Introduction to critical point theory and applications to boundary value problems. Stepan Tersian,. University of Ruse,. 2014 ...
Introduction to critical point theory and applications to boundary value problems Stepan Tersian, University of Ruse, 2014
1. Historical notes Brachistrochrone problem, 1696
A brachistochrone curve (Gr. βράχιστος, brachistos - the shortest, χρόνος, chronos - time) or curve of fastest descent, is the path that will carry a point-like body from one place A to another B in the least amount of time.
Johann Bernoulli (6 August [O.S. 27 July] 1667 – 1 January 1748)
Given two points A and B, with A not lower than B, only one upside down cycloid passes through both points
Johann Bernoulli posed the problem of the brachistochrone to the readers of Acta Eruditorum in June, 1696. Six months were allowed by Bernoulli for the solution of the problem, and in the event of none being sent to him he promised to publish his own. Bernoulli received a letter from Leibniz, and requesting that the period for their solution should be extended to Christmas next
Five mathematicians responded with solutions: Isaac Newton, Jakob Bernoulli (Johann's brother), Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus and Guillaume de l'Hôpital.
On 29 January 1697 Newton returned at 4pm from working at the Royal Mint and found in his post the problems that Bernoulli had sent to him directly; two copies of the printed paper containing the problems. Newton stayed up to 4am before arriving at the solutions; on the following day he sent a solution of them to Montague. The solution was published, unsigned in “Philosophical Transactions”, May, 1697. Solutions were also obtained from Leibniz and the Marquis de l’Hopital; and, although Newton's solution was anonymous, he was recognized by Bernoulli as its author; "tanquam ex ungue leonem" (we recognize the lion by his claw).
b 1 + y' 2
T=∫
2 gy
0
dx → min
y (0 ) = 0 > y( b ) = B , y ∈ C 1 ([0 ,b]) Energy functional
b
E( y ) = ∫ F ( x , y , y' )dx → min, a
y ∈ C 1 ([a ,b]), y (a ) = A, y (b ) = B.
For (1) EL- equation is
((
))
1 + y' 2 2 2 ' y' ' = − ⇔ 2 yy' ' +1 + y' = 0 ⇒ y 1 + y' =0 2y
(
)
⇔ y 1 + y' 2 = C
C x = (t − sin t ), 2 C y = (1 − cos t ). 2
C b = (t − sin t ), 2 C B = (1 − cos t ). 2
B 1 − cos t = b t − sin t
Example: y(0)=0, y(2)=-2, y(0)=0, y(2)=-4. -5 -10 -15 -20 -25 -30 -2-1.8 -1.6 -1.4 -1.2
-1 -0.8 -0.6 -0.4 -0.2 t
1 − Cos@XD X − Sin@XD 1 − Cos@YD FindRootB Y − Sin@YD FindRootB
−1, 8X, −0.1 is duality pairing between X and its dual space X ∗ . < ϕ0 (u) , v >= lim
t→0
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
The problem (1) has a variational structure if there exists an appropriate reflexive Banach space X and energy functional ϕ : X → R such that < ϕ0 (u) , v >=< F (x, u, Du), v > . In this case, critical points of ϕ, i.e. elements u ∈ X : ϕ0 (u) = 0 are weak (non smooth or classical) solutions of (1). Under some assumptions they are classical or strong solutions of (1). The equation ϕ0 (u) = 0 is called Euler-Lagrange equation of (1). The existence of critical points of the functional ϕ : X → R depends on specific geometric and compactness conditions, which are matter of various critical point theorems.
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
Simple examples and four problems: ½ 00 u + 2u = sin t, 0 < t < π, (P1 ) : u (0) = u (π) = 0. (P1 ) has the unique solution u = sin t. ½ 00 u + u = sin t, 0 < t < π, (P2 ) : u (0) = u (π) = 0. (P2 ) has no solution.
2. Introduction
3. One dimensional Sobolev spaces and inequalities
½ (P3 ) :
4. Basic critical point theorems
u 00 − u 3 = 0, 0 < t < π, u (0) = u (π) = 0.
