Mikhail A. Sychev

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1213–1218http://france.elsevier.com/direct/CRASS1/Calculus of Variations/Mathematical Problems in MechanicsLocal minimizers of one-dimensional variational problemsand obstacle problems ✩Mikhail A. SychevLaboratory of Differential Equations and Related Problems in Analysis, Sobolev Institute for Mathematics,Koptuyg Avenue 4, Novosibirsk 630090, RussiaReceived 1 July 2007; accepted 8 July 2008Available online 14 October 2008Presented by John M. BallAbstractIn this Note we suggest a direct approach to study local minimizers of one-dimensional variational problems. To cite this article:M.A. Sychev, C. R. Acad. Sci. Paris, Ser. I 346 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméMinimiseurs locaux de problèmes variationnels en une dimension et de problémes d’obstacle. Dans cette Note nous sug-gérons une approche directe pour étudier les minimiseurs locaux de problèmes variationnels monodimensionnels. Pour citer cetarticle : M.A. Sychev, C. R. Acad. Sci. Paris, Ser. I 346 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeOn considère un problème de minimisation de la forme (1), (2). Nous considérons en premier lieu le problèmed’obstacle (3). On suppose que la fonction L est convexe en v, et satisfait les hypothèses (H1) et (H2). Nos résultatsconcernant le problème d’obstacle sont les deux théorèmes suivants :Théorème 1. Soit L vérifiant (H1)–(H2) et soient c, ε > 0. Supposons f > g tels que ‖f ‖W 1,∞,‖g‖W 1,∞  c.Alors il existe δ = δ(ε) > 0 tel que le problème (1)–(3) avec f (a)  A  g(a), f (b)  B  g(b) ait une solutiondans W 1,∞[a, b] dès que ‖f − g‖C  δ. De plus, chaque solution u vérifie‖u̇‖L∞  max{‖ḟ ‖L∞,‖ġ‖L∞} + ε.Une plus grande regularité de f,g (C1 ou C1,α) se reporte sur u (C1 ou C1,γ ). Ce résultat est uniforme enles fonctions L qui satisfont (H1)–(H2), pourvu que ces hypothèses soient verifiées uniformément dans l’ensembleKν = [a, b] × [−c − ν, c + ν]2 pour un certain ν > 0.✩ This work was supported by RFBR (project 06-08-00386) and by SB RAS (project 1.6).E-mail address: masychev@math.nsc.ru.1631-073X/$ – see front matter © 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2008.07.0231214 M.A. Sychev / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1213–1218Ce théorème peut-être précisé lorsque L est C3, voir Theorem 5.Théorème 2. Soit L satisfaisant (H1)–(H2) et soient c,α > 0. On suppose ‖f ‖C1,α ,‖g‖C1,α  c, f > g. Il existe δ > 0et c̄, γ > 0 tels que chaque problème (1)–(3) où f (a)  A  g(a), f (b)  B  g(b) ait une solution C1 pourvu que‖f −g‖C  δ. De plus, ‖u‖C1,γ  c̄. Ce résultat est uniforme en les fonctions L satisfaisant (H1)–(H2) uniformémentdans un ν-voisinage, ν > 0, d’un compact K ⊂ R3 contenant les graphes des fonctions f,g.Une première application du Théorème 2 est la correspondance entre minimiseurs locaux faibles et forts.Théorème 3. Si L satisfait (H1)–(H2), tout minimiseur local faible est C1,α pour un α > 0 adéquat et est un minimi-seur local fort. De plus tout minimiseur local faible isolé est fort.Une autre application du Théorème 2 est la suivante :Théorème 4. Soit L satisfaisant (H1)–(H2) et u un minimiseur local isolé de (1), (2). Supposons que Ln vérifie(H1)–(H2) uniformément en n dans un voisinage Vε du graphe de u (voir Theorem 4). Si ‖Ln − L‖C(Vε) → 0 alors ilexiste des minimiseurs locaux un des problèmes (1), (2)n tels que ‖un − u‖C1 → 0.Dans le cas L ∈ C3, la théorie classique montre sous des hypothèses de convexité adéquate que l’équation d’Euler–Lagrange est vérifiée et implique (5). On montre finalement l’alternative suivante :Théorème 5. Soit L ∈ C3 tel que Lvv > 0 et