Positive harmonic functions in union of chambers
aa r X i v : . [ m a t h . A P ] M a r HARMONIC FUNCTIONS IN UNION OF CHAMBERS.
LAURA ABATANGELO, SUSANNA TERRACINI
Abstract.
We characterize the set of harmonic functions with Dirichlet bound-ary conditions in unbounded domains which are union of several differentchambers. We analyze the asymptotic behavior of the solutions in connectionwith the changes in the domain’s geometry. Finally we classify all (possiblysign-changing) infinite energy solutions having given asymptotic frequency atthe infinite ends of the domain. Introduction
In this paper we are concerned with solutions to the following problem(1) (cid:26) ∆ u = 0 in Ω u = 0 on ∂ Ω,where Ω is a particular unbounded domain defined as the union of two or moreinfinite cylinders. In this context the term chamber stands exactly for cylinder. Webecame interested in these issues in connection with the problem of the interplayof the geometry of the domain with the transmission of frequencies of solutions,as it will appear in the sequel. As a matter of facts, problems of type (1) mayarise, for example, from a blow-up analysis for eigenvalues equations in boundeddomains with varying geometries. This type of equations may describe the possibletransmission of frequency from a chamber to another one, when passing througha certain number of other chambers, connected by thin tubes (whose section isnegligible with respect to its own length), (see e.g. [7, 2]).As this is the simplest case where the domain presents a sensitive change ofgeometry, one may expect the domain’s geometry and solutions’ shape to be strictlyrelated to each other. We mean that such geometric changes in the domain affectsthe solutions’ shape as well as, from the opposite point of view, that solutions maycarry some information about the domain’s geometry.We are now going to specify the context and the notation that we will usethroughout the paper. Let U R and U L two open regular connected domains in R N − for N ≥
2, possibly unbounded and let C R := { ( x, y ) ∈ R × R N − s.t. x > y ∈ U R } ; C L := { ( x, y ) ∈ R × R N − s.t. x < y ∈ U L } ;Ω := C R ∪ C L ∪ Γ being Γ := ∂C R ∩ ∂C L . We stress that in our setting positive solutions can not have finite energy at bothends of the domain. In the same way, uniqueness of solutions of inhomogoneousLaplace equations does not hold, unless the energy is supposed to be finite. There-fore, in order to classify solutions of (1), we need to waive the energy boundednessand to allow infinite energy solutions. As it will appear in the proofs, we can handle
Date : May 5, 2018.2010
Mathematics Subject Classification.
Keywords. harmonic functions, unbounded domains, asymptotic estimates.Partially supported by the PRIN2009 grant “Critical Point Theory and Perturbative Methodsfor Nonlinear Differential Equations”. infinite energy solutions by imposing suitable thresholds to the so-called Almgrenquotient. Being v any solution of (1) we define its Almgren frequency function:(2) N ( v )( x ) := R Ω x |∇ v | R Γ x v , where Γ x := { ( x, y ) : y ∈ U R } , Ω x := { ( ξ, η ) ∈ Ω : ξ ∈ (0 , x ) } if x > x := { ( ξ, η ) ∈ Ω : ξ ∈ ( x, } if x <
0. We set N (0)( x ) ≡ i ) ( − ∆ ψ Lk = λ Lk ψ Lk on U L ψ Lk = 0 on ∂U L , ( ii ) ( − ∆ ψ Rk = λ Rk ψ Rk on U R ψ Rk = 0 on ∂U R ;here ∆ denotes the ( N − U L and U R respectively.For what concerns our first aim about possible characterization of solutions, ourmain result relies on the following lemma Lemma 1.1.
Let u be any nontrivial solution to the problem (1) . Then there exist lim x → + ∞ N ( x ) = l R ∈ nq λ Rj o + ∞ j =1 ∪ { + ∞} lim x →−∞ N ( x ) = l L ∈ nq λ Lj o + ∞ j =1 ∪ { + ∞} . We fix two numbers d R ∈ (cid:8) λ Rj (cid:9) + ∞ j =1 and d L ∈ (cid:8) λ Lj (cid:9) + ∞ j =1 and define the followingset S = (cid:26) u C solution to (1) such that if Z C R,L |∇ u | = + ∞ then l R,L ≤ √ d R,L (cid:27) . (4)Now we can state our main result as follows Theorem 1.2.
The set S defined in (4) is a linear space of dimensiondim S = m ( d R ) + m ( d L ) where m denotes the Morse index.In particular, if we restrict to those solutions with finite energy on one hand ofthe domain, namely C L , the set (5) S L = (cid:26) u C solution to (1) with Z C L |∇ u | < + ∞ and l R ≤ √ d R (cid:27) is a linear space of dimension m ( d ) , where m denotes again the Morse index. For the reader’s convenience we recall that in this context the Morse index of theeigenvalue λ L,Rk is the sum of the multiplicity of the eigenvalues λ L,Rj with j ≤ k .If we focus our attention just on positive solutions, as a byproduct of the previousresult we obtain the following theorem, which corresponds to the particular case d R = λ R in Equation (5): Theorem 1.3.
There exists a unique (up to a multiplicative constant) positive C solution v L to the problem (1) , provided (6) Z C L (cid:12)(cid:12) ∇ v L (cid:12)(cid:12) < ∞ . ARMONIC FUNCTIONS IN UNION OF CHAMBERS 3
Moreover, • if U R is bounded, then v L is asymptotic to e √ λ x ψ ( y ) as x → ∞ uniformlywith respect to y ∈ U R , being ψ and λ the first eigenfunction and eigen-value respectively to the problem (ii) in Equation (3) with a little abuse ofnotation; • if C R is the whole right halfspace of R N , then v L is asymptotic to x as x → ∞ uniformly with respect to y ∈ R N − .An analogous statement defines v R . Finally, all positive solutions to problem (1) are positive convex combinations of v L and v R . Remark 1.4.
