →
0, we finally get the desiredinequality − u (˜ x + ˆ x ) ≥ κe − κ ˜ x w (ˆ x ) > e − κ (˜ x +ˆ x ) . (cid:3) Proof of Lemma 2.3.
The function w is positive and increasing up to a value ˆ x ∈ (0 , + ∞ ]. In the interval (0 , ˆ x ), w satisfies w (cid:48)(cid:48) + βw (cid:48) + γw = 0. We treat the differenttypes of solutions of this equation separately. Case β > γ . In this case the solution w is given in (0 , ˆ x ) by w ( x ) = 12 ω e − β x (cid:0) e ωx − e − ωx (cid:1) , with ω := (cid:112) β − γ > . If γ = 0 then ω = β and therefore w is increasing, which immediately entails theconclusion of the lemma. If γ > ω < β and we find that ˆ x is a critical pointfor w , characterised by β (cid:0) e ω ˆ x − e − ω ˆ x (cid:1) = ω (cid:0) e ω ˆ x + e − ω ˆ x (cid:1) . Using this equivalence, we derive w (ˆ x ) e κ ˆ x = 12 ω e ( κ − β ) ˆ x (cid:0) e ω ˆ x − e − ω ˆ x (cid:1) = 1 β e ( κ − β ) ˆ x (cid:0) e ω ˆ x + e − ω ˆ x (cid:1) > β e ( κ − β − ω ) ˆ x . Because κ ≥ β + ω , the above right-hand side is larger than κ . The proof of thelemma is thereby achieved in this case. Case β = 4 γ . Observe preliminarily that β, γ (cid:54) = 0, because otherwise κ = 0. The solution w isgiven by w ( x ) = xe − β x . We see that w (cid:48) (ˆ x ) = 0, with ˆ x = β . Direct computation reveals that w (ˆ x ) e κ ˆ x = 2 β e − β κ , which is larger than κ − because κ > β . Case β < γ . The solution w is now given by w ( x ) = 1 ω e − β x sin( ωx ) , with ω := (cid:112) γ − β . Then, ˆ x = π ω , and there holds w (ˆ x ) e κ ˆ x = 1 ω e ( κ − β ) ˆ x , which is larger than κ − because κ ≥ ω as well as κ > β . (cid:3) Proof of Theorem 1.1.
Take x >
0. Suppose that | u ( x ) | (cid:54) = 0, otherwise the resulttrivially holds. Fix c ∈ (0 , | u ( x ) | ) and define¯ x := sup (cid:8) x ≥ x : | u ( x ) | > ce − κ ( x − x ) (cid:9) . By continuity we know that ¯ x > x . Assume by way of contradiction that ¯ x < + ∞ .This means that | u (¯ x ) | = ce − κ (¯ x − x ) and there exists a sequence ( x n ) n ∈ N such that x n (cid:37) ¯ x as n → ∞ and | u ( x n ) | > ce − κ ( x n − x ) . Up to replacing u with − u if need be,it is not restrictive to assume that u (¯ x ) >
0, and thus u > x ¯ n , ¯ x ] for some ¯ n ∈ N . HE LANDIS CONJECTURE WITH SHARP RATE OF DECAY Since u and ce − κ ( x − x ) are respectively a supersolution and a solution of L ∗ = 0,it follows from the maximum principle of Proposition 2.1 that u ( x ) ≥ ce − κ ( x − x ) for x ∈ ( x ¯ n , ¯ x ). Actually, the second statement of the proposition implies that theinequality is strict, because it is strict at x ¯ n , and in addition there holds that u (cid:48) (¯ x ) < − κce − κ (¯ x − x ) = − κu (¯ x ) . We can finally apply the ‘bouncing’ Lemma 2.2, which provides us with some h > − u (¯ x + h ) > u (¯ x ) e − κh = ce − κ (¯ x − x + h ) . This contradicts the definition of ¯ x .We have thereby shown that ¯ x = + ∞ , that is,lim x → + ∞ | u ( x ) | e κx ≥ ce κx . This concludes the proof, due to the arbitrariness of c ∈ (0 , | u ( x ) | ). (cid:3) We conclude the study of the 1-dimensional case with an estimate of the distancebetween points where u ‘does not decay too fast’. Proposition 2.4.
