Manuscript submitted to
AIMS’ Journals
Volume 00, Number 0, Xxxx XXXX
Website: http://AIMsciences.org
pp. 000–000
A DIRECT PROOF OF THE TONELLI’S PARTIAL
REGULARITY RESULT
Alessandro Ferriero
uam – icmat
Departamento de Matemáticas
Universidad Autónoma de Madrid
Campus de Cantoblanco
28049 Madrid, Spain
Abstract. The aim of this work is to give a simple proof of the Tonelli’s
partial regularity result which states that any absolutely continuous solution
to the variational problem
Z b
ff
min
L(t, u(t), u̇(t))dt : u ∈ W1,1
(a,
b)
0
a
has extended-values continuous derivative if the Lagrangian function L(t, u, ξ)
is strictly convex in ξ and Lipschitz continuous in u, locally uniformly in ξ
(but not in t). Our assumption is weaker than the one used in [2, 4, 5, 6, 13]
since we do not require the Lipschitz continuity of L in u to be locally uniform
in t, and it is optimal as shown by the example in [12].
1. Introduction. A solution ū in the Sobolev space W01,1 (a, b) to the variational
problem
(
)
Z b
1,1
min I(u) :=
L(t, u(t), u̇(t))dt : u ∈ W0 (a, b)
a
is said to be regular in the sense of Tonelli if its derivative ū˙ is extended-values
continuous on [a, b]. The main advantage of this type of regularity is that the set
of singularities of ū˙ is a closed set of [a, b]. In other words, the interval [a, b] can
be covered a.e. by the numerable union of disjoint open intervals such that ū has
continuous derivative on each of them.
The purpose of this paper is to give a simple proof of the Tonelli’s partial regularity result for Lagrangian functions L(t, u, ξ) which are strictly convex in ξ and
Lipschitz continuous in u, locally uniformly in ξ. To be more precise, we require
that, for each R > 0, there exists an integrable function CR : [a, b] → R+ such that
|L(t, u, ξ) − L(t, v, ξ)| ≤ CR (t)|u − v|,
(1)
for a.e. t in [a, b] and any vectors u, v, ξ in Rd with modulus smaller than R.
Up to now [2, 4, 5, 6, 13], the Lipschitz continuity of L on u, locally uniform in t
and u, has been the weakest condition imposed for proving this kind of regularity.
2000 Mathematics Subject Classification. Primary: 49B05, 49A05, 49C05, 35J20.
Key words and phrases. Variational problems, regularity, quasi-linear elliptic differential
equations.
1
2
A DIRECT PROOF OF THE TONELLI’S PARTIAL REGULARITY RESULT
In [12], the author give an example of a Lagrangian which is Lipschitz continuous
in u, locally uniformly in ξ, but with non-integrable Lipschitz constant CR , which
admits solutions which do not enjoy the Tonelli’s regularity. Our result is therefore
optimal in this sense.
In [10], we relax the Lipschitz continuity of L in u and we prove the Tonelli’s
partial regularity assuming an invariance property on the functional I as in the
Noether’s theorem. In [9], we prove a weaker regularity result than the Tonelli’s
partial regularity, namely the local Lipschitz continuity of any solutions ū, releasing
the strict convexity of L in ξ and assuming a weaker hypothesis called the bounded
slope condition (which roughly speaking means that the convexified function of L
in ξ does not admit unbounded affine pieces).
The proof we propose here is based on the following direct approach. Suppose
˙ i.e.
that t0 in (a, b) is a jump point for ū,
˙
˙
ū˙ − (t0 ) := lim ū(t)
6= lim ū(t)
=: ū˙ + (t0 ).
t→t−
0
t→t+
0
We can then show that the competitor w̄ of ū obtained by taking the average
2−1 [ū˙ − (t0 ) + ū˙ + (t0 )] of the right and left limit values of ū at t0 on a neighbourhood
of t0 and identically equal to ū elsewhere has, by the strict convexity of L, energy
strictly lower than ū. Hence, ū cannot be a minimizer for I. That is, if the
Lagrangian L is strictly convex on ξ and ū is a minimizer, the derivative of ū
cannot present jumps or oscillations but it must be extend-values continuous. Strict
convexity of L and discontinuities of ū˙ are linked by the minimality of I(ū) through
the convexity of the integral operation.
Our proof has the great value to be simple, direct and brief. It is based on the
idea of averaging afore sketched and a local generalized weak form of the Jensen
inequality.