As appearing in Theorem 1.2, the dimension of the space S L evenin Theorem 1.3 is related to the multiplicity of the first eigenvalue λ R . Therefore,we stress the uniqueness stated in Theorem 1.3 relies essentially on the assumption U R is a connected domain in R N − . In the framework of positive solutions, Theorem 1.3 is not a brand new result,as we can find it within the so-called
General Martin Theory . This is a quitegeneral theory which provides a one-to-one correspondence between regular positive solutions and the points of the so-called minimal Martin boundary by means of finiteharmonic measures supported on the minimal Martin boundary. We then foreseethat the Martin boundary is useful to gain some information about the number ofthe (linearly independent) positive solutions to a differential equation, whenever thedifferential operator satisfies several minimal assumptions. We then find worthwhilerecalling some general concepts of the
General Martin Theory . To this aim we referto the book by Ross Pinsky [17], chapter 7.In order to state the known results, let us consider a quite general differentialoperator L on a domain D ⊆ R N satisfying the following Assumptions 1.5.
For any D ′ ⊂⊂ D the operator L is of the form L = ∇ · a ∇ + b · ∇ + V , with a i,j , b i ∈ C ,α ( D ′ ) , V ∈ C α ( D ′ ) and P i,j ( x ) v i v j > for all v ∈ R N \ { } and for all x ∈ D ′ . It is defined P L ( D ) = { u ∈ C ,α ( D ) : Lu = 0 and u > D } the set of all regular positive solutions; and, fixed a point x ∈ D and, denoting G the Green’s function, define the Martin kernel as k ( x, y ) = G ( x, y ) G ( x , y ) y = x, y = x y = x , x = x y = x = x . Definition 1.6.
A sequence { y n } n ⊂ D for which the limit lim n →∞ k ( x, y n ) ∈P L ( D ) is called a Martin sequence . Two Martin sequences which have the samelimit are called equivalent . The collection of such equivalence classes is called the
Martin boundary for L on D . We briefly mention that the Martin boundary does not depend on the choiceof the fixed point x in the Martin kernel and it can be endowed with a suitabletopology; we do not enter into the details, since they go beyong our specific aim.More related to our work, we find the following Definition 1.7.
A function u ∈ P L ( D ) is called minimal if whenever v ∈ P L ( D ) and v ≤ u then in fact v = cu for some constant c ∈ (0 , . LAURA ABATANGELO, SUSANNA TERRACINI
Given a point ξ of the Martin boundary, the notation k ( x ; ξ ) means that, up topositive multiples, k ( x ; ξ ) = lim n → + ∞ k ( x ; y n ) where y n is any representative of the equivalence class ξ .A point ξ on the Martin boundary is called a minimal Martin boundary point if k ( x, ξ ) is minimal. Theorem 1.8 (Martin Representation Theorem) . Let L satisfy the Assumptions1.5 on a domain D ⊆ R N and assume that L is subcritical. Then for each u ∈P L ( D ) there exists a unique finite measure µ u supported on the minimal Martinboundary Λ such that u ( x ) = Z Λ k ( x, ξ ) µ u ( dξ ) . Conversely, for each finite measure µ supported on the minimal Martin boundary Λ , u ( x ) := Z Λ k ( x, ξ ) µ u ( dξ ) ∈ P L ( D ) . As already mentioned this is the basic theorem in order to state a one-to-one cor-respondence between the elements of the set P L ( D ) and the points of the minimalMartin boundary by means of finite harmonic measures supported on the minimalMartin boundary.The General Martin Theory covers even our case of domains formed by differentchambers by means of the following theorems Theorem 1.9 (Theorem 6.6 in [17]) . Let D be a non-compact open N -dimensional C ,α -Riemannian manifold with m ends, i.e. it can be represented in the form D = F ∪ E ∪ . . . ∪ E m with F is bounded and closed, E i is open, E i ∩ E j = ∅ for i = j and F ∩ E i = ∅ . Let L on D satisfy the assumptions 1.5 and be subcritical.Then the Martin boundary for L decomposes into m components in the followingsense: if { x n } n ⊂ D is a Martin sequence, then all but a finite number of its termslie in E i for some i = 1 , . . . , m . Corollary 1.10 (Corollary 6.7 in [17]) . Let L satisfy assumptions 1.5 and be sub-critical on the domain D = ( α, β ) where −∞ ≤ α < β ≤ + ∞ . Then the Martinboundary of L on D consists of two points. More specifically, a sequence { x n } n withno accumulation points in D is a Martin sequence if and only if lim n →∞ x n = α or lim n →∞ x n = β . Moreover (see Proposition 5.1.3 in [17]), in this case P L ( D ) is 2-dimensional, whichmeans that, via the Martin Representation Theorem, the minimal Martin boundaryconsists exactly of two points.In particular, a suitable N -dimensional generalization of the previous Corollarycovers the case of Theorem 1.3 in the present paper. More precisely, the Martinboundary of our domain Ω consists in its topological boundary together with theunion of two points, which can be identified, roughly speaking, with the two endsof the domain. Taking into account Dirichlet boundary conditions in our problem(1), this means that problem (1) has got exactly two linearly independent positivesolutions, as we are able to show, too. Thus, our Theorem 1.3 does not provideany additional information to the known results provided by the General MartinTheory, except maybe by gaining greater understanding of the positive solutions’space P L ( D ): under Dirichlet boundary conditions, a basis of P L ( D ) is formed bytwo positive regular functions which have finite energy on one end of the domain,whereas on the other end they tend to infinity. Further, we are able to describeexactly the divergent behavior, as well as to prove every element in P L ( D ) to ARMONIC FUNCTIONS IN UNION OF CHAMBERS 5 be minimal according to Definition 1.7. Then, our original contribution does nolonger refer strictly to the result, but rather to the method: an upper bound forthe Almgren quotient of possible solutions is the key ingredient for existence ofsolutions. In the case of positive solutions the specific threshold for the Almgrenfrequency is set by the positivity assumption of the solutions, but the method canbe extended even to sign-changing solutions, which are not included in the GeneralMartin Theory, providing a stronger result that is Theorem 1.2.As already mentioned, we are enforced to consider infinite energy solutions. Asa second point of our work, we follow the idea that normalizing their necessarydivergent asymptotic behavior, we force the asymptotic (vanishing) behavior of thesolution even at the other hand of the domain. For a pair of cylinders, the rate ofgrowth at + ∞ can be related with the rate of vanishing at −∞ by means of theevaluation of a transfer operator (see Section § § C R either when its sectionis bounded or when it is a whole hyperplane, and we investigate their possiblebehavior at infinity; in Section 3 we collect the previous results in order to proveTheorem 1.3 (see also Theorem 3.2) and Theorem 1.2. In Section 4 we study therelation between the asymptotic behavior of positive solutions at + ∞ and −∞ ,generalizing our results to domains which are union of more than two chambers inthe very last subsection.2. Existence and uniqueness of a positive harmonic function on C R . We claim the following
Theorem 2.1.