(P3 ) has unique zero solution. Zπ 0=
¡ 00 ¢ u − u 3 udx = −
0
½ (P4 ) :
Zπ
¡ 02 ¢ u + u 4 dx ⇒ u = 0.
0
u 00 + u 3 = 0, 0 < t < π, u (0) = u (π) = 0.
(P4 ) has not unique zero solution.
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
Denote by Lp (0, T ) for p ≥ 1, the Lebesgue space of p−integrable functions over the interval (0, T ) , endowed with the usual norm Rb ||u||pp = |u(t)|p dt, and by || · || and || · ||∞ the corresponding a
norms in L2 (0, T ) and C ([0, T ]) ZT 2
||u||
|u(t)|2 dt, u ∈ L2 (0, T ),
= 0
||u||∞ =
max |u(t)|, u ∈ C ([0, T ]).
t∈[0,T ]
Denote by H 1 (0, T ) , Hp1 (0, T ) , H01 (0, T ) and H 2 (0, T ) the Sobolev’s spaces H 1 (0, T ) = {u ∈ L2 (0, T ) : u 0 ∈ L2 (0, T )},
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
H01 (0, T ) = {u ∈ L2 (0, T ) : u 0 ∈ L2 (0, T ), u(0) = u(T ) = 0}, Hp1 (0, T ) = {u ∈ L2 (0, T ) : u 0 ∈ L2 (0, T ), u(0) = u(T )}, and H 2 (0, T ) = {u ∈ L2 (0, T ) : u 0 ∈ L2 (0, T ), u 00 ∈ L2 (0, T )}. H 1 (0, T ) , Hp1 (0, T ) , H01 (0, T ) are Hilbert spaces with inner product and norm ZT (u 0 v 0 + uv )dt, ||u||21 = (u, u).
(u, v ) = 0
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
H 2 (0, T ) is a Hilbert space with inner product and norm ZT (u 00 v 00 + u 0 v 0 + uv )dt, ||u||22 =< u, u > .
< u, v >= 0
Let CT∞ be the space of C ∞ , T −periodic functions. Fundamental Lemma. Let u, v ∈ L1 (0, T ) . If for every f ∈ CT∞ ZT
ZT 0
u (t) f (t) dt = − 0
then
RT
v (t) f (t) dt, 0
v (t) d = 0 and there exists c, such that
0
u (t) = c +
Rt 0
v (s) ds a.e. on [0, T ] .
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
The function v is called a weak derivative of u; Rt b (t) = c + v (s) ds is a continuous function. If u ∈ L2 (0, T ) and u 0
¶ ∞ µ a0 X 2πkt 2πkt u (t) = + ak cos + bk sin 2 T T k=1
is the Fourier series expansion, then the weak derivative is ¶ µ ∞ 2π X 2πkt 2πkt u (t) = k −ak sin + bk cos . T T T 0
k=1
Denote a0 1 u= = 2 T
ZT e (t) = u (t) − u. u (t) dt, u 0
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
Theorem I1. (a) If u ∈ Hp1 (0, T ) ZT 0
T2 |u (t) − u|2 dt ≤ 2 4π
ZT |u 0 (t) |2 dt, 0
(Poincare inequality). If u ∈ Hp1 (0, T ) and ZT
T2 |u (t) | dt ≤ 2 4π
RT
u (t) dt = 0, then
0
ZT
2
|u 0 (t) |2 dt,
0
0
(Wirtinger inequality) and ||u||2∞ ≤
T 12
RT 0
|u 0 (t) |2 dt, (Sobolev
inequality). (b) The inclusions Hp1 (0, T ) ⊂ C ([0, T ]) and H01 (0, T ) ⊂ C ([0, T ]) are compact.
