On the connection problem for the p-Laplacian system for potentials with several global minima
aa r X i v : . [ m a t h . C A ] N ov ON THE CONNECTION PROBLEM FOR THE p -LAPLACIANSYSTEM FOR POTENTIALS WITH SEVERAL GLOBALMINIMA NIKOLAOS KARANTZAS
Abstract.
We study the existence of solutions to systems of ordinary differ-ential equations that involve the p -Laplacian for potentials with several globalminima. We consider the connection problem for potentials with two minimain arbitrary dimensions and with three or more minima on the plane. Introduction
We consider the problem of existence of solutions to systems of ordinary dif-ferential equations that involve the p -Laplacian operator, that is, systems of theform ( | u x | p − u x ) x − q ∇ W ( u ) = 0for vector-valued functions u and potentials W that possess several global minima.The corresponding problem was considered in the papers Alikakos and Fusco [4]and Alikakos, Betel´u, and Chen [3] for the standard Laplacian and here we provideextensions of these results for p > R N for potentials with two global minima and state withoutproof an existence theorem together with a variational characterization of the con-necting solutions (for the full proofs we refer to [9]). The problem for potentialspossessing three or more global minima (even for the case p = 2) is significantlyharder and essentially open. In Section 3, by restricting ourselves to N = 2, we areable to exhibit a class of potentials for which we have reasonably complete results.In particular, we establish a uniqueness theorem and also give some examples ofpotentials that exhibit non-existence and non-uniqueness properties.2. The connection problem for potentials possessing two globalminima
Let Ω be an open and connected subset of R N and W : Ω → R be a C nonneg-ative potential function with two minima, that is, W > R N \ A , with W = 0on A = { a + , a − } . In [4], Alikakos and Fusco analyze the existence of solutions to The author was partially supported through the project PDEGE – Partial Differential Equa-tions Motivated by Geometric Evolution, co-financed by the European Union – European SocialFund (ESF) and national resources, in the framework of the program Aristeia of the ‘Opera-tional Program Education and Lifelong Learning’ of the National Strategic Reference Framework(NSRF). the Hamiltonian system(1) u xx − ∇ W ( u ) = 0 , with lim x →±∞ u ( x ) = a ± , where u : R → R N is a vector-valued function. Such solutions are called heteroclinicconnections . The system (1) represents the motion of N material points of equalmass under the potential − W ( u ), with x standing for time and u for position. Theapproach in [4] is variational and is based on Hamilton’s principle of least action,that is, on the minimization of the action functional A : W , ( R , R N ) → R , definedas A ( u ) = 12 Z R ( | u x | + W ( u )) dx. The method depends on the introduction of a constraint leading to the existenceof local minimizers. The constraint can later be removed and therefore provide asolution to (1).In this section, we state without proof an extension of the results in [4] to the p -Laplacian operator. To this end, we consider the system(2) ( | u x | p − u x ) x − q ∇ W ( u ) = 0 , with lim x →±∞ u ( x ) = a ± where u : R → R N is again a vector-valued function and p, q > /p + 1 /q = 1. Alternatively, by Hamilton’s principle of leastaction, the motion from one minimum of the potential to another is a critical pointof the action functional A p : W ,p ([ t , t ] , R N ) → R , defined as A p ( u, ( t , t )) := Z t t (cid:18) | u x | p p + W ( u ) q (cid:19) dx. Therefore, the system (2) is the associated Euler–Lagrange equation with t = −∞ and t = + ∞ . To state our results, we assume that the following hypothesis holds. Hypothesis 1.
The potential W is such that lim inf | u |→∞ W ( u ) > and also thereexists R > such that the map r W ( a ± + rξ ) has a strictly positive derivative forevery r ∈ (0 , R ) and for every ξ ∈ S N − := { u ∈ R N : | u | = 1 } , with R < | a + − a − | . Then, under the above hypothesis, we have the following theorems.
Theorem 1.
Let W : R N → R be a non-negative C potential function and let a − = a + ∈ R N be such that W ( a ± ) = 0 . Also assume that Hypothesis 1 holds.Then, there exists a connection U between a − and a + . Theorem 2.