There exists a unique (up to a multiplicative constant) positivesolution u to the problem (7) (cid:26) ∆ v = 0 in C R v = 0 on ∂C R if U R is bounded or it is a whole hyperplane. In the first case it will be (8) v ( x, y ) = (cid:16) e √ λ x − e −√ λ x (cid:17) ψ ( y ) , being λ and ψ the first eigenvalue and the first eigenfunction respectively of theproblem (3) item (ii); whereas in the second case it will be (9) v ( x, y ) = x denoting x the first variable in R N . Remark 2.2.
We stress the aforementioned solutions have an infinite energy.
In order to prove this theorem, we will study the two cases separately.2.1.
The case U R bounded. It is quite simple to prove that the function v definedin (8) is a solution to the problem (7). Moreover, we stress it is asymptotic toe √ λ x ψ ( y ) as x → ∞ . We aim to prove it is in fact the unique solution. Proposition 2.3.
The function v defined in (8) is the unique solution up to mul-tiplications by constants. LAURA ABATANGELO, SUSANNA TERRACINI
The proof relies essentially on three different tools: the so-called “Phragm`en-Lindel¨of Principle”, which may be read as a comparison principle on unboundeddomains, a boundary version of the Harnack inequality, and an Almgren–type ar-gument. For similar arguments, see [14, 16].Let us recall the well-known Phragm´en–Lindel¨of Principle stated for the Laplaceoperator:
Theorem 2.4 (Phragm´en–Lindel¨of Principle, [18]) . Let D be a domain, boundedor unbounded, and let u satisfy − ∆ u ≤ in D , u ≤ on Γ ,where Γ is a subset of ∂D . Suppose that there is an increasing sequence of boundeddomains D ⊂ D ⊂ · · · ⊂ D k ⊂ · · · with properties (1) each D k is contained in D ; for each point x ∈ D there is an integer N suchthat x ∈ D N ; (2) the boundary of each D k consists in two parts Γ k and Γ k ′ where Γ k is asubset of Γ and Γ k ′ is a subset of D .Further, suppose there exists a sequence { w k } which satisfies w k ( x ) > on D k ∪ ∂D k , − ∆ w k ≥ in D k .Assume there is a function w ( x ) with the property that at each point x ∈ D theinequality w k ( x ) < w ( x ) holds for all k above a certain integer N x . If u satisfies the growth condition lim inf k →∞ (cid:26) sup Γ k ′ u ( x ) w k ( x ) (cid:27) ≤ then u ≤ in D . Lemma 2.5 (Boundary Harnack inequality, [10]) . Let D ⊂ R N , N ≥ , be aLipschitz domain and let V an open set such that V ∩ ∂D = ∅ . Suppose W is adomain such that W ⊂ D , W ⊂ V and let P be a point in W . Then there is aconstant C > such that if u and v are nonnegative harmonic functions in D whichvanish on V ∩ ∂D and satisfy u ( P ) ≤ v ( P ) then u ( P ) ≤ Cv ( P ) for all P ∈ W . Thanks to these two preliminary results, we can state
Proposition 2.6.
Let u and v be two different positive solutions to the problem (7) . Then u = O ( v ) .Proof. According to the notation in Theorem (2.4), let D k denotes the rectangle { ( x, y ) ∈ C R , k − < x < k and y ∈ U R } and Γ ′ k := { ( k − , y ) , y ∈ U R } ∪{ ( k, y ) , y ∈ U R } . We can claim that(10) lim inf k → + ∞ sup Γ ′ k u ( x, y ) v ( x, y ) > . If not, Theorem (2.4) would apply with w k = vχ D k + ε where ε is any positiveconstant. Thus, we would obtain u ≤ u = 0, a contradiction.We define b k = max Γ ′ k u ( x, y ) v ( x, y ) a k = min Γ ′ k u ( x, y ) v ( x, y ) . ARMONIC FUNCTIONS IN UNION OF CHAMBERS 7
Then, Equation (10) implies b k ≥ C > k large enough and then, by Lemma(2.5) u ( x,y ) v ( x,y ) ≤ C in C R . We can rewrite the previous inequality as (cid:12)(cid:12)(cid:12) u ( x ,y ) v ( x ,y ) − u ( x ,y ) v ( x ,y ) (cid:12)(cid:12)(cid:12) ≤ C u ( x ,y ) v ( x ,y ) for any ( x , y ) , ( x , y ) ∈ C R in order to obtain(11) 1 ≤ b k a k ≤ C. This means that the two sequences a k and b k share the same asymptotic behavior.Moreover, their divergence to ∞ cannot occur. If they diverged to + ∞ , then theinverse quotient vu would be uniformly convergent to zero as x → + ∞ , Theorem(2.4) would apply and provide the contradiction v ≤ C ≤ uv ≤ C for some positive constants C and C . (cid:3) Proposition 2.7.