The paper is organized as follows. In section 2, we present the generalized form
of the Jensen inequality, the main result concerning the Tonelli’s partial regularity
of solutions and a counterexample showing that strict convexity of L in ξ is a
necessary hypothesis. We also show that the result is not true under Carathéodory
condition on L, i.e. L measurable in t and continuous with respect to u and ξ.
2. Main Result. In the present paper we shall deal with Lagrangian functions
L(t, u, ξ) : ([a, b] \ ΣL ) × Rd × Rd → R,
where ΣL is a closed set of zero Lebesgue measure in [a, b], which are continuous
with respect to t, u and ξ. In case ΣL = ∅, L is continuous on [a, b] × Rd × Rd . We
further assume that, for each R > 0, there exists a function CR : [a, b] → R+ in
L1 (a, b) such that
|L(t, u, ξ) − L(t, v, ξ)| ≤ CR (t)|u − v|,
(2)
for a.e. t in [a, b] and every vector u, v, ξ in Rd with modulus smaller than R. Thus,
L is Lipschitz continuous in u, locally uniformly in ξ but not in t. Notice that this is
weaker than the assumption in [6] where it is required that the Lipschitz continuity
of L in u is locally uniform also in t.
The variational problem we are interested in is
(
)
Z
b
L(t, u(t), u̇(t))dt : u ∈ W01,1 (a, b) ,
min I(u) :=
a
(3)
A DIRECT PROOF OF THE TONELLI’S PARTIAL REGULARITY RESULT
3
where W01,1 (a, b) denotes as usual the Sobolev space of functions u from [a, b] to
Rd , d ≥ 1, with zero boundary conditions.
The general Dirichlet boundary data u(a) = A, u(b) = B, that is taking r +
W01,1 (a, b) as space of admissible functions, where r(t) := (t−a)(B −A)/(b−a)+A,
can be deduced by the zero boundary case (preserving the assumptions on L) by
considering the modified Lagrangian L̃(t, u, ξ) := L(t, u − r(t), ξ − ṙ(t)). For any
solution ū in W01,1 (a, b) to I corresponds a solution ũ = r + ū in r + W01,1 (a, b) to
Ĩ, and conversely.
In what follows,
• R denotes the two points compactification R ∪ {+∞} ∪ {−∞} of R,
• Rd denotes the one point compactification Rd ∪ {∞} of Rd , for d ≥ 2,
C([a, b]\ΣL ; Rd ) denotes the space of continuous functions from [a, b]\ΣL to Rd . We
will also refer to these functions as extended-values continuous functions. Notice
that, for any ξ in C([a, b]\ΣL ; Rd ), the preimage ξ −1 (Rd ) is an open set of [a, b]\ΣL .
Definition 1. We say that a solution ū in W01,1 (a, b) to (3) is regular in the sense
of Tonelli if its derivative ū˙ belongs to C([a, b] \ ΣL ; Rd ).
For a function ū that enjoys the properties of the definition, recalling that ΣL is
a closed set of zero measure, there exist numerable many disjoint
Sopen intervals In
of [a, b] such that ū|In belongs to C1 (In ), for any n, and also | n In | = b − a. In
other words, the set Sū˙ of singular points of the derivative of ū, i.e.
Sū˙ := {t ∈ [a, b] \ ΣL : ū˙ is not continuous at t},
is a closed set of [a, b] \ ΣL of zero measure.
We present the proof of the Tonelli’s partial regularity result in Theorem 3 and
4; in Theorem 3 we deal with dimensions d greater than 1, meanwhile in Theorem
4 we deal with dimension d = 1. Notice that our result for d = 1 is finer than the
one for d ≥ 2 since C([a, b] \ ΣL ; R) is included in C([a, b] \ ΣL ; R ∪ {∞}).
Throughout all the paper, {on (1)} denotes any sequence which converges to 0,
as n goes to ∞, and B(u0 ; R) ⊂ Rd is the closed ball with center u0 and radius R.
We start showing a Lemma that will be important in the proofs of Theorem 3
and 4. It is a local weak version of the Jensen inequality adapted to our problem.
Lemma 2. Let δ > 0, t0 ∈ (a, b) \ ΣL and {En } be a sequence of measurable sets
in the interval Iδ := [t0 − δ, t0 + δ] ⊂ [a, b] \ ΣL such that the diameter of En ∪ {t0 }
converges to 0, as n → ∞.