Let U be the minimizer provided by Theorem 1 above and let R beas defined in Hypothesis 1. Also let A be the set that consists of all functions u ∈ W ,p loc ( R ; R N ) for which there exist x − u < x + u (depending on u ) such that ( | u ( x ) − a − | ≤ R/ , for all x ≤ x − u , | u ( x ) − a + | ≤ R/ , for all x ≥ x + u . Then, A p ( U ) = min u ∈A A p ( u ) . N THE CONNECTION PROBLEM FOR THE p -LAPLACIAN SYSTEM 3 The above theorems yield the existence of the desired heteroclinic connection andalso a variational characterization. Our approach (for full details, see [9]) follows thelines of the method established in [4] and therefore it is based on the minimizationof the action functional A p over the whole real line. The proof of existence ofa connection is generally straightforward, since the test functions constructed inLemmas 3.1 and 3.2 in [4] succeed in the pointwise reduction of both the gradientand the potential terms.3. The connection problem on the complex plane for potentialspossessing several global minima
In this section we extend to the p -Laplacian operator the existence and unique-ness results in Alikakos, Betel´u, and Chen [3]. Here, we consider potentials possess-ing three or more global minima and restrict ourselves to the planar case N = 2for which we identify R with the complex plane C .We tackle the problem by utilizing Jacobi’s principle, which deals with curvesand detects geodesics. Specifically, one considers the length functional L p ( u ) = Z t t q p W ( u ) | u x | dx, which is independent of parametrizations and hence is more properly denoted by L p (Γ) = Z Γ q p W (Γ) d Γ . Notice that the functional A p is defined on functions while the functional L p isdefined on curves. The relationship between the two is that critical points of L p parametrized under the equipartition parametrization (that is, a parameter t issuch that | u t | p = W ( u ), for all t in an interval ( a, b )) render critical points of A p .In this case, we study the system of ordinary differential equations( | u x | p − u ix ) x − q ∂W ( u , u ) ∂u i = 0 , for i = 1 , , where u = ( u , u ) : R → R . We identify u = ( u , u ) with the complex number z = u + iu , and similarly we write W ( u ) as W ( z ). Since W ( · ) is non-negative,we can write W ( z ) = | f ( z ) | q for some analytic function f . We can also verify thatthe equations in (2) are equivalent to( | z x | p − z x ) x = ( f f ) q − f f ′ , where the bar represents complex conjugation.We begin by sketching the method for the standard triple-well potential W ( z ) = | z − | q , the minima of which are taken at the points 1, e πi , and e − πi . We willconstruct the connection between a = 1 and b = e πi , by considering the variationalproblem min L p ( u )along embeddings on the plane connecting a to b . Since the functional L p is inde-pendent of parametrizations, we choose u : (0 , → R with u (0) = a , u (1) = b ,and set z ( t ) = u ( t ) + u ( t ). Then, L p ( u ) = Z | z ′ ( τ ) || z ( τ ) − | dτ = Z (cid:12)(cid:12)(cid:12)(cid:12) ddτ g ( z ( τ )) (cid:12)(cid:12)(cid:12)(cid:12) dτ = Z | w ′ ( τ ) | dτ, NIKOLAOS KARANTZAS where w = g ( z ) = z − z /
4. It is clear that minimizing L p over the set of curvesconnecting a to b is reduced to the simple problem of minimizing the length func-tional on the w -plane for curves connecting g ( a ) = 3 / g ( b ) = 3 e πi /
4, which ofcourse is minimized by the line segment connecting these image points. Now, bychoosing the following parametrization for the line segment g ( z ( τ )) = τ g ( a ) + (1 − τ ) g ( b ) = 34 ( τ + (1 − τ ) e πi ) , for 0 ≤ τ ≤ , we can show that the curve z ( τ ) = r ( τ ) e iθ ( τ ) satisfies the parameter-free equation4 r cos ( θ − π r cos (4 θ − π π , for 0 ≤ θ ≤ π < r < , which is exactly the same equation presented in [3] for p = q = 2. The dependenceon p is through the parametrization dtdx = p p W ( z ( t )) | z ′ ( t ) | , with t (0) = 12 , which leads to the connection u ( x ) = z ( t ( x )). So, naturally, one would expect thatthe theory presented in [3] could be extended for any p >
1, and although this istrue, it is not without some extra technical effort.
Theorem 3.
Let W ( u , u ) = | f ( z ) | q , where f = g ′ is holomorphic in an opensubset D of R , and let the point ( u , u ) be identified with the complex number z ( x ) = u ( x ) + iu ( x ) . Additionally, let γ = { u ( x ) : x ∈ ( a, b ) } be a smooth curvein D and x an equipartition parameter, that is, | u x | p = W ( u ) . Also, set α = u ( a ) and β = u ( b ) . Then, u is a solution to (3) ( | u x | p − u x ) x − q ∇ W ( u ) = 0 , in ( a, b ) if and only if (4) Im (cid:18) g ( z ) − g ( α ) g ( β ) − g ( α ) (cid:19) = 0 , for all z ∈ γ. Proof.