u un minimiseur local de (1), (2). La situation est l’une de celles décriteci-dessous :(i) il existe des solutions u+n , u−n de (5) telles que u+n > u > u−n , avec u+n décroissante en n, u−n croissante en n,telles que ‖u+n − u‖C1,‖u−n − u‖C1 → 0 lorsque n → ∞ ;(ii) Pour un certain ε > 0, toutes les solutions des problèmes de Cauchy w(a) = A, ẇ(a) = α, α ∈ [u̇(a), u̇(a) + ε]de (5) vérifient à la fois (2) et l’égalité J (w) = J (u), et il existe une suite croissante u−n de solutions de (5) tellesque u−n < u et ‖u−n − u‖C1 → 0 ;(iii) Pour un certain ε > 0, toutes les solutions des problèmes de Cauchy w(a) = A, ẇ(a) = α, α ∈ [u̇(a) − ε, u̇(a)]de (5) satisfont à la fois (2) et l’egalité J (w) = J (u), et il existe une suite décroissante u+n de solution de (5)telles que u+n > u et ‖u+n − u‖C1 → 0 ;(iv) Pour un certain ε > 0, toutes les solutions des problèmes de Cauchy w(a) = A, ẇ(a) = α, α ∈ [u̇(a) − ε,u̇(a) + ε] de (5) satisfont à la fois (2) et l’identité J (w) = J (u).1. IntroductionConsider a minimization problemJ (u) =b∫aL(x,u(x), u̇(x))dx → min, L(x,u, v) : [a, b] × R2 → R, (1)u(a) = A, u(b) = B, u : [a, b] → R. (2)Recall that u ∈ C1[a, b] is called a weak local minimizer (an isolated weak local minimizer) of the problem (1),(2) if (2) holds and there exists ε > 0 such that J (u)  J (w) (J (u) < J(w)) for any w ∈ C1[a, b] that satisfies (2),w = u and ‖u − w‖C1[a,b]  ε. A function u ∈ C1[a, b] is called a strong local minimizer (an isolated strong localminimizer) of the problem (1), (2) if (2) holds and for some ε > 0 we can guarantee that J (u)  J (w) (J (u) < J(w))for all w ∈ C1[a, b] such that w satisfies (2), w = u and ‖u − w‖C[a,b]  ε. Any strong local minimizer is obviouslya weak local minimizer. Strong local minimizers are solutions of the problem (1)–(3) in the class C1, whereg  u  f, f,g : [a, b] → R, (3)for appropriate f,g, say for g = u − ε, f = u + ε with ε > 0 sufficiently small.M.A. Sychev / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1213–1218 1215We emphasize that in this paper weak and strong local minimizers are C1-regular functions.The problems (1)–(3) are known as obstacle problems and their theory repeat the general existence and regularitytheory for Sobolev solutions of the problem (1), (2), see e.g. [3]. Therefore it requires a number of assumptions ongrowth of the first and second derivatives of L. It turns out that the case of thin strips, i.e. when ‖f − g‖C is sufficientlysmall, admits a better solvability theory which involves only standard assumptions of the classical local theory, i.e.certain requirements on regularity of L and on ellipticity in v. We will recall the results of this theory below.We assumeL(x,u, v) : [a, b] × R2 → R is locally Holder continuous, (H1)L is convex in v and for each compact subset K of R3 there exists μ = μ(K) > 0 such thatL(x,u, v2) − L(x,u, v1) − l(v2 − v1)  μ(v2 − v1)2 (H2)provided (x,u, v1), (x,u, v2) ∈ K , l ∈ ∂vL(x,u, v1), where ∂vL(x,u, v1) is the subgradient of the function L(x,u, ·) :R → R at v1.The following two theorems state our results for the obstacle problem (1)–(3).Theorem 1 (an a priori estimate). Let L satisfy (H1)–(H2) and let c, ε > 0. Let also f,g : [a, b] → R be such that‖f ‖W 1,∞,‖g‖W 1,∞  c, f > g in ]a, b[.There exists δ = δ(ε) > 0 such that each such problem (1)–(3) with f (a)  A  g(a), f (b)  B  g(b) has asolution in the class W 1,∞[a, b] provided ‖f − g‖C  δ, moreover for each such solution u we have‖u̇‖L∞  max{‖ḟ ‖L∞,‖ġ‖L∞} + ε;in case f,g ∈ C1 the solutions are also C1-regular, in case f,g ∈ C1,α we can assert that u ∈ C1,γ for some γ > 0.The result is uniform with respect to L that satisfy (H1)–(H2) provided