If L(w, ·) is convex and L(w, ξ) ≥ |ξ|, for every w in Iδ × B(0; R) and any ξ in
Rd , then, for anyR continuous function w : Iδ → Iδ × B(0; R) and any ξ in L1 (Iδ )
such that |En |−1 En ξ(t)dt → ξ¯ ∈ Rd , there exists a subsequence {kn } for which
Z
1
¯ + on (1).
L(w(t), ξ(t))dt ≥ L(w(t0 ), ξ)
|Ekn | Ekn
R
Proof. If there exists a subsequence {kn } such that |Ekn |−1 Ek |ξ(t)|dt → ∞, then
n
Z
Z
1
1
¯ + on (1).
L(w(t), ξ(t))dt ≥
|ξ(t)|dt ≥ L(w(t0 ), ξ)
|Ekn | Ekn
|Ekn | Ekn
R
Otherwise, assume that {|En |−1 En |ξ(t)|dt} is bounded.
Since convex functions are a.e. differentiable and since the subgradients of convex
functions are bounded on bounded sets, there exists a sequence {ξn } ⊂ Rd , which
4
A DIRECT PROOF OF THE TONELLI’S PARTIAL REGULARITY RESULT
¯ such that, for every n, the subgradient of L(w(t0 ), ·) at ξn is a
converges to ξ,
¯
singleton, i.e. ∂ξ L(w(t0 ), ξn ) = {pn (t0 )}, and pn (t0 ) → p(t0 ), p(t0 ) in ∂ξ L(w(t0 ), ξ).
By Proposition 2 in [3], limt→t0 pn (t) = pn (t0 ), where pn (t) ∈ ∂ξ L(w(t), ξn ), and
therefore there exists a subsequence {kn } for which kpn − pn (t0 )kL∞ (Ekn ) → 0.
Then, kpn − p(t0 )kL∞ (Ekn ) → 0.
By the convexity of L in ξ, we have
L(w(t), ξ(t)) ≥ L(w(t), ξn ) + hpn (t), ξ(t) − ξn i,
for every t in Ekn and any n. By taking the integral over Ekn of the inequality
above, we obtain
Z
Z
¯ +
¯
L(w(t), ξ(t))dt ≥ |Ekn |L(w(t0 ), ξ)
[L(w(t), ξn ) − L(w(t0 ), ξ)]dt
Ekn
Ekn
Z
+
[hpn (t) − p(t0 ), ξ(t) − ξn i + hp(t0 ), ξ(t) − ξn i]dt.
Ekn
By the boundedness of {ξn } and of the sequence of the averages of |ξ| on En , the
two integrals at the second member of the inequality above are equal to |Ekn |on (1).
That concludes the proof.
Theorem 3. Let d ≥ 2. If L(t, u, ·) is strictly convex, for any t in [a, b] \ ΣL and
any u in Rd , then any solution ū in W01,1 (a, b) to the variational problem (3) is
regular in the sense of Tonelli.
Proof. Let ū in W01,1 (a, b) be a solution to (3).
Part 1. Let t0 be any point in [a, b) \ ΣL , I() be the closed interval [t0 , t0 + ],
> 0, and fix C to be a compact set of [a, b] \ ΣL big enough to contain t0 as
interior point. We claim that
Z
1
˙
∃ lim
ū(t)dt
=: ξ¯ in Rd ∪ {∞}.
→0 I()
Observe that the claim is trivially true at a.e. point t0 in [a, b) \ ΣL , that is at the
˙ What we claim is that this is true for all points of [a, b) \ ΣL .
Lebesgue points of ū.
Here, for ξ¯ = ∞, we mean that
Z
1 ˙
lim ū(t)dt
= ∞.
→0 I()
Suppose, on the contrary, that there exist two sequences of positive numbers
{1n }, {2n }, 2n+1 < 1n < 2n , for every n, which converge to 0, such that
Z
1
˙
lim 1
ū(t)dt
=: ξ¯1 in Rd ,
n→∞ n I(1 )
Z n
1
˙
lim
ū(t)dt
=: ξ¯2 in Rd ,
n→∞ 2
n I(2n )
and
ξ¯1 6= ξ¯2 .
(4)
¯
¯
¯
Since ξ1 and ξ2 cannot be both ∞ by (4), one of the two, say ξ1 , belongs to Rd .