Let u be a solution of L u = 0 in R + satisfying u (0) = 1 . Then,for every κ (cid:48) > sup | q | α + (cid:114) sup | q | α + sup | V | α , there exists h > depending on (cid:107) q/α (cid:107) ∞ , (cid:107) V /α (cid:107) ∞ and κ (cid:48) such that sup ¯ x
Assume by way of contradiction that there exist some functions ( α n ) n ∈ N ,( q n ) n ∈ N , ( V n ) n ∈ N , ( u n ) n ∈ N such that α n ( x ) u (cid:48)(cid:48) n + q n ( x ) u (cid:48) n + V n ( x ) u n = 0 in R + , with | q n | α n ≤ β, | V n | α n ≤ γ, and moreover u n (0) = 1 and | u n ( x ) | ≤ e − κ (cid:48) x for x ∈ [ x n , x n + n ] , for some x n ≥
0. Up to replacing u n with − u n and decreasing x n if need be, wecan assume without loss of generality that u n ( x n ) = e − κ (cid:48) x n . Consider the functions( v n ) n ∈ N defined by v n ( x ) := u n ( x n + x ) e κ (cid:48) x n . They satisfy some linear equations of the form v (cid:48)(cid:48) n + ˜ q n ( x ) v (cid:48) n + ˜ V n ( x ) v n = 0 in R + , with | ˜ q n | ≤ β , | ˜ V n | ≤ γ , together with v n (0) = 1 and | v n ( x ) | ≤ e − κ (cid:48) x for x ∈ [0 , n ] . We now use standard elliptic estimates. They imply that the ( v n ) n ∈ N are uniformlybounded in W ,p ((0 , R )), for all p < + ∞ and R >
0, and thus in C ,δ ([0 , R ]), δ ∈ (0 , L ∗ v n ≤ ≤ L ∗ v n and we find that (up to subsequences) ( v n ) n ∈ N converges locallyuniformly in [0 , + ∞ ) to a function v ∈ W ,ploc ( R + ) ∩ C ([0 , + ∞ )) satisfying L ∗ v ≤ ≤L ∗ v and moreover v (0) = 1 and | v ( x ) | ≤ e − κ (cid:48) x for x >
0. We deduce in particularthat v (cid:48) (0) ≤ − κ (cid:48) < − κ . It then follows from Lemma 2.2 that v ( h ) < − e − κh forsome h >
0, which is impossible because | v ( h ) | ≤ e − κ (cid:48) h . (cid:3) The radial cases
We now turn to the N -dimensional case, considering first radial solutions. Proof of Corollary 1.2.
Assume by contradiction that there exists a nontrivial, radialsolution u ( x ) = φ ( | x | ) such that φ ( r ) ≺ e − κr for some κ satisfying κ > lim | x |→ + ∞ | q | α + (cid:115) lim | x |→ + ∞ | q | α + lim | x |→ + ∞ | V | α . The function φ belongs to W ,Nloc (( R , + ∞ )), where R > R N \ B R ⊂ Ω.Let e be the first vector of the canonical basis of R N . For r > R , we compute L u ( re ) = A ( re ) φ (cid:48)(cid:48) ( r ) + (cid:18) q ( re ) + Tr A ( re ) − A ( re ) r (cid:19) φ (cid:48) ( r ) + V ( re ) φ ( r ) = 0 . Namely, φ satisfies the equation ˜ L φ = 0 in ( R , + ∞ ), where˜ L φ := ˜ α ( r ) φ (cid:48)(cid:48) + ˜ q ( r ) φ (cid:48) + ˜ V ( r ) φ, with˜ α ( r ) := A ( re ) , ˜ q ( r ) := q ( re ) + Tr A ( re ) − A ( re ) r , ˜ V ( r ) := V ( re ) . We have that ˜ α ( r ) ≥ α ( re ), where α ( x ) is the smallest eigenvalue of A ( x ). Therefore, | ˜ q ( r ) | α ( r ) ≤ sup R N \ B r | q | α + Cr − , | ˜ V ( r ) | ˜ α ( r ) ≤ sup R N \ B r | V | α , where C only depends on N and the L ∞ norm of the coefficients of A . In particular,for R > R sufficiently