the assumptions (H1)–(H2) hold uniformlyin the set Kν = [a, b] × [−c − ν, c + ν]2 for certain ν > 0.Theorem 2 (compactness in C1). Let L satisfy (H1)–(H2) and let c,α > 0. We assume ‖f ‖C1,α , ‖g‖C1,α  c, f > gin ]a, b[.There exists δ > 0 and c̄, γ > 0 such that each such problem (1)–(3) with f (a)  A  g(a), f (b)  B  g(b) issolvable in the class C1 provided ‖f − g‖C  δ, moreover each solution u satisfies the inequality ‖u‖C1,γ  c̄.The result is uniform with respect to L that satisfy (H1)–(H2) provided (H1)–(H2) hold uniformly in aν-neighbourhood, ν > 0, of a compact set K ⊂ R3 which contains graphs of the functions f,g.We can use these results to study a number of open problems for local minimizers and to complete this way theclassical local theory, which results we recall for convenience.2. Applications of Theorems 1 and 2Recall the standard results of the classical local theory for the problems (1), (2), cf. e.g. [3].In case u is a weak local minimizer and in case L ∈ C1 the Euler–Lagrange equation holdsddxLv(x,u(x), u̇(x)) = Lu(x,u(x), u̇(x)). (4)If L ∈ C2, Lvv > 0 then u ∈ C2 and differentiating (4) we obtainü = Lu − Lxv − Luvu̇Lvv, (5)see [4] for precise results on necessity of convexity in v.The next necessary requirement for u to be a weak local minimizer is the Jacobi conditionuα(x,α0) > 0, x ∈]a, b[, α0 = u̇(a), (6)where for each α the function u(·, α) solves (5) with u(a,α) = A, u̇(a,α) = α; here we need to assume that L ∈ C3.1216 M.A. Sychev / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1213–1218In caseuα(x,α0) > 0, x ∈]a, b], α0 = u̇(a) (7)Weierstrass has proved that u is a strong local minimizer. In this caseuα(x,α0) > 0, x ∈]a − ε, b],for a sufficiently small ε > 0 and for all α0 close to u̇(a − ε); here u(a − ε,α) = u(a − ε), u̇(a − ε,α) = α. The graphsof solutions of such Cauchy problems for (5) form a neighbourhood U of the graph of u in [a, b] and do not intersect.Therefore for each (x,u) ∈ U we can find α such that u(a − ε,α) = u(a − ε), u(x,α) = u and we can defineΦ(x,u) :=x∫a−εL(y,u(y,α), u̇(y,α))dy. (8)A marvelous observation by Weierstrass says thatL(x,u, v)  Φx(x,u) + Φu(x,u)v, (x,u) ∈ U, v ∈ R, (9)where the equality holds only for v = u̇(x,α).Therefore J (u) = Φ(b,B) − Φ(a,A) and J (w) > Φ(b,B) − Φ(a,A) for any other function w ∈ C1 that satisfies(2) with the graph lying in U .Further developments were connected with so-called generalized solutions, i.e. solutions in the class of Sobolevfunctions. What could be proved is that the functional J (u) is lower semicontinuous with respect to weak convergencein W 1,1 provided L(x,u, v) is convex in v. Then the superlinear growth of L in v implies weak compactness of theminimizing sequences and then the existence of a global minimizer. Ellipticity Lvv > 0 implies partial regularity ofthese solutions, i.e. u(·) has classical derivative, finite or infinite, everywhere and the derivative u̇(·) is continuous asfunction with values in R̄ = R ∪ {∞} ∪ {−∞}. In particular u(·) satisfies the Euler equation (5) in an open set of fullmeasure. Therefore full regularity of solutions follows provided there no solutions of the Euler equation bounded inC-norm with unbounded derivative. There results are due to Tonelli and his theory remained unchanged until recentdays, cf. [3].However this theory of generalized solution is insufficient to deal with the obstacle problems in generality requiredand we need to develop another direct approach. Moreover Theorems 1, 2 allow to resolve a number of questions,stated below, for local minimizers that could not be treated in context of the classical local theory or of Tonelli’s directarguments. This