Furthermore, by the continuity of the integral function
Z
1
˙
ū(t)dt,
for in [1n , 2n ],
M (I()) :=
|I()| I()
A DIRECT PROOF OF THE TONELLI’S PARTIAL REGULARITY RESULT
5
we can assume that ξ¯2 belongs to Rd , too, by moving 2n closer to 1n .
We can also suppose, by replacing L(t, u, ξ) with α[L(t, u, ξ) + hp̄1 , ξ − ξ¯1 i] + β,
for suitable α, β ≥ 0 and p̄1 in Rd , that
L(t, u, ξ) ≥ |ξ|, for any (t, u, ξ) in C × B(0; kūkL∞ (a,b) + 1) × Rd ,
(5)
where recall that C is the compact set fixed at the beginning of Part 1 (see [4, 6]).
˙ 0 + 2n t)}n converges in measure, for
We claim that the sequence of functions {ū(t
t in (0, 1), to the constant function ξ¯2 .
Indeed, on the contrary, there exists δ > 0 for which the set En (δ) := {t ∈ (0, 1) :
˙ 0 + 2n t) − ξ¯2 | ≥ δ} is such that |En (δ)| ≥ λ > 0, for every n. Define
|ū(t
˙ − ξ¯2 | ≥ δ}.
En2 (δ) := t0 + 2n En (δ) = {t ∈ I(2n ) : |ū(t)
Analogously as in the proof of Lemma 2, there exists a sequence {ξn } ⊂ Rd ,
which converges to ξ¯2 , such that, for every n, ∂ξ L(t0 , ū(t0 ), ξn ) = {pn (t0 )}, and
pn (t0 ) → p(t0 ), p(t0 ) in ∂ξ L(t0 , ū(t0 ), ξ¯2 ). Therefore, for pn (t) in ∂ξ L(t, ū(t), ξn ),
there exists a subsequence {kn } for which kpn − pn (t0 )kL∞ (I(2k )) → 0. Then,
n
kpn − p(t0 )kL∞ (I(2k
n
))
→ 0.
Reindex {I(2n )} and {En2 (δ)} with {kn }.
By the strict convexity of L on ξ, the functions L(t, ū(t), ξ) − hpn (t), ξ − ξn i and
L(t0 , ū(t0 ), ξ) − hp(t0 ), ξ − ξ¯2 i have their unique minima over ξ in Rd respectively at
˙ over I(2n ) is bounded1
ξ = ξn and ξ = ξ¯2 . Since the sequence of the averages of |ū|
2
2
˙
and |En (δ)|/n ≥ λ, there exist h > |ξ¯2 | + δ, λ̃ > 0 for which |{t ∈ En2 (δ) : |ū(t)|
≤
h}|/2n ≥ λ̃, for every n. Let c > 0 be such that
L(t, ū(t), ξ) − hpn (t), ξ − ξn i
= L(t0 , ū(t0 ), ξ) − hp(t0 ), ξ − ξ¯2 i + on (1) ≥ L(t0 , ū(t0 ), ξ¯2 ) + c + on (1),
for every |ξ − ξ¯2 | ≥ δ, with |ξ| ≤ h, and any t in I(2n ). Therefore,
Z
˙
˙ − ξn i]dt
[L(t, ū(t), ū(t))
− hpn (t), ū(t)
I(2n )
˙
≥ |{t ∈ En2 (δ) : |ū(t)|
≤ h}|[L(t0 , ū(t0 ), ξ¯2 ) + c + on (1)]
˙
+|I(2 ) \ {t ∈ E 2 (δ) : |ū(t)|
≤ h}|[L(t0 , ū(t0 ), ξ¯2 ) + on (1)]
n
(6)
n
≥ 2n [L(t0 , ū(t0 ), ξ¯2 ) + λ̃c + on (1)].
On the other hand, by the minimality of ū,
Z
˙
L(t, ū(t), ū(t))dt
2)
I(
Zn
≤
L(t, ū(t0 ) + M (I(2n ))(t − t0 ), M (I(2n )))dt
(7)
I(2n )
= 2n [L(t0 , ū(t0 ), ξ¯2 ) + on (1)].
Comparing (7) and (6), we reach a contradiction since the integral of hpn , ū˙ − ξn i
over I(2n ) is equal to 2n on (1) and λ̃c > 0. This proves the claim.
1 Observe that in general, by the minimality of ū, (5) and (7) imply that the sequence of
˙ over intervals In ⊂ [a, b] \ ΣL , |In | → 0, is bounded as long as {M (In )}, i.e. the
averages of |ū|
sequence of the averages of ū˙ over In , is bounded.