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
Proof. (a) We have by Parseval’s identity
2 T
ZT |u (t) − u|2 dt =
∞ ∞ X ¡ 2 ¢ ¢ T 2 X 4π 2 ¡ 2 2 ak + bk2 = 2 a + b k k 4π T2 k=1
0
≤
=
k=1
∞ T2 X
4π 2
k=1
2 T2 T 4π 2
¢ 4π 2 k 2 ¡ 2 ak + bk2 2 T
ZT |u 0 (t) |2 dt. 0
If u = 0, we obtain the Wirtinger inequality. By Euler identity and Cauchy-Schwartz inequality
2. Introduction
3. One dimensional Sobolev spaces and inequalities
à ||u||2∞ ≤
∞ X
!2 (|ak | + |bk |)
k=1
≤
à =
4. Basic critical point theorems
∞ X 2πk k=1
T (|ak | + |bk |) . T 2πk
!2
ZT ∞ ∞ ¢X 1 2T 2 X 4π 2 k 2 ¡ 2 T2 2 ak + bk = |u 0 (t) |2 dt. 4π 2 T2 k2 12 k=1
k=1
0
(b) The inclusion Hp1 (0, T ) ⊂ C ([0, T ]) is continuous by inequality
2. Introduction
3. One dimensional Sobolev spaces and inequalities
|u(t)| ≤
1 T
ZT
T 1/2 Z √ |u (t) |dt + T |u 0 (t) |2 dt
0
≤
4. Basic critical point theorems
(2)
0
T T 1/2 1/2 Z Z √ 1 √ |u (t) |2 dt + T |u 0 (t) |2 dt T 0
0
Let (uk ) be a bounded sequence in Hp1 (0, T ) . Then by (2) , (uk ) be a bounded sequence in C ([0, T ]). Since (uk ) ∈ Hp1 (0, T ), we obtain, for every t1 , t2 ∈ [0, T ], the following inequalities Zt2 |uk (t2 )−uk (t2 ) | ≤ t1
√ |uk0 (t) |dt ≤ t2 − t1
ZT 0
√ |uk0 (t) |2 dt ≤ C t2 − t1 ,
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
which shows that the sequence (uk ) is equi-continuous. By Ascoli-Arzela theorem (uk ) is relatively compact in C ([0, T ]).¤ Let X = H01 (0, T ) ∩ H 2 (0, T ) be the Hilbert space endowed with the usual scalar product. Suppose that a and b are positive continous functions on [0, T ]. ZT ||u||2X
(|u 00 |2 + a|u 0 |2 + buv )dt
= 0
is an equivalent norm in X . It is known, that imbedding inequality holds for all u ∈ X : ||u||C 1 = max{||u||∞ , ||u 0 ||∞ } ≤ M||u||X ,
(3)
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
Proposition I1. The inclusion X ⊂ C 1 ([0, T ]) is compact. Proof. Let (uk ) ⊂ X be a bounded sequence, ||uk ||X ≤ C . Since the inclusion H01 (0, T ) ⊂ C ([0, T ]) is compact, we have that there is (uk ) a subsequence of (uk ) , still denoted by (uk ), such that uk → u strongly in C ([0, T ]) . By (3), we know that (||uk0 ||∞ ) is bounded and, since (uk ) ∈ H 2 (0, T ), we obtain, for every t1 , t2 ∈ [0, T ], the following inequalities Zt2 |uk0
(t2 )−uk0
|uk00 (t) |dt
(t2 ) | ≤ t1
√ ≤ t2 − t1
ZT
√ |uk00 (t) |2 dt ≤ C t2 − t1 ,
0
which shows that the sequence (uk0 ) is equi-continuous. Then, by Arzela-Ascoli theorem uk0 → v strongly in C ([0, T ]) and u 0 = v , which shows that uk → u strongly in C 1 ([0, T ]).
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
To this end, we introduce classical notations and results. Let E be a reflexive real Banach space. Recall that a functional I : E → R is lower semi-continuous (resp. weakly lower semi-continuous (w.l.s.c.)) if uk → u (resp. uk * u ) in E implies lim inf I (uk ) ≥ I (u) k→∞
(see [MW]). We have the following well known Minimization theorem Let I be a weakly lower semi-continuous and bounded below functional on reflexive real Banach space E . Then I has a minimum c = minu∈E I (u) = I (u0 ). If I : E → R is a differentiable functional, u0 is a critical point of I .