way, in addition to independent interest from the point of view of obstacle problems Theorems 1, 2suggest a direct approach to attack general unconstrained problems.Question 1. Is each weak local minimizer also a strong local minimizer?This question was stated recently by Brezis and Nirenberg in [2]. When in the multi-dimensional case the questionwas addressed for special classes of integrands L, see [8,9] in the one-dimensional case one can hope to resolve itunder the standard assumptions of the classical local theory. In fact, the Weierstrass result says that this is true incase (7). The only open case if (6). However nothing is known if L /∈ C3.We can apply Theorem 2 with f = u + ε, g = u − ε to obtainTheorem 3. Let L satisfy (H1)–(H2). Then each weak local minimizer is C1,α-regular for an appropriate α > 0 andis a strong local minimizer. Moreover each isolated weak local minimizer is an isolated strong local minimizer.Therefore everywhere below we do not specify the cases of weak and strong minimizers saying simply “a localminimizer”.Question 2. Let u be a local minimizer for the problem (1), (2) with L ∈ C∞, Lvv > 0. Let Ln ∈ C∞, (Ln)vv > 0 andlet ‖Ln − L‖C → 0. Is this correct that there exist local minimizers un associated with Ln such that ‖un − u‖C1 → 0.M.A. Sychev / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1213–1218 1217This question belongs to Ulam [10, Ch. 6, §1] who remarked that to apply the classical local theory we need‖Ln − L‖C3 → 0, when the variational problem (1), (2) is posed only by L, but the derivatives.Again due to Theorem 2 we can assertTheorem 4. Let L satisfy (H1)–(H2) and let u be an isolated local minimizer for (1), (2). Let Ln also satisfy(H1)–(H2), moreover let Ln satisfy (H1)–(H2) uniformly with respect to n in the set Vε for some ε > 0, whereVε :={(x,u, v): x ∈ [a, b], ∣∣u − u(x)∣∣  ε, ∣∣v − u̇(x)∣∣  ε}.If ‖Ln − L‖C(Vε) → 0 then there exist local minimizers un of the problem (1), (2)n such that ‖un − u‖C1 → 0.Remark. The result remains valid for a more general case when u is no longer an isolated minimizer provided thereexist positive numbers ε+n → 0, ε−n → 0 such that the functions u+ ε+n , u− ε−n , n ∈ N, do not touch other minimizersof the problem.Question 3. What are necessary and sufficient conditions for the problem (1), (2) admit a local minimizer?The classical local theory indicates that (7) is a sufficient condition for u to be a local minimizer and involvesrequirements on a family of solutions of (5) of one parameter. One could look for a less demanding condition thatincludes also the situation (6).Theorem 1 can be precised in case L ∈ C3.Theorem 5. Let L ∈ C3, Lvv > 0. Assume f and g to be such solutions of (5) that f > g in [a, b], f (a) > A > g(a),f (b) > B > g(B) and ‖f − g‖C < 1/8M , whereM := max{∣∣F(x,u, v)∣∣: x ∈ [a, b], g(x)  u  f (x), |v|  max{‖ḟ ‖C,‖ġ‖C} + 1}with F being the right-hand side of (5).Then there exists a local minimizer u of (1), (2) such that g < u < f in [a, b].Theorem 5 uses in fact the necessary condition which excludes only the case when u could be embedded in afamily of minimizers of one parameter.Theorem 6. Let L ∈ C3, Lvv > 0 and let u be a local minimizer for the problem (1), (2). Then one the followingsituations hold:1) there exist solutions u+n , u−n of (5) such that u+n > u, u−n < u in [a, b] and u+n is a decreasing sequence, u−n is anincreasing sequence with ‖u+n − u‖C1 ,‖u−n − u‖C1 → 0 as n → ∞;2) for certain ε > 0 all solutions of the Cauchy problemsw(a) = A, ẇ(a) = α, α ∈ [u̇(a), u̇(a) + ε]for (5) satisfy both (2) and the equality J (w) = J (u), there exists an increasing sequence u−n of solutions of (5)such