Any solution to (7) is asymptotic to e √ λ k x ψ k ( y ) as x → ∞ uniformly with respect to y ∈ U R for some k ∈ N , where ψ k denotes one of eigen-functions relative to the k -th eigenvalue of the problem (13) (cid:26) ∆ ψ k = λ k ψ k in U R ψ k = 0 on ∂U R . To prove this last step we need several preliminary results, which are stated inLemma (2.8), Lemma (2.11) and Lemma (2.10).Being v any solution to (7), we recall the Almgren frequancy function:(14) N ( v )( x ) := R Ω x |∇ v | R Γ x v , where Ω x := { ( ξ, η ) ∈ Ω : 0 < ξ < x } and Γ x := { ( x, y ) : y ∈ U R } . Lemma 2.8.
Given a solution v to (7) , the function N ( v )( x ) − C √ λ e −√ λ x ismonotone increasing with respect to x .Proof. It is simple to see that D ′ ( x ) = Z Γ x |∇ v | H ′ ( x ) = Z Γ x v v x . Multiplying the equation by v x and integrating by parts we obtain Z Ω x ∇ v ∇ v x = Z Γ x v x − Z Γ v x ;whereas differentiating it and multiplying it by v we obtain Z Ω x ∇ v ∇ v x = Z Γ x v v xx = − Z Γ x v ∆ y v = Z Γ x v y ;from which Z Γ x v x = Z Γ x v y + Z Γ v x . LAURA ABATANGELO, SUSANNA TERRACINI
Let us compute the derivative ddx N ( x ) = R Γ x v x + v y R Γ x v − (cid:16)R Γ x v v x (cid:17) (cid:16)R Γ x v (cid:17) = 2 R Γ x v x − R Γ v x R Γ x v − (cid:16)R Γ x v v x (cid:17) (cid:16)R Γ x v (cid:17) ≥ − R Γ v x R Γ x v ≥ − C e √ λ x for some positive C : the first inequality is given by the H¨older inequality and thesecond one is implied by Proposition (2.6). (cid:3) Remark 2.9.
Under our hypothesis we can claim N ( v )( x ) admits a finite limitas x → ∞ . Indeed, it admits a limit in view of Lemma (2.8) , and such a limit isfinite since v is O (e √ λ x ψ ( y )) from Proposition (2.6) , so that N ( v ) is a boundedfunction from above. In order to detect lim x → + ∞ N ( v )( x ) we introduce the sequence of normalizedfunctions v ξ ( x, y ) := v ( x + ξ, y ) (cid:16)R Γ ξ v ( ξ, y ) (cid:17) / for ξ ∈ R , x ∈ (0 , y ∈ U R . Lemma 2.10. As ξ → ∞ the sequence { v ξ } ξ converges C -uniformly on compactsets of the cylinder { ( x, y ) ∈ R N : x ∈ R and y ∈ U R } to a function harmonic onthe cylinder whose N ( x ) is identically constant.Proof. First we observe N ( v ξ )( x ) = N ( v )( x + ξ ) ≤ N for all x ∈ (0 ,
1) and forall ξ ∈ R , thanks to the definition of v ξ and to Remark (2.9). Thus, R Ω x |∇ v ξ | ≤ N R Γ x v ξ where we recall Z Γ x v ξ = R U R v ( x + ξ, y ) dy R U R v ( ξ, y ) dy . Via Harnack inequality, if x ranges in a compact set, the previous ratio is boundedfrom above by a fixed constant, then also the H -norm is uniformly bounded fromabove. Thus, there exists a subsequence at least C -uniformly convergent to afunction w which is harmonic on the whole cylinder. It holds for any fixed x ∈ R N ( v ξ )( x ) = N ( v )( x + ξ ) → N as ξ → ∞ , and thenlim ξ →∞ N ( v ξ )( x ) = N ∀ x ∈ R . Moreover this happens for any convergent subsequence. Then we can conclude thewhole sequence v ξ is C -uniformly convergent to a function w which is harmonic onthe whole cylinder and has N ( x ) identically constant. (cid:3) Lemma 2.11.
Let w be a solution to (cid:26) ∆ w = 0 on { ( x, y ) ∈ R N : x ∈ R and y ∈ U R } w = 0 if y ∈ ∂U R ARMONIC FUNCTIONS IN UNION OF CHAMBERS 9 with R y ∈ URx ≤ x |∇ w | < ∞ for all x . Then N ( w )( x ) is identically constant in x ifand only if w ( x, y ) = e √ λ k x ψ k ( y ) for some k ∈ N , being λ k the k -th eigenvalue ofproblem (13) and ψ k one of its relative eigenfunctions.Proof. Note for such solutions it holds R Γ x w x = R Γ x w y , so that ddx N ( w )( x ) = 2 R Γ x w x R Γ x w − (cid:16)R Γ x w w x (cid:17) k w k L (Γ x ) k w x k L (Γ x ) . Thus, N is identically constant in x if and only if we have an equality in the H¨olderinequality, that is (cid:18)Z Γ x w w x (cid:19) = Z Γ x w Z Γ x w x . This happens if and only if w x ( x, y ) = λ ( x ) w (0 , y ), which leads to w ( x, y ) = w (0 , y ) (cid:26) Z x λ ( t ) dt (cid:27) . If we substitute this expression in N ( w )( x ) ≡ N we obtain λ ( x ) = N (cid:26) Z x λ ( t ) dt (cid:27) which is a differential equation whose solution is λ ( x ) = N e Nx ; then w ( x, y ) =e Nx w (0 , y ), from which w ( x, y ) = e √ λ k x ψ k ( y ) imposing w is harmonic and zero onthe boundary. (cid:3) Proof of Proposition 2.7 . We exploit the following chain of equalities:lim x → + ∞ N ( v )( x ) = lim ξ → + ∞ N ( v )( x + ξ ) = lim ξ → + ∞ N ( v ξ )( x ) = N ( w )( x ) ≡ p λ k . Therefore Lemma 2.11 gives immediately the proof. (cid:3)
Proof of Proposition 2.3 . By Remark 2.9 we need to prove A = B . This is astraightforward consequence of Proposition 2.7 where positivity of solutions forces λ k = λ . (cid:3) The case U R hyperplane. The existence of a positive solution in this caseis immediately proved by considering the function ¯ v ( x, y ) := x , where we recall x denotes the first variable in R N .We aim to prove this is in fact the unique solution to the problem (7) when U R is a whole hyperplane of R N , namely { x = 0 } . To do this, we follow the sameoutline as before.Let B r be the ball in R N centered in the origin with radius r , we denote C r := C R ∩ B r and Γ r := ∂B r ∩ C R . Proposition 2.12.