large, there holds κ > sup r>R | ˜ q | α + (cid:115) sup r>R | ˜ q | α + sup r>R | ˜ V | ˜ α . As a consequence, applying Theorem 1.1 to the operator ˜ L , we infer that φ = 0in ( R, + ∞ ). We would like to conclude from this that u ≡ A being only in L ∞ (Ω), we arenot in the regularity framework where such result applies. We overcome this difficultyby reducing to the 1-dimensional case. Consider any ˆ R > φ is defined in( ˆ R, + ∞ ). It satisfies there | φ (cid:48)(cid:48) | ≤ C (cid:48) (1 + ˆ R − ) | φ (cid:48) | + C (cid:48)(cid:48) | φ | , for some C (cid:48) , C (cid:48)(cid:48) >
0. We can now apply the unique continuation property of [3], oreven the classical Carath´eodory theorem for ODEs, to deduce that φ ≡ R, + ∞ ). HE LANDIS CONJECTURE WITH SHARP RATE OF DECAY By the arbitrariness of ˆ R , this means that u ≡ (cid:3) Next, we consider radial operators. In the sequel, S = ∂B stands for the unitsphere in R N centred at the origin and we let dS denote its surface element. Proof of Theorem 1.3.
Assume by contradiction that there exists a nontrivial solutionsatisfying u ( x ) ≺ e − κ | x | with κ > lim r → + ∞ | q | (cid:114) lim r → + ∞ | q | r → + ∞ | V | . It is convenient to rewrite the equation in spherical coordinates. Let R be such that R N \ B R ⊂ Ω and let ˜ u : ( R , + ∞ ) × S → R be the expression for u in sphericalcoordinates, i.e., ˜ u ( r, σ ) := u ( rσ ). The Laplace operator rewrites as follows:∆ u ( rσ ) = 1 r N − ∂ r ( r N − ∂ r ˜ u ) + 1 r ∆ σ ˜ u, with ∆ σ indicating the Laplace-Beltrami operator on the sphere S . Then, using theidentity ∂ r ˜ u ( r, σ ) = σ · ∇ u ( rσ ), we find that L u ( rσ ) = 1 r N − ∂ r ( r N − ∂ r ˜ u ) + 1 r ∆ σ ˜ u + q ( r ) ∂ r ˜ u ( r, σ ) + V ( r )˜ u = 0 , r > R , σ ∈ S. The eigenvalues of − ∆ σ (counted with their multiplicity) are given by0 = λ < λ ≤ λ ≤ · · · Let ϕ ≡ , ϕ , ϕ , . . . be the corresponding eigenfunctions, with L ( S ) norm equalto 1. We would like to multiply the equation for ˜ u by ϕ j , j = 1 , . . . , and integrateit on the sphere, in order to get an ODE for the projections u j defined by u j ( r ) := (cid:90) S ˜ u ( r, σ ) ϕ j ( σ ) dS σ . This cannot directly be done because ∂ rr ˜ u , ∆ σ ˜ u are just in L ploc (( R , + ∞ ) × S ),for all p < + ∞ . The lower order terms do not pose any problem because ˜ u ∈ C (( R , + ∞ ) × S ) by Morrey’s inequality, and thus u (cid:48) j ( r ) = (cid:90) S ∂ r ˜ u ( r, σ ) ϕ j ( σ ) dS σ . In order to derive the equation for u j , we consider ψ ∈ C ∞ c (( R , + ∞ )) and compute (cid:90) R N \ B R (∆ u ) ϕ j (cid:18) x | x | (cid:19) ψ ( | x | ) dx = (cid:90) R N \ B R u ∆ (cid:18) ϕ j (cid:18) x | x | (cid:19) ψ ( | x | ) (cid:19) dx = (cid:90) + ∞ R dr (cid:90) S ˜ u (cid:16) ∂ r ( r N − ψ (cid:48) ( r )) − r N − λ j ψ ( r ) (cid:17) ϕ j ( σ ) dS σ = (cid:90) + ∞ R u j ( r ) (cid:16) ∂ r ( r N − ψ (cid:48) ( r )) − r N − λ j ψ ( r ) (cid:17) dr = (cid:90) + ∞ R (cid:16) − u (cid:48) j ( r ) r N − ψ (cid:48) ( r ) − u j ( r ) r N − λ j ψ ( r ) (cid:17) dr. On the other hand, using the equation L u = 0, we get (cid:90) R N \ B R (∆ u ) ϕ j (cid:18) x | x | (cid:19) ψ ( | x | ) dx = − (cid:90) + ∞ R dr (cid:90) S (cid:0) q ( r ) ∂ r ˜ u + V ( r )˜ u (cid:1) ϕ j ( σ ) ψ ( r ) r N − dS σ = − (cid:90) + ∞ R r N − (cid:0) q ( r ) u (cid:48) j ( r ) + V ( r ) u j ( r ) (cid:1) ψ ( r ) dr. This means that the following equalities hold in the distributional sense in ( R , + ∞ ): − q ( r ) u (cid:48) j − (cid:18) V ( r ) − λ j r (cid:19) u j = 1 r N − ( r N − u (cid:48) j ) (cid:48) = u (cid:48)(cid:48) j + N − r u (cid:48) j . It follows in particular that u (cid:48)(cid:48) j ∈ L ∞ (( R , + ∞ )) and thus the equation is satisfieda.e. The coefficients of this equation fulfil12 sup r>R (cid:12)(cid:12)(cid:12)(cid:12) q ( r )2 + N − r (cid:12)(cid:12)(cid:12)(cid:12) + (cid:115) sup r>R (cid:12)(cid:12)(cid:12)(cid:12) q ( r )2 + N − r (cid:12)(cid:12)(cid:12)(cid:12) + sup r>R (cid:12)(cid:12)(cid:12)(cid:12) V ( r ) − λ j r (cid:12)(cid:12)(cid:12)(cid:12) < κ, provided R is larger than some R j . Therefore, because | u j ( r ) | ≤ (cid:107) ˜ u ( r, · ) (cid:107) L ( S ) ≤ (cid:112) | S | (cid:107) ˜ u ( r, · ) (cid:107) L ∞ ( S ) ≺ e − κr , the 1-dimensional UCI property (Theorem 1.1) entails that u j ( r ) = 0 for r > R j .Then, owing to the unique continuation property for ODEs, we have that u j ≡ R , + ∞ ). Finally, being ( ϕ n ) n ∈ N an Hilbert basis for L ( S ), we know that˜ u ( r, σ ) = (cid:80) + ∞ j =1 u j ( r ) ϕ j ( σ ) in the L sense, and thus a.e. We have thereby shownthat ˜ u ( r, σ ) = 0 for r > R . Applying again the unique continuation property, thistime for equations in dimension N with leading term given by the Laplace operator,see [3, 11], we eventually conclude that u ≡ (cid:3) Remark 1.
Looking at the proof of Theorem 1.3, one realizes that more generalsecond order terms than the Laplace operator are allowed. Namely, those expressedin spherical coordinates by 1 r N − ∂ r ( r N − ∂ r ˜ u ) + ϑ ( r ) r ∆ σ ˜ u, for a given function ϑ , not necessarily positive.4. Positive supersolutions
In this section, we consider a parabolic equation associated with the operator P u = ∂ t u − Tr( A ( x, t ) D u ) + q ( x, t ) · Du + V ( x, t ) u. We always assume that
A, q, V are in L ∞ and that A is continuous and uniformlyelliptic, i.e., that the function α ( x, t ) := min ξ ∈ R N \{ } A ( x, t ) ξ · ξ | ξ | is bounded from below away from 0. Solutions, subsolutions and supersolutions arenow assumed to be in L N +1 loc with respect to the ( x, t ) variable, together with theirderivatives Du, D u, ∂ t u . HE LANDIS CONJECTURE WITH SHARP RATE OF DECAY Here is our main result concerning positive supersolutions, from which the otherresults of the section readily follow. It is achieved using a refinement of the argumentof the proof of [19, Lemma 3.1].
Theorem 4.1.