Let u = ( u , u ) : ( a, b ) → D be a solution to (3) with | u x | p = W ( u ). Also,let L be the total arclength and l the arclength parameter defined by L = Z ba | u x | q p W ( u ) dx and l = Z xa | u x ( y ) | q p W ( u ( y )) dy. We will show that g ( γ ) is the line segment [ g ( α ) , g ( β )]. We begin by modifyingequation (3). Here we note that the equipartition relationship gives(5) | z x | p = W ( u ) = ( f f ) q and that since ∂W∂u = ( | f ( z ) | q ) u = [( f f ) q ] u = q f f ) q − ( f ′ f + f f ′ ) ,∂W∂u = ( | f ( z ) | q ) u = q f f ) q − ( if ′ f − if f ′ ) , we have(6) ∂W∂u + i ∂W∂u = q f f ) q − ( f ′ f + f f ′ − f ′ f + f f ′ ) = q ( f f ) q − f f ′ . N THE CONNECTION PROBLEM FOR THE p -LAPLACIAN SYSTEM 5 Hence, based on (6), equation (3) can be written as0 = ( | u x | p − u x ) x − q ∇ W ( u )= p | z x | p − z xx + ( p − | z x | p − z x z xx − ( f f ) q − f f ′ . Now, by multiplying the above equation by | z x | − p and by further simplifying, wesee that equation (3) is equivalent to(7) p | z x | z xx + ( p − z x z xx − | z x | − p ) f f ′ = 0 . In addition, by differentiating the equipartition relationship | z x | p = ( f f ) q , weobtain the equation p | z x | p − ( z xx z x + z x z xx ) = q ( f f ) q − ( f ′ f z x + f f ′ z x ) , which simplified, is equivalent to(8) ( p − | z x | p − ( z xx z x + z x z xx ) = f ′ f z x + f f ′ z x . Finally, differentiating the function g , we have dg ( z ) dl = g ′ ( z ) z x ( f f ) q = f z x ( f f ) q = z x f q − f q = z x f q − f q and d g ( z ) dl = f f z xx + − q f f ′ z x − q | z x | f f ′ f q f q +1 = f f z xx + p − p − f f ′ z x − p p − | z x | f f ′ f q f q +1 = 2( p − f f z xx + ( p − f f ′ z x − p | z x | f f ′ p − f q f q +1 = 2( p − f f z xx + z x (( p − f f ′ z x − pz x f f ′ )2( p − f q f q +1 = 2( p − f f z xx + z x (( p − f f ′ z x + z x f f ′ ) − p − z x f f ′ )2( p − f q f q +1 . (9)Now utilizing (8), equation (9) becomes d g ( z ) dl = 2 f f z xx + ( p − | z x | | z x | p − z xx f q f q +1 + ( p − | z x | p − z x z xx − | z x | f f ′ f q f q +1 (10)and by virtue of (5), equation (10) becomes d g ( z ) dl = 2 | z x | p − z xx + ( p − | z x | p − z xx f q f q +1 + ( p − | z x | p − z x z xx − | z x | f f ′ f q f q +1 , NIKOLAOS KARANTZAS which after simplification can be written as d g ( z ) dl = p | z x | p − z xx + ( p − | z x | p − z x z xx − | z x | f f ′ f q f q +1 = p | z x | z xx + ( p − z x z xx − | z x | − p ) f f ′ f q f q +1 | z x | − p ) = 0 , as a result of (7). Thus dg ( z ) /dl = C is constant. Integrating this equation andevaluating it at l = L gives respectively g ( z ) = g ( α ) + Cl and Cl = g ( β ) − g ( α ). Inaddition, we have (cid:12)(cid:12)(cid:12)(cid:12) dg ( z ) dl (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z x f q − f q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | z x || z x | pq | z x | p = | z x || z x | p − | z x | p = 1 = | C | , hence L = | g ( β ) − g ( α ) | and C = g ( β ) − g ( α ) | g ( β ) − g ( α ) | , which means that g ( z ( l )) = L − lL g ( α ) + lL g ( β ) . Thus, g is the desired line segment.For the converse, assume that γ = u (( a, b )) satisfies (4) and the parameter x isan equipartition parameter for u . Then, equation (4) can be written as g ( z ) − g ( α ) = s ( x )( g ( β ) − g ( α )) , where s ( x ) is a real-valued function. Upon differentiation, we obtain s x ( g ( β ) − g ( α )) = g ′ ( z ) z x = f ( z ) z x . This equation implies that | s x | = | f ( z ) || z x || g ( β ) − g ( α ) | = | f ( z ) | q | g ( β ) − g ( α ) | > s ( x ) ∈ R , with s ( a ) = 0 and s ( b ) = 1, we must have s x ( x ) >