Any positive solution to the problem (7) is O ( x ) as x → ∞ uniformly with respect to y . The proof of this proposition is essentially the same as in the previous case,provided the domains D k are now defined as C k . Proposition 2.13.
Any solution to (7) is asymptotic to r N v (1 , θ ) as r → ∞ uni-formly with respect to θ ∈ S N − in such a way that [ N ( N −
1) + N ( N − is aneigenvalue for the spherical Laplacian and v (1 , θ ) is one of its relative eigenfunc-tions. To prove this last step we need several preliminary results, which we state in theLemma (2.14), Lemma (2.16) and Lemma (2.15).We aim to pursue again an Almgren-type argument on the domains C r . Being v any solution to (7), let us introduce the following Almgren-type quotient(15) N ( v )( r ) := r − N R C r |∇ v | r − N R Γ r v =: D ( r ) H ( r ) . Lemma 2.14.
Given a solution v to (7) , the quotient N ( v )( r ) is monotone in-creasing with respect to r .Proof. It is quite simple to see(16) H ′ ( r ) = 2 r − N Z Γ r v v r . Testing the equation by v we obtain(17) H ′ ( r ) = 2 r − N Z C r |∇ v | = 2 r D ( r );from which D ( r ) = ( r/ H ′ ( r ).On the other hand we claim(18) D ′ ( r ) = 2 r − N Z Γ r v r . Indeed,(19) D ′ ( r ) = (2 − N ) r − N Z C r |∇ v | + r − N Z Γ r |∇ v | ;testing the equation with ∇ v · ( x, y ) and integrating by parts we obtain(20) Z C r ∇ v · ∇ ( ∇ v · ( x, y )) = r Z Γ r v r , which is in fact(21) Z C r ∇ v · ∇ ( ∇ v · ( x, y )) = − N − Z C r |∇ v | + r Z Γ r |∇ v | via integration by parts. From (19), (20) and (21) we immediately obtain (18).Now, the derivative of N is of course N ′ ( r ) = D ′ ( r ) H ( r ) − D ( r ) H ′ ( r ) H ( r ) , and werecall that D ( r ) H ′ ( r ) = ( r/ H ′ ( r )) , so that N ′ ( r ) = 2 r − N H ( r ) (Z Γ r v r Z Γ r v − (cid:18)Z Γ r v r v (cid:19) ) ≥ (cid:3) Now we introduce the sequence of normalized functions v r ( x, y ) := v ( rx, ry ) (cid:16)R Γ / v ( rx, ry ) (cid:17) / for r > . Lemma 2.15. As r → ∞ the sequence { v r } r converges C -uniformly on C to afunction which is harmonic on the whole halfspace and whose N ( x ) is identicallyconstant.Proof. Here the proof is essentially the same as in Lemma (2.10). (cid:3)
ARMONIC FUNCTIONS IN UNION OF CHAMBERS 11
Lemma 2.16.
Let v any non-trivial solution to the problem (7) . Then its Almgren’sfrequency function is identically constant equal to N if and only if v ( r, θ ) = r N v (1 , θ ) in such a way that [ N ( N − N ( N − is an eigenvalue for the spherical Laplacianand v (1 , θ ) is one of its relative eigenfunctions.Proof. If the derivative of the frequency function is identically zero, then an equalitymust hold in the H¨older inequality, so that v r ( r, θ ) = λ ( r ) v (1 , θ ), that is v ( r, θ ) = v (1 , θ ) { Z r λ ( t ) dt } . Imposing D ( r ) /H ( r ) = ( r/ H ′ ( r ) /H ( r )) = N we obtain N = r Z Γ r v v r Z Γ r v = r Z Γ r v (1 , θ ) λ ( r ) (cid:18) Z r λ ( t ) dt (cid:19) dθ Z Γ r v (1 , θ ) (cid:18) Z r λ ( t ) dt (cid:19) dθ = r λ ( r )1 + Z r λ ( t ) dt . The solution of the ordinary differential equation r λ ( r ) = N (cid:26) Z r λ ( t ) dt (cid:27) is indeed R r λ ( t ) dt = r N −
1, which leads to v ( r, θ ) = r N v (1 , θ ). Imposing v isharmonic on the whole halfspace, we deduce the conditions on N and v (1 , θ ). (cid:3) Corollary 2.17.
The solution ¯ v defined in (9) is the unique positive solution tothe problem (7) up to multiplication by constants.Proof. Positivity assumption forces N = 1 in Proposition (2.13). This homogeneitydegree together with v (0 , y ) = 0 implies v ( x, y ) = x . (cid:3) Solutions on
Ω3.1.
Positive solutions on Ω with finite energy on C L . The following propo-sition can be easily proved.