6
A DIRECT PROOF OF THE TONELLI’S PARTIAL REGULARITY RESULT
˙ 0 + 2n ·)}n , by Egorov’s theorem and a
From the convergence in measure of {ū(t
diagonal argument, there exists a positive sequence {δn } such that, setting Zn2 :=
En2 (δn ), |Zn2 |/2n → 0 and
δn = kū˙ − ξ¯2 kL∞ (I(2n )\Zn2 ) → 0,
(8)
˙ I(2 )\Z 2 converges uniformly to ξ¯2 . This also implies that
that is, ū|
n
n
lim 1n /2n = 0.
n→∞
We proceed with the proof separating three cases: the first case in which ū˙ is
bounded close to t0 , i.e. in I(2n ), the second case in which ū˙ is unbounded but the
˙ i.e. M (En2 (δ)) with δ big, have a limit different
averages of the high values of ū,
¯
from ξ2 and the third case in which the averages of the high values of ū˙ converge
always to ξ¯2 .
Case 1. If there exists δ̄ > 0 such that |En2 (δ̄)| = 0, for every n, then we choose
a measurable set Bn2 in I(2n ) \ [I(1n ) ∪ Zn2 ] of the same measure as I(1n ), and we
Rt
define the competitor wn in W01,1 (a, b) by wn (t) := a ẇn (τ )dτ where
(
τ in I(1n ) ∪ Bn2 ,
[M (I(1n )) + M (Bn2 )]/2,
ẇn (τ ) :=
˙ ),
ū(τ
otherwise.
Observe that wn coincides with ū on [a, b] \ I(2n ).
We claim that
γn := kwn − ūkL∞ (I(2n )) = 1n On (1),
where On (1) is a bounded sequence. Indeed, by (5), the minimality of ū and the
Lipschitz continuity of L in u, we have that
Z
Z
˙
˙
|ū(t)|dt
≤
L(t, ū(t), ū(t))dt
2
2
I(1n )∪Bn
I(1n )∪Bn
Z
Z
˙
≤
L(t, wn (t), ẇn (t))dt −
L(t, ū(t), ū(t))dt
≤
I(2n )
21n [L(t0 , ū(t0 ), (ξ¯1
2]
I(2n )\[I(1n )∪Bn
+ ξ¯2 )/2) + on (1)]
Z
˙
˙
[L(t, wn (t), ū(t))
− L(t, ū(t), ū(t))]dt
Z
≤ 21n [L(t0 , ū(t0 ), (ξ¯1 + ξ¯2 )/2) + on (1)] + γn
+
2]
I(2n )\[I(1n )∪Bn
2]
I(2n )\[I(1n )∪Bn
C|ξ̄2 |+δ̄ (t)dt.
Observing that
Z
γn ≤
2
I(1n )∪Bn
˙
|ū(t)|dt
+ 1n [|ξ¯1 + ξ¯2 | + on (1)]
and, by the integrability of C|ξ̄2 |+δ̄ , that the integral over I(2n ) \ [I(1n ) ∪ Bn2 ] of
C|ξ̄2 |+δ̄ converges to 0, as n goes to ∞, we infer that
γn [1 − on (1)] ≤ 21n [L(t0 , ū(t0 ), (ξ¯1 + ξ¯2 )/2) + |ξ¯1 + ξ¯2 |/2 + on (1)],
which implies the claim.
A DIRECT PROOF OF THE TONELLI’S PARTIAL REGULARITY RESULT
7
By Lemma 2 on I(1n ) and on Bn2 , we have
Z
˙
L(t, ū(t), ū(t))dt
≥ 1n [L(t0 , ū(t0 ), ξ¯1 ) + L(t0 , ū(t0 ), ξ¯2 ) + on (1)]
I(2n )
Z
˙
+
L(t, ū(t), ū(t))dt.
2]
I(2n )\[I(1n )∪Bn
By the strict convexity of L in ξ, there exists a positive constant c > 0 such
that L(t0 , ū(t0 ), ξ¯1 ) + L(t0 , ū(t0 ), ξ¯2 ) ≥ 2L(t0 , ū(t0 ), (ξ¯1 + ξ¯2 )/2) + c. Hence, by the
Lipschitz continuity of L in u, and since γn = 1n On (1),
Z
˙
L(t, ū(t), ū(t))dt
≥ 1n [2L(t0 , ū(t0 ), (ξ¯1 + ξ¯2 )/2) + c + on (1)]
I(2n )
Z
˙
+1n on (1) +
L(t, wn (t), ū(t))dt
2]
I(2n )\[I(1n )∪Bn
= 1n [c + on (1)] +
Z
L(t, wn (t), ẇn (t))dt.