0. Hence, s x ( x ) = | f ( z ) | q | g ( β ) − g ( α ) | . Consequently,(11) z x = s x ( g ( β ) − g ( α )) f ( z ) = C ( f f ) q f , where C = g ( β ) − g ( α ) | g ( β ) − g ( α ) | .Lastly, utilizing (11), we construct the differential equation (7) as follows. First,we have | z x | = ( f f ) q − ,z xx = q − C f q − f q f ′ + q f q − f q − f ′ ,z x = C f q − f q ,z xx = q − C f q − f q f ′ + q f q − f q − f ′ , N THE CONNECTION PROBLEM FOR THE p -LAPLACIAN SYSTEM 7 so from the above relations it follows that p | z x | z xx = pq − p C f q − f q − f ′ + pq f q − f q − f ′ (12) ( p − z x z xx = pq − p − q + 42 f q − f q − f ′ (13) + pq − q C f q − f q − f ′ . Adding (12) and (13) gives p | z x | z xx + ( p − z x z xx = ( pq − p − q ) C f q − f q − f ′ (14) + ( pq − p − q + 2) f q − f q − f ′ , and utilizing the fact that | z x | − p ) = ( f f ) q − , equation (14) becomes p | z x | z xx + ( p − z x z xx = 2( f f ) q − f f ′ = 2 | z x | − p ) f f ′ . This equation is equivalent to u being a solution to q ( | u x | p − u x ) x = W u ( u ) and theproof is complete. (cid:3) The proof implies that (3) is equivalent to the first-order ordinary differentialequation(15) z x = C ( f f ) q f , for C ∈ C with | C | = 1 . Multiplying (15) by f ( z ) C gives ddx g ( z ) C = | f ( z ) | q = W > (cid:18) g ( z ) − g ( α ) C (cid:19) = 0and g ( z ) − g ( α ) C = Z xa W ( z ( t )) dt = l. This in particular implies that the map x g ( z ( x )) is a one-to-one map. Also, inthe case of u being a solution, the theorem states that the set g ( γ ) = { g ( z ) : z ∈ γ } is a line segment with end points g ( α ) and g ( β ) and that the partial transitionenergy is given by Z ya (cid:18) | u x | p p + W ( u ) q (cid:19) dx = Z ya | u x | q p W ( u ) dx = Z ya (cid:12)(cid:12)(cid:12)(cid:12) ddx g ( u ) (cid:12)(cid:12)(cid:12)(cid:12) dx = | g ( u ( y )) − g ( α ) | , for all y ∈ ( a, b ] . Theorem 4.
There exists at most one trajectory connecting any two minima of aholomorphic potential, that is, if W ( z ) = | f ( z ) | q , where f is holomorphic on C , thenthere exists at most one solution of (2) that connects any two roots of W ( z ) = 0 . NIKOLAOS KARANTZAS
Proof.
Let g be an antiderivative of f and suppose that γ and γ are two trajecto-ries to (2) with the same end points α , β . Since the energy | g ( β ) − g ( α ) | is positive,it follows that g ( β ) = g ( α ) and we can define the functionˆ g = | g ( β ) − g ( α ) | g ( β ) − g ( α ) ( g ( z ) − g ( α )) , for all z ∈ C . Then, ˆ g is real on γ ∪ γ . If γ = γ , then γ and γ will enclose an open domain D in C . As the imaginary part of ˆ g on ∂D = γ ∪ γ ∪ { α, β } is zero, it has tobe identically zero in D . This implies that ˆ g is a constant function in C , which isimpossible. Thus, γ = γ . (cid:3) Finally, we present some specific examples of non-existence and non-uniquenessof connections between the minima of various potentials. Specifically, it can beproved that for both the potentials W ( z ) = | z n − | q , where n ≥ W ( z ) = | (1 − z )( z + ε ) | q , where 0 < ε < ∞ , there always exists a unique connection between each pair oftheir minima. We also refer to a non-existence and non-uniqueness phenomenonfor the potentials W ( z ) = | (1 − z )( z − iε ) | q , where 0 ≤ ε < ∞ , and W ( z ) = | ( z − z + a ) /z | q , where 0 < a <
1, respectively. In the first case it can be proved that there exists aconnection between − | ε | > p √ −
3, while in the second casethere exist exactly two connections between − a and 1, one in the upper half-planeand one in the lower half-plane. We conclude by stating that all examples given in[3] have exact analogs since the only modification needed for the transformation tothe p -case is the change from potentials of the form W = | f | (cf. [3]) to potentialsof the form W = | f | q . Acknowledgments
The author would like to thank Professor Nicholas D. Alikakos for his valuablehelp and guidance.
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