Proposition 3.1.
Let us consider the case
Ω := C L ∪ C R where U R is the hy-perplane { x = 0 } . Let Φ be unique normalized positive solution of (7) , extendedas vanishing outside the semicylinder. There exists a unique positive solution v toproblem (1) such that u = v − Φ has finite energy on Ω : it is the solution of theminimum problem (22) min u ∈D , (Ω) Z Ω |∇ u | − Z Γ ∂ Φ ∂x | x =0 u. We note that the minimizer u is not a C solution. Indeed, on one hand for every ϕ ∈ D , (Ω) we have(23) Z Ω ∇ u ∇ ϕ = Z Γ ∂ Φ ∂x | x =0 ϕ ;whereas on the other hand, multiplying the equation by ϕ nd integrating by partsover C L and C R we obtain Z Ω ∇ u ∇ ϕ = Z C L ∪ C R ∇ u ∇ ϕ = Z Γ ϕ (cid:18) − ∂u R ∂x | x =0 + ∂u L ∂x | x =0 (cid:19) where u L := uχ C L and u R is defined similarly. Thus,(24) ∂u L ∂x | x =0 = ∂u R ∂x | x =0 + ∂ Φ ∂x | x =0 , in the sense that must be specified yet (see Section 4). In order to abtain a C solution, we need to consider the sum v = u + Φ instead of u .Furthermore, if the test function ϕ has compact support far away from Γ, Equa-tion (23) shows that the minimum is a harmonic function in Ω \ Γ. In this way, ifwe are looking for a harmonic function u + Φ on the whole Ω, Φ must be the unique(up to multiplication by constants) solution to the problem (7) (see the previoussection). In other words, given the function Φ solution to the problem (7), thefunction u + Φ is the unique solution to the problem (1) with finite energy on theleft. Furthermore, it is possible to prove that any positive solution to the problem(1) with finite energy on the left takes the form u + Φ for a certain Φ solution tothe problem (7), in order to state the following Theorem 3.2.
There exists a unique (up to multiplicative constants) solution tothe problem (1) having finite energy on C L and satisfying lim x → + ∞ N ( x ) = q λ R . It is asymptotic to a multiple of (8) if U R is bounded, whereas it is asymptotic toa multiple of (9) if U R is a whole hyperplane.Proof. The proof follows the same outline as the proof of Theorem (2.1).Propositions (2.6) and (2.12) can be stated and proved in the same way choosing D k = { ( x, y ) ∈ Ω , − k < x < k } in the first case and D k = { ( x, y ) ∈ Ω , − k < x ≤ } ∪ C k in the second case.We conclude the proof throughout an Almgren type argument on the domainsΩ x = { ( ξ, η ) ∈ Ω : ξ < x } (but now Γ = { x = 0 } ∩ ∂ Ω) in the first case andΩ r = { ( x, y ) ∈ Ω : x ≤ } ∪ C r in the second case. In both cases the computationsare the same. (cid:3) Remark 3.3.
As already highlighted in [1] , the minimum in Equation (22) isstrictly related to the concept of compliance . We define (25) C (Γ) := max w ∈D , (Ω) (cid:18) Z Γ ∂ Φ ∂x | x =0 w − Z Ω |∇ w | dx (cid:19) the compliance functional associated to a force concentrated on the section Γ in theflavor of [4, 5] . In general, the compliance functional measures the rigidity of amembrane subject to a given (vertical) force: the maximal rigidity is obtained byminimizing the compliance functional C (Γ) in a certain class of admissible regions Γ . Infinite energy solutions.
Up to now, we have proved that given a positiveprofile φ on U R , there exist at least two positive solutions to the problem ∆ w = 0 , in C R ; w = φ, on U R ; w = 0 , on ∂C R \ U R .(26)Indeed, one has finite energy and it is the minimum of the Dirichlet realization on C R , we name it u ; whereas the second one is obtained from the previous simplyadding a multiple of the solution v of the Theorem (2.1). Theorem 3.4.
Any positive solution to the problem (26) is a linear combination u + cv with c ≥ , being u and v as mentioned above. ARMONIC FUNCTIONS IN UNION OF CHAMBERS 13
Proof.
Let w > u since in this case we have uniqueness of solution.If w has an infinite energy, consider the difference w − u . Then, we can immedi-ately state that lim inf x → + ∞ sup Γ ′ x w − uv > w − u ≤
0, a contradic-tion. As in the proof of Proposition (2.6) we obtain(27) c ≤ w − uv ≤ c . We follow the same outline as before and study the Almgren quotient N ( x ) onΩ x := { ( ξ, η ) ∈ R N : ξ ∈ (0 , x ) , η ∈ U R } . As before, N ( x ) = D ( x ) H ( x ) where D ( x ) = R Ω x |∇ w | and H ( x ) = R Γ x w being Γ x = { ( x, η ) : η ∈ U R } . Multiplyingthe Laplace equation by w itself, we obtain Z Ω x |∇ w | = Z Γ x w w x − Z Γ w w x . Multiplying the Laplace equation by w x we obtain Z Ω x ∇ w · ∇ w x = Z Γ x w x − Z Γ w x where Z Ω x ∇ w · ∇ w x = Z ∂ Ω x |∇ w | ν · e = Z Γ x |∇ w | − Z Γ |∇ w | so that Z Γ x |∇ w | = Z Γ |∇ w | + 2 Z Γ x w x − Z Γ w x . Thus, the derivative N ′ ( x ) = D ′ ( x ) H ( x ) − D ( x ) H ′ ( x ) H ( x )= (cid:18)Z Γ |∇ w | + 2 Z Γ x w x − Z Γ w x (cid:19) Z Γ x w − (cid:18)Z Γ x w x − Z Γ w x (cid:19) Z Γ x w w x (cid:18)Z Γ x w (cid:19) = 2 (Z Γ x w x Z Γ x w − (cid:18)Z Γ x w w x (cid:19) ) + Z Γ w y Z Γ x w − Z Γ w x Z Γ x w + 2 Z Γ w w x Z Γ x w w x (cid:18)Z Γ x w (cid:19) ≥ Z Γ w y − w x Z Γ x w + 2 Z Γ w w x Z Γ x w w x (cid:18)Z Γ x w (cid:19) via H¨older inequality. Thanks to the estimate (27) the function Z Γ w y − w x Z Γ x w + 2 Z Γ w w x Z Γ x w w x (cid:18)Z Γ x w (cid:19) ∈ L (0 , + ∞ ) , so that N ( x ) admits a limit as x → + ∞ . Moreover, such a limit is finite since thequantities a k and b k cannot diverge to infinity via Lemma (2.5) and Theorem (2.4)as in the proof of Proposition (2.6). We conclude the proof invoking Proposition(2.7). (cid:3) Theorem 3.5.