I(2n )
Thus, if n̄ is such that on̄ (1) < c, I(ū) > I(wn̄ ), in contradiction with the minimality of ū.
Case 2. If, for every δ > 0, |En2 (δ)| > 0, for every n, and there exists δ̄ > 0 such
that M (En2 (δ̄)) converges to ξ˜ 6= ξ¯2 in Rd , possibly ξ˜ = ∞, then, by the regularity
of the Lebesgue measure and the continuity of the integral, there exists a sequence
of measurable sets A2n ⊃ En2 (δ̄) in I(2n ) \ I(1n ) such that M (A2n ) converges to ξ¯
˙ 0 + 2n ·)}n converges in measure to ξ¯2 , then
in Rd finite and ξ¯ 6= ξ¯2 . Since {ū(t
2
2
lim |An |/n = 0.
Letting Bn2 be a measurable subset of I(2n ) \ [I(1n ) ∪ Zn2 ∪ A2n ] such that |Bn2 | =
Rt
2
|An | =: αn2 , we define the competitor wn in W01,1 (a, b) by wn (t) := a ẇn (τ )dτ
where
(
[M (A2n ) + M (Bn2 )]/2,
τ in A2n ∪ Bn2 ,
ẇn (τ ) :=
˙ ),
ū(τ
otherwise.
Observe that wn coincides with ū on [a, b] \ [I(2n ) \ I(1n )].
Analogously as in Case 1, one can prove that
γn := kwn − ūkL∞ (I(2n )) = αn2 On (1).
Indeed, this claim depends only on (5), the minimality of ū, the Lipschitz continuity
˙ over I(2n ) \ [I(1n ) ∪ A2n ∪ Bn2 ].
of L in u and the boundedness of |ū|
Again, as in Case 1, one can prove that, by Lemma 2 on A2n and on Bn2 , the
strict convexity of L in ξ and the Lipschitz continuity of L in u,
Z
¯ + L(t0 , ū(t0 ), ξ¯2 ) + on (1)]
˙
L(t, ū(t), ū(t))dt
≥ αn2 [L(t0 , ū(t0 ), ξ)
2
1
I(n )\I(n )
Z
˙
+
L(t, w(t), ū(t))dt
≥
2]
I(2n )\[I(1n )∪A2n ∪Bn
2
αnZ[c + on (1)]
+
L(t, wn (t), ẇn (t))dt.
I(2n )\I(1n )
Thus, if n̄ is such that on̄ (1) < c, I(ū) > I(wn̄ ), in contradiction with the minimality of ū.
8
A DIRECT PROOF OF THE TONELLI’S PARTIAL REGULARITY RESULT
Case 3. If, for every δ > 0, |En2 (δ)| > 0, for every n, and M (En2 (δ)) → ξ¯2 , we
choose δ̄ := L(t0 , ū(t0 ), ξ¯2 ) + 1 + |ξ¯2 |, En2 := En2 (δ̄) \ I(1n ).
Rt
Define the competitor wn in W01,1 (a, b) by wn (t) := a ẇn (τ )dτ where
(
M (En2 ),
τ in En2 ,
ẇn (τ ) :=
˙ ),
ū(τ
otherwise.
Observe that wn coincides with ū on [a, b] \ [I(2n ) \ I(1n )].
˙ is bounded by |ξ¯2 | + δ̄ over
Analogously as in Case 1 and Case 2, since |ū|
I(2n ) \ [I(1n ) ∪ En2 ], one can prove that
γn := kwn − ūkL∞ (I(2n )) = |En2 |On (1).
By (5) and the Lipschitz continuity of L in u,
Z
˙
L(t, ū(t), ū(t))dt
≥ |En2 |[L(t0 , ū(t0 ), ξ¯2 ) + 1 + on (1)]
I(2n )\I(1n )
Z
˙
+
L(t, w(t), ū(t))dt
≥
2]
I(2n )\[I(1n )∪En
2
|EZn |[1 + on (1)]
+
L(t, wn (t), ẇn (t))dt.