Let u satisfy P u ≥ , x ∈ Ω , t < , where Ω is an exterior domain in R N , together with inf x ∈ Kt< u ( x, t ) > , for any compact set K ⊂ Ω . Then, u ( x, (cid:31) e − κ | x | , for all κ satisfying κ > lim | x |→∞ (cid:32) sup t< (cid:18) | q | α + (cid:114) | q | α + | V | α (cid:19)(cid:33) . (9) Proof.
Let R > R N \ B R ⊂ Ω. Take κ satisfying (9)and consider the function χ : R + × R + → R defined by χ ( r, s ) := (cid:40)(cid:16) − rs (cid:17) κs if 0 ≤ r < s r ≥ s. Then define the function η as follows: η ( x, t ) := χ ( | x | − R , δt + δ − + h ) , where the parameters R ≥ R and δ, h > η are that it is compactly supported in space and that it converges to the function e − κ ( | x |− R ) as the parameter δ tends to 0. We now proceed in two steps: first showingthat η is a (generalised) subsolution of P = 0, next comparing it with u . Step 1.
The function η is a generalised subsolution of P = 0 provided R, h aresufficiently large and δ is sufficiently small.For − δ − < t < R < | x | < R + δt + δ − + h , we compute P η = δ ∂ s χ − Ax · x | x | ∂ rr χ − (cid:18) q · x | x | + Tr A | x | − Ax · x | x | (cid:19) ∂ r χ − V χ.
Here and in what follows, the functions
A, q, V and α are evaluated at ( x, t ), whereas χ and its derivatives at ( r, s ) = ( | x | − R , δt + δ − + h ), which satisfy0 < r < s and h < s < h + δ − . We impose h ≥ /κ , so that ∂ rr χ ≥
0. We then obtain P η n ≤ δ ∂ s χ − α ∂ rr χ − (cid:18) | q | + C | x | (cid:19) ∂ r χ + | V | χ, with C only depending on N and the L ∞ norm of the coefficients of A . Observing that ∂ r χ = − κ ss − r χ, ∂ rr χ = κ (cid:18) κ − s (cid:19) (cid:18) ss − r (cid:19) χ, ∂ s χ = κχ (cid:18) log (cid:16) − rs (cid:17) + rs − r (cid:19) ≤ κχ ss − r , we eventually derive P ηχ ≤ δκ ss − r − ακ (cid:18) κ − s (cid:19) (cid:18) ss − r (cid:19) + κ (cid:18) | q | + C | x | (cid:19) ss − r + | V | = (cid:18) ss − r (cid:19) (cid:32) − ακ + κ (cid:18) | q | + C | x | + δ (cid:19) s − rs + αs κ + | V | (cid:18) s − rs (cid:19) (cid:33) ≤ (cid:18) ss − r (cid:19) (cid:18) − ακ + κ (cid:18) | q | + CR + δ + αh (cid:19) + | V | (cid:19) We need to show that the right-hand side is less than or equal to 0. For this, weuse (9) which allows us to rewrite κ = ˜ κ + ε , with ε > κ := sup | x | >R t< (cid:18) | q | α + (cid:114) | q | α + | V | α (cid:19) , for some sufficiently large R > R . The quantity ˜ κ is the supremum with respect to | x | > R , t < Q x,t ( X ) := α ( x, t ) X − | q ( x, t ) | X − | V ( x, t ) | . It follows that Q x,t (˜ κ ) ≥ | x | > R and t <
0, whence Q x,t ( κ ) = Q x,t (˜ κ ) + α ( x, t )(2˜ κε + ε ) − | q ( x, t ) | ε ≥ (2 α ( x, t )˜ κ − | q ( x, t ) | ) ε + α ( x, t ) ε ≥ α ( x, t ) ε , where the last inequality follows from the explicit expression for ˜ κ . As a consequence,for | x | > R > R and t <
0, we have that − ακ + κ (cid:18) | q | + CR + δ + αh (cid:19) + | V | ≤ − (cid:0) inf α ) ε + κCR + κδ + καh , which is a negative constant provided R, h are large enough and δ is small enough.In the end, under such conditions, there holds P η ≤ − δ − < t < , R < | x | < R + δt + h. Step 2.