Any positive solution to the problem (1) is a linear combination c L v L + c R v R with c L , c R ≥ (at least one of the two constants must be differentfrom zero), where v L and v R are the solutions in the Theorem (3.2) with finiteenergy on C L and C R respectively.Proof. The proof relies essentially on the Phragmen-Lindel¨of Principle. Let w >
0a solution to the problem (1). We simply apply the aforementioned principle on w − ( c L v L + c R v R ) comparing it with c L v L + c R v R + 1. In this case we choosethe sequence of domains D k as the union { ( ξ, η ) : ξ ∈ (0 , k ) η ∈ U R } ∪ { ( ξ, η ) : ξ ∈ ( − k, η ∈ U L } whenever U R is bounded, whereas { ( ξ, η ) : ξ ∈ ( − k, η ∈ U L } ∪ ( C R ∩ B k ( x )) where x is the junction point between C L and C R whenever U R is the whole hyperplane. (cid:3) We stress that such solutions have lim x →±∞ N ( x ) lowest as possible in order tobe nontrivial, that is p λ R and p λ L respectively. We note that v L is asymptoticto a multiple of e − √ λ L x ψ L as x → −∞ . Does the reverse implication hold true?Not exactly, but we can state Theorem 3.6.
The function set S := (cid:26) w solution to (1) s.t. lim x → + ∞ N ( x ) ≤ q λ R or lim x →−∞ N ( x ) ≤ q λ L (cid:27) is a linear space of dimension 2 and { v L , v R } is a basis, being v L , v R as in theprevious theorem. We remark that in this case no positivity assumption can be made on solutions,but we can state that they change their sign at most just one time.
Remark 3.7.
The procedure presented up to now works even in the case the upperbound for the Almgren frequency is set to be a k -th eigenvalue of the problem (13) with k ≥ , up to minor modifications. This allows us to extend Theorem 3.2providing the following Theorem 3.8.
Let λ Rk be the k -th eigenvalue of the problem (13) and let us denote m Rk its multiplicity. Then, there exist exactly m Rk linearly independent solutions tothe problem (1) having finite energy on C L and satisfying lim x → + ∞ N ( x ) = q λ Rk . Each of them is asymptotic to a multiple of e √ λ Rk x ψ Rk ( y ) , being ψ Rk one the eigen-functions relative to λ Rk , if U R is bounded, whereas each of them is asymptotic toa multiple of r N v (1 , θ ) if U R is a whole hyperplane, in such a way that [ N ( N −
1) + N ( N − is an eigenvalue for the spherical Laplacian and v (1 , θ ) is one of itsrelative eigenfunctions.Thus, Theorem 1.2 is finally proved. Frequency transfer from two consecutive cylinders
Let us focus our attention on the unique solution which has finite energy at −∞ .We are talking about u + Φ, where u is the minimum of (22) and Φ the uniquesolution of the problem (7). Thanks to the uniqueness of such a solution, whenever ARMONIC FUNCTIONS IN UNION OF CHAMBERS 15 we impose the exact behavior at x → + ∞ , the asymptotic behavior for x → −∞ is determined. We aim to investigate how such a fact occurs. Remark 4.1.