I(2n )\I(1n )
Thus, if n̄ is such that on̄ (1) < c, I(ū) > I(wn̄ ), in contradiction with the minimality of ū. This concludes the three cases and the proof of the claim at the beginning
of the proof, that is, for every t0 in [a, b) \ ΣL ,
Z
1
˙
ū(t)dt
= ξ¯2 =: ξ¯ in Rd ∪ {∞},
∃ lim
→0 I()
In the same way one can show that, if I() is the closed left interval [t0 − , t0 ]
then
Z
1
˙
∃ lim
ū(t)dt
=: ξ¯0 in Rd ,
→0 I()
and also, slightly modifying the proof above, that the limit of the right and the left
averages of ū˙ at t0 , i.e. ξ¯ and ξ¯0 , must be equals (replacing I(1n ) with [t0 − 1n , t0 ]
and I(2n ) with [t0 − 1n , t0 + 2n ])2 .
Part 2. Let {tn } ⊂ [a, b] \ ΣL be any sequence of Lebesgue points of ū˙ which
converges to t0 .
We claim that
¯
˙ n ) = ξ,
∃ lim ū(t
n→∞
where ξ¯ is defined in Part 1 as the limit of the averages of ū˙ at t0 . Indeed, suppose
on the opposite that (by passing to a subsequence)
¯
˙ n ) =: ξ¯1 in Rd , ξ¯1 6= ξ.
lim ū(t
n→∞
Assume also for simplicity that tn < t0 , for every n (the other case tn > t0 , for
˙ there exists
every n, can be proved analogously). Being tn a Lebesgue point for ū,
2 The contradiction argument of Part 1 is based on the existence of two sequences of sets of
positive measure, which converge to a point t0 and which union is an interval, such that the limit
of the average of ū˙ on those sets are different.
A DIRECT PROOF OF THE TONELLI’S PARTIAL REGULARITY RESULT
1n in (0, 2n ) such that
Z tn +1n
1
˙
˙ n ) + on (1) = ξ¯1 + on (1).
ū(t)dt
= ū(t
1n tn
9
(9)
Proceeding similarly as in Part 1 (replacing I(1n ) with [tn , tn + 1n ] and I(2n ) with
[tn , t0 ]), one can prove the claim.
We have therefore obtained by Part 1 and Part 2 that, for every point t0 in
[a, b] \ ΣL and for any sequence of Lebesgue points {tn } of ū˙ which converges to t0 ,
˙ n ) = ξ¯ in Rd ,
lim ū(t
n→∞
¯ 0 ). Indeed, by Part 1, denoting by E
where ξ¯ depends only on t0 , i.e. ξ¯ = ξ(t
¯
˙
the set of Lebesgue points of ū, the functions ū˙ restricted to E coincides with ξ.
¯
Therefore, by Part 2, the function ξ can be extended by continuity to [a, b] \ ΣL .
Hence, ū˙ is a.e. equal on [a, b] \ ΣL to an extended-values continuous functions on
[a, b] \ ΣL .
From Theorem 3, the set [a, b] \ (ΣL ∪ Sū˙ ) is numerable union ofSdisjoint open
intervals In of [a, b] such that ū|In belongs to C1 (In ), for any n, and | n In | = b−a.
Notice that the proof of Theorem 3 is valid also for d = 1 and it shows that
ū˙ belongs to C([a, b] \ ΣL ; R ∪ {∞}). Nevertheless, in Theorem 4, we propose a
finer Tonelli’s partial regularity result for the scalar case. Before, we present two
examples that show the optimality of our result.
First, strict convexity is a necessary condition for Theorem 3. Indeed, consider
the Lagrangian
L(t, u, ξ) := |u − f (t)|(ξ + 1)2
where f is a function in W01,1 (0, 1) with Sf˙ a non-closed set of [0, 1]. Notice that
the function L is continuous on t and ξ, smooth with respect to u, with ΣL = ∅,
and convex with respect to ξ but not strictly convex as u = f (t). The variational
problem
Z 1
min
|u(t) − f (t)|[u̇(t) + 1]2 dt,
0
for u in W01,1 (0, 1), admits the unique solution ū = f which is not regular in the
sense of Tonelli.