that u−n < u and ‖u−n − u‖C1 → 0;3) for certain ε > 0 all solutions of the Cauchy problemsw(a) = A, ẇ(a) = α, α ∈ [u̇(a) − ε, u̇(a)]for (5) satisfy both (2) and the equality J (w) = J (u), there exists a decreasing sequence u+n of solutions of (5)such that u+n > u and ‖u+n − u‖C1 → 0;4) for certain ε > 0 all solutions of the Cauchy problemsw(a) = A, ẇ(a) = α, α ∈ [u̇(a) − ε, u̇(a) + ε]for (5) satisfy both (2) and the equality J (w) = J (u).Remark. In the situation 2 there exists δ > 0 such that there are no solution u+ of (5) with u+ > u in [a, b],‖u+ − u‖C  δ.1218 M.A. Sychev / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1213–1218Method of the proofs of Theorems 1, 2 develops an elementary direct approach suggested in [6]; however theproofs require an independent scheme. Therefore Theorem 5 presents also a direct proof of sufficiency of (7) for thesolution u of (5) to be a local minimizer. The results of Theorems 1, 2 fail in the case of thick strips since there areno solvability theorem in W 1,∞ or in C1 for problems (1), (2) even in the case L(x,u, v)  |v|2, Lvv > 0, in which aSobolev solution always exists. First examples when Sobolev solutions fail to be W 1,∞-regular were suggested in [1].In [7] we constructed examples of problems (1), (2) without solutions in the class C1 ]a, b[∩C[a, b] even for the caseL ∈ C∞,μ1|v|2  L  μ2|v|2, 0 < μ1  Lvv  μ2. The paper [5] indicates a necessary and sufficient condition forthe Sobolev solutions to be W 1,∞, and then C1-regular, though it is not that effective.The multi-dimensional analogue of Theorems 1, 2, i.e. the a priori estimate and compactness in C1 of solutions ofthe minimization problems in thin strips, is an important and difficult open problem.AcknowledgementsThe author is grateful to the head of Laboratory Prof. V.S. Belonosov for his support and to the unknown refereefor useful suggestions.References[1] J.M. Ball, V.J. Mizel, One-dimensional variational problems whose minimizers do not satisfy the Euler–Lagrange equation, Arch. RationalMech. Anal. 90 (1985) 325–388.[2] H. Brezis, L. Nirenberg, H 1 versus C1 local minimizers, C. R. Acad. Sci. Paris Ser. I Math. 317 (1993) 465–472.[3] G. Buttazzo, M. Giaquinta, S. Hildebrandt, One-dimensional Variational Problems. An Introduction, Oxford University Press, 1998.[4] C. Marcelli, E. Outkine, M. Sychev, Remarks on necessary conditions for minimizers of one-dimensional variational problems, NonlinearAnal. 48 (2002) 979–993.[5] V.J. Mizel, M. Sychev, A condition on the value function both necessary and sufficient for full regularity of minimizers of one-dimensionalvariational problems, Trans. Amer. Math. Soc. 350 (1998) 119–133.[6] M. Sychev, On the question of regularity of the solutions of variational problems, Russian Acad. Sci. Sb. Math. 75 (1993) 535–556.[7] M. Sychev, Examples of classically unsolvable regular scalar variational problems satisfying standard growth conditions, Siberian Math. J. 37(1996) 1212–1227.[8] A. Taheri, Sufficiency theorems for local minimizers of the multiple integrals of the calculus of variations, Proc. Roy. Soc. EdinburghSect. A 131 (2001) 155–184.[9] A. Taheri, Strong versus weak local minimizers for the perturbed Dirichlet functional, Calc. Var. 15 (2002) 215–235.[10] S. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, vol. 8, Interscience Publishers, NewYork–London, 1960.

Local minimizers of one-dimensional variational problems and obstacle problems

Sychev, Mikhail A.

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Local minimizers of one-dimensional variational problems and obstacle problems

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