Via the Phragm`en-Lindel¨of Theorem, the restrictions u L := uχ C L and u R := uχ C R are u L ( x, y ) = O (e √ λ L x ϕ L ( y )) whereas u R ( x, y ) = O (e − √ λ R x ϕ R ( y )) .Indeed, given the particular domain’s geometry, u L and u R can be written as P k c Lk ( x ) ϕ Lk ( y ) and P k c Rk ( x ) ϕ Rk ( y ) respectively. Then, imposing that ∆ u i = 0 for i = L, R andthat their energy is finite, they take the form (28) u L ( x, y ) = X k α k e √ λ Lk x ϕ Lk ( y ) u R ( x, y ) = X k β k e − √ λ Rk x ϕ Rk ( y ) where the eigenfunctions { ϕ Lk } and { ϕ Rk } are basis for L ( U L ) and L ( U R ) respec-tively. The key points for this analysis are Equation (24) together with the fact thatthe two profiles of u L and u R coincides on the boundary { ( x, y ) ∈ Ω , x = 0 } .In particular, Equation (24) makes sense in a distributional sense, so that itshould be read in the dual space H − / ( U L ). Indeed, both u L and u R are D , func-tions on C L and C R respectively, then their traces on { x = 0 } are H / functionsand then their partial derivatives on { x = 0 } are in H − / ( U L ) and H − / ( U R )respectively. In order to specify these concepts, we introduce the following spaces h / L := { ( α j ) j s.t. X j (cid:0) λ Lj (cid:1) / α j < + ∞} , h / R := { ( α j ) j s.t. X j (cid:0) λ Rj (cid:1) / α j < + ∞} , being λ Lj and λ Rj the eigenvalues of ∆ N − on U L and U R respectively, and operators U : h / L −→ h / R α = ( α j ) j ( U ( α )) k = U kj α j e U : H / ( U L ) −→ H / ( U R ) u = α j ϕ Lj e U u = ( U kj α j ) ϕ Rk . Moreover, e U ∗ : H − / ( U R ) −→ H − / ( U L ) will be the adjoint operator.These mean that Equation (24) is correctly read as(29) ∂u L ∂x | x =0 = e U ∗ (cid:18) ∂u R ∂x | x =0 (cid:19) + e U ∗ (cid:18) ∂ Φ ∂x | x =0 (cid:19) in H − / ( U L ) . which is(30) α j q λ jL ϕ jL − e U ∗ (cid:18) β k q λ kR ϕ kR (cid:19) = e U ∗ (cid:0) γ k ϕ kR (cid:1) where γ k are the coefficients of ∂ Φ ∂x | x =0 . Thus the equation for the coefficientsbecomes α j q λ j L − U ∗ (cid:18) β k q λ kR (cid:19) = U ∗ ( γ k ) α j q λ jL − U ∗ (cid:18)q λ kR U jk α j (cid:19) = U ∗ ( γ k )(31)since β k = α j U jk from the fact u L (0 , y ) = u R (0 , y ) = P k β k ψ Rk ( y ). Equation (31) becomes Λ L α − U ∗ Λ R U α = α (cid:0) Λ L − U ∗ Λ R U (cid:1) α = α (cid:0) I − (Λ L ) − U ∗ Λ R U (cid:1) α = α (32)where α = (Λ L ) − U ∗ ( γ k ), Λ R the diagonal operator between h / R and h − / R whichmultiplies by the square root of the eigenvalues q λ jR , which is in fact an isometrybetween those two spaces, whereas (Λ L ) − is analogously an isometry from h − / L into h / L . Proposition 4.2.
The operator T = (Λ L ) − U ∗ Λ R U is a contraction on h / L .Proof. Proving that U ∗ Λ R U has got the same eigenvalues of Λ R will be sufficientto our aim. Once we have that, we apply the well-known Weyl’s law: being λ j the j -th eigenvalue of the Laplacian on a bounded regular domain Ω of dimension n ,the following asymptotic behavior holds λ j ∼ C n j /n | Ω | − /n as j → + ∞ and C n is a constant depending only on the dimension n . Then, not only the ratio λ Rj λ Lj < T is a contraction at every point, but also the ratio is uniformly far awayfrom 1, so that T is a contraction on the whole space h / L .Let us study the eigenvalues of U ∗ Λ R U . First of all we note that U is abounded operator from h / L into h / R with operator norm less or equal to 1. Infact, e U is an isometry from L ( U L ) into L ( U R ) as well as from H ( U L ) into H ( U R ). Being H / ( U L ) and H / ( U R ) intermediate spaces [ L ( U L ) , H ( U L )] / and [ L ( U R ) , H ( U R )] / respectively, the operator e U : H / ( U L ) → H / ( U R ) hasoperator norm (cid:13)(cid:13)(cid:13) e U (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) e U (cid:13)(cid:13)(cid:13) L ( U L ) ,H ( U L ) · (cid:13)(cid:13)(cid:13) e U (cid:13)(cid:13)(cid:13) L ( U R ) ,H ( U R ) ≤ h / L ⊂ h / R thanks to the relation between the eigenvalues mentionedabove.Then, U is a partially isometric operator from h / R into h / R , since it is anisometry on the subspace h / L . So, UU ∗ = I on h / L (see [11]), and multiplying theeigenvalue equation ( U ∗ Λ R U ) α = µα by U we obtainΛ R U α = U µα = µ U α, the thesis. (cid:3) Thanks to the previous proposition, Equation (32) has a unique solution whichis nontrivial since α = 0.We note that whenever Φ is the solution to the problem (7), then the firstcomponent α of the solution α to Equation (32) is for sure different from zero. Thisis implied by the uniqueness of a positive solution to the problem (1). Moreover,from Remark (4.1) it describes the asymptotic behavior of u L for x → −∞ .4.1. Generalization to union of many chambers.
Let us consider a domainwhich is a union of several different chambers, such that the width of each chamberis negligible with respect to the corresponding length. We mean Ω = C ∪ . . . ∪ C N .The previous case Ω = C L ∪ C R is obviously covered by this type of domains.The proof of existence and uniqueness of a C positive harmonic function in sucha domain is a straightforward consequence of Theorem (1.3). As a matter of fact, ARMONIC FUNCTIONS IN UNION OF CHAMBERS 17 we can merely iterate its proof N − N denotes the number of the chambers.Moreover, suppose not to know the number of the chambers, but rather theasymptotic behavior of the solution for x → −∞ , that is(33) u ( x, y ) x →−∞ ∼ κ e √ λ x ϕ ( y )where λ denotes the first eigenvalue for ∆ N − for the first chamber and ϕ ( y ) itsrelative eigenfunction. Then it will be(34) κ = α · α · . . . · α N − , where α j are the analogues of α in Equation (28) for the couple of chambers( C j , C j +1 ). In this way we can deduce the number of the chambers from κ , i.e.from the solution’s asymptotic behavior at −∞ .Conversely, if the domain consists in the union of N chambers, we can imme-diately state that the asymptotic behavior of the unique C positive harmonicfunction for x → −∞ is (33) with κ given by (34). References [1] L. Abatangelo, V. Felli, S. Terracini,
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Laura Abatangelo: Dipartimento di Matematica e Applicazioni, Universit`a di MilanoBicocca, Piazza Ateneo Nuovo, 1, 20126 Milano (Italy)
E-mail address : [email protected] Susanna Terracini: Dipartimento di Matematica “Giuseppe Peano”, Via Carlo Al-berto 10, 10123 Torino (Italy)
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