Second, the continuity of L with respect to t in [a, b] \ ΣL , with ΣL closed, is
an optimal condition and the continuity assumption on L cannot be replaced by a
Carathéodory condition. In fact, consider the Lagrangian
L(t, u, ξ) := [ξ − f (t)]2
R1
where f is a function in C([0, 1] \ ΣL ), 0 f = 0, ΣL is a non-closed set in [a, b],
|ΣL | = 0. The function L is strictly convex with respect to ξ and continuous on
[a, b] \ ΣL with respect to t. The variational problem
Z 1
min
[u̇(t) − f (t)]2 dt,
0
Rt
for u in
admits the unique solution ū(t) = 0 f which is not regular in
the sense of Tonelli.
Third, the integrability of the Lipschitz continuity constant CR of L in u is an
optimal condition as the example in [12] shows.
W01,1 (0, 1),
10
A DIRECT PROOF OF THE TONELLI’S PARTIAL REGULARITY RESULT
Theorem 4. Let d = 1. If L(t, u, ·) is strictly convex, for any t in [a, b] \ ΣL and
any u in R, then any solution ū in W01,1 (a, b) to the variational problem (3) is
regular in the sense of Tonelli.
Proof. The key feature of the scalar case is that any continuous path in R that goes
from ±∞ to ∓∞ is forced to pass by a finite point of R, which is not the case in
the multidimensional setting. The proof is very similar to the one of Theorem 3.
We present here the main differences.
Part 1. Let t0 be a point in [a, b) \ ΣL and let I() be the closed right interval
[t0 , t0 + ], > 0.
We claim that
Z
1
˙
∃ lim
ū(t)dt
=: ξ¯ in R ∪ {+∞} ∪ {−∞}.
→0 I()
Indeed, suppose, on the contrary, that there exist two sequences of positive numbers
{1n }, {2n }, 2n+1 < 1n < 2n , which converge to 0, such that
Z
1
˙
lim
ū(t)dt
=: ξ¯1 in R,
n→∞ 1
n ZI(1n )
1
˙
lim 2
ū(t)dt
=: ξ¯2 in R,
n→∞ n I(2 )
n
and
ξ¯1 6= ξ¯2 .
Suppose, without loss of generality, that ξ¯1 = +∞ and ξ¯2 = −∞. Then, by the
continuity of the integral function M (I()), for in [1n , 2n ] (see the proof of Theorem
3), we can choose 2n such that
Z
1
˙
ū(t)dt
= 1.
lim
n→∞ 2
n I(2n )
The proof therefore follows as in Part 1 of Theorem 3.
Part 2. Let {tn } ⊂ [a, b] \ ΣL be a sequence of Lebesgue points of ū˙ which
converges to t0 .
We claim that
¯
˙ n ) = ξ,
∃ lim ū(t
n→∞
where ξ¯ is defined in Part 1 as the limit of the averages of ū˙ at t0 . Indeed, suppose
on the opposite that (by passing to a subsequence)
¯
˙ n ) =: ξ¯1 in R, ξ¯1 6= ξ.
lim ū(t
n→∞
Assume also for simplicity that tn < t0 , for every n (the other case tn > t0 , for
˙ there exists
every n, can be proved analogously). Being tn a Lebesgue point for ū,
1n in (0, 2n ) such that
Z tn +1n
1
˙
˙ n ) + on (1) = ξ¯1 + on (1).
ū(t)dt
= ū(t
(10)
1n tn
Suppose, without loss of generality, that ξ¯ = +∞ and ξ¯1 = −∞. Then, by the
continuity of M (I()), for in [1n , 2n ], we can choose 1n such that
Z
1
˙
lim 1
ū(t)dt
= 1.
n→∞ n I(1 )
n
A DIRECT PROOF OF THE TONELLI’S PARTIAL REGULARITY RESULT
11
Proceeding similarly as in Part 1 of Theorem 3, one can prove the claim.
We then obtain by Part 1 and Part 2 that ū˙ is a.e. equal to a continuous function
from [a, b] \ ΣL to R.
As a Corollary of Theorem 3 and 4, we obtain that any Lipschitz solution to the
variational problem (3) is actually C1 .
Corollary 5. Let d ≥ 1. If L(t, u, ·) is strictly convex, for any t in [a, b] \ ΣL and
any u in Rd , then any solution ū in W01,∞ (a, b) to the variational problem (3) is
C1 ([a, b] \ ΣL ).
Proof. It follows directly from Theorem 3 and 4.
Acknowledgement. The author wishes to thank the Spanish Ministry of Science
and Innovation (MICINN) for supporting the research contained in this work.
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E-mail address: [email protected]