Comparison between u and η .By the previous step, we can take R, h > η is a generalised subsolution of P = 0 for − δ − < t < | x | > R , provided δ is sufficiently small. By hypothesis,we can renormalise u in such a way thatinf R ≤| x |≤ R + ht< , u = 1 . At the time t = − δ − , we have that η ( x, δ − ) = χ ( | x | − R , h ) , which is bounded from above by 1 and vanishes for | x | ≥ R + h . Hence, u ( x, δ − ) ≥ η ( x, δ − ) for | x | ≥ R . We further have that u ( x, t ) ≥ η ( x, t ) for | x | = R , t < HE LANDIS CONJECTURE WITH SHARP RATE OF DECAY Thus, for δ small enough, applying the parabolic comparison principle in the set( R N \ B R ) × ( − δ − , u ≥ η in this set, and then in particular u ( x, ≥ η ( x,
0) = χ ( | x | − R , δ − + h ) . Recalling the expression of χ , for R ≤ | x | ≤ R + δ − + h we compute the aboveright-hand side getting χ ( | x | − R , δ − + h ) = (cid:18) − | x | − Rδ − + h (cid:19) κ ( δ − + h ) , which tends to e − κ ( | x |− R ) as δ → + . This shows that u ( x, ≥ e − κ ( | x |− R ) for | x | ≥ R ,with R depending on κ . Since this is true for every κ satisfying (9), the proof iscomplete. (cid:3) Proof of Theorem 1.4.
One just applies Theorem 4.1 to the stationary supersolution u of the equation ∂ t u −L u = 0. Observe that inf K u ( x ) > K ⊂ Ω,because u is positive and continuous. (cid:3) Proof of Theorem 1.5.
We know from [5, Theorem 1.4] that the generalised principaleigenvalue λ admits a positive eigenfunction ϕ , that is, a positive solution of −L ϕ = λ ϕ in Ω. Moreover, ϕ = 0 on ∂ Ω if Ω is smooth ( (cid:54) = R N ). Because λ ≥
0, we have −L ϕ = λ ϕ ≥ . Theorem 1.4 then implies that ϕ ( x ) (cid:31) e − κ | x | , for all κ satisfying (4). Assume by con-tradiction that there exist a nontrivial solution u of L u = 0 in Ω and κ satisfying (4)such that lim | x |→ + ∞ | u ( x ) | e κ | x | < + ∞ . Then, for κ (cid:48) < κ still satisfying (4), we have that u ( x ) ≺ e − κ (cid:48) | x | . We now distinguishthe two different hypotheses of the theorem. Case
Ω = R N .Because u ( x ) ≺ e − κ (cid:48) | x | ≺ ϕ ( x ), the quantity C := max R N | u | ϕ is a well defined positive number. It follows that either min R N ( Cϕ − u ) = 0 ormin R N ( Cϕ + u ) = 0. The strong maximum principle then yields Cϕ ≡ ± u , which isimpossible because u ≺ ϕ . Case Ω (cid:54) = R N and lim x → ∂ Ω u ( x ) ≥ δ > δ .Consider the sequence of open sets (Ω n ) n ∈ N defined byΩ n := { x ∈ Ω : dist( x, ∂ Ω) > /n } . Since Ω is an exterior domain, we have that | Ω \ Ω n | → n → ∞ , whence | Ω \ Ω ¯ n | < δ for some ¯ n ∈ N . Next, using the fact that u ≺ ϕ and that ϕ > in Ω ⊃ Ω ¯ n , we can find C > Cϕ ≥ u in Ω ¯ n . Finally, in the open set O := Ω \ Ω ¯ n , the function w := Cϕ − u is a supersolution of L = 0 satisfyinglim x → ∂ O w ( x ) ≥ . Applying the maximum principle [4, Theorem 2.6] in every connected componentof O , we find that Cϕ ≥ u in O too. We have shown that Cϕ ≥ u in Ω.Define C ∗ := inf { C > Cϕ ≥ u in Ω } . Assume by contradiction that C ∗ >
0. On one hand, there holds C ∗ ϕ ≥ u in Ω.On the other, for ε >
0, we necessarily have that inf Ω ¯ n (( C ∗ − ε ) ϕ − u ) <
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