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
The existence of a bounded minimizing sequence appears if the functional I is coercive, i.e. I (u) → +∞ as ||u|| → ∞. Note that a functional I : E → R is w.l.s.c. on I if I (u) = I1 (u) + I2 (u), I1 is convex and I2 is sequentially weakly continuous (i.e. uk * u in E implies limk→∞ I2 (uk ) = I2 (u)) (see [Be] pp. 301-302 ). Next, recall the notion of Palais-Smale (PS) condition, mountain-pass theorem and Clarke’s theorem.
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
We say that I satisfies (PS) condition if any sequence (uk ) ⊂ E for which I (uk ) is bounded and I 0 (uk ) → 0 as k → ∞ possesses a convergent subsequence. I satisfies Cerami’s (C ) condition if any sequence (uk ) ⊂ E for which I (uk ) is bounded and (1 + ||uk ||E )I 0 (uk ) → 0 as k → ∞ possesses a convergent subsequence. Clearly (C ) condition implies (PS) condition.
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
Mountain-pass theorem. (Ambrosetti, Rabinowitz, 1973). Let E be a real Banach space and I ∈ C 1 (E , R) satisfying (PS). Suppose I (0) = 0 and (i) there are constants ρ, α > 0 such that I (u) ≥ α if ||u|| = ρ, (ii) there is an e ∈ E , ||e|| > ρ such that I (e) ≤ 0. Then I possesses a critical value c ≥ α. Moreover c can be characterized as c = inf{max{I (u) : u ∈ γ ([0, 1])} : γ ∈ Γ} where Γ = {γ ∈ C ([0, 1] , E ) : γ (0) = 0, γ (1) = e}.
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
Clarke’s theorem. Let E be a real Banach space and I ∈ C 1 (E , R) with I even, bounded from below, and satisfying (PS). Suppose I (0) = 0, there is a set K ⊂ E such that K is homeomorphic Sm−1 by an odd map, and sup{I (u) : u ∈ K } < 0. Then I possesses at least m distinct pairs of critical points.
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
Let f (x, t) be a function on Ω × R, where Ω is either bounded or unbounded domain. f is a Carath´eodory function (CF) if f (x, t) is continuous in t, a.e. x ∈ Ω and measurable in x for every t ∈ R. (see [ZS]): Theorem I2. Assume p ≥ 1, q ≥ 1 and f (x, t) is CF on Ω × R and |f (x, t) | ≤ a + b|t|p/q , ∀ (x, t) ∈ Ω × R, a > 0, b > 0. Then if Ω is bounded, the Carath´eodory (superposition or Nemytskii) operator Bu := f (x, u (x)) : Lp (Ω) → Lq (Ω) is continuous and bounded mapping, i.e. if (uk ) ⊂ Lp (Ω) , ||uk − u||Lp → 0, then ||Buk − Bu||Lq → 0 as k → ∞. The same is true if Ω is unbounded and a = 0.
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
Theorem I3. Assume p ≥ 1, q ≥ 1 and f (x, t) is CF on Ω × R and |f (x, t) | ≤ a|t|q + b|t|p , ∀ (x, t) ∈ Ω × R, a > 0, b > 0, Ω is either bounded or unbounded. Define the functional Z I (u) :=
Zu F (x, u (x)) dx, F (x, u) :=
f (x, t) dt. 0
Ω
Assume that (E , ||.||) is a Sobolev space such that the inclusions E ⊂ Lq (Ω) , E ⊂ Lp (Ω) are continuous. Then I ∈ C 1 (E , R) and Z 0 < I (u) , v >= f (x, u (x)) v (x) dx, ∀v ∈ E . Ω
Moreover, if the inclusions E ⊂ Lq (Ω) , E ⊂ Lp (Ω) are compact,then I 0 : E → E ∗ is a compact operator.
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
M. Berger, Nonlinearity and Functional Analysis, Academic Press, 1977. J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989. M.R. Grossinho and S.A. Tersian. An Introduction to Minimax Theorems and their Applications to Differential Equations, Kluwer, 2001. P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf., Vol. 65, Amer. Math. Soc., RI, 1986. W.Zou, M.Schechter. Critical Point Theory and its Applications, Springer, 2006.
2. Introduction
3. One dimensional Sobolev spaces and inequalities
4. Basic critical point theorems
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