Proceedings of the International Congress of Mathematicians
Berkeley, California, USA, 1986
The Problem of the Regularity of Minimizers
MARIANO GIAQUINTA
1. Introduction. Since the origin of Calculus of Variations, and for a long
time, all the information concerning the minima of variational functionals
f[u; fi] = / F(x, u, Du) dx
Jn
were obtained by means of its Euler-Lagrange equation
(1.1)
dx dp
du
It was in this century, after the pioneering works of Riemann, Arzelà, Hilbert,
Lebesgue, and the strong contributions of Leonida Tonelli and Charles B. Morrey,
that the so-called direct methods of Calculus of Variations established themselves
as the main tool to deal with the problem of the existence of minima.
As a result of the use of direct methods, a sufficiently general theorem that
we may now state because of the contributions of many authors is
THEOREM 1.1. Let fi be a bounded domain in Rn; let F(x, u,p):Qx RN x
R
—> R be measurable in x, continuous in (u,p), and quasiconvex; i.e., for
almost every XQ E fi and for all UQ E RN and p0 E RnN and all <ß(x) E
Cg°(ü,RN) we have
nN
/ F(x0,uo,po)dx < / F(xn,UQ,pQ + D<j))dx.
(1.2)
Jn
Jn
Moreover suppose that \p\m < F(x,u,p) < A(l + |p| 2 ) m / 2 where m > 1 and
A > 0. Then the functional 7 attains its minimum in the class of functions
u E Wlim(Q,RN)
with prescribed value at the boundary dii.
The notion of quasiconvexity, which seems to be a global condition, was introduced by C. B. Morrey in 1952 [45], and Theorem 1.1, under some stronger
^assumptions—goesœback^to4iim"(fo^
recall that in the scalar case N = 1, the condition of quasiconvexity is equivalent
to convexity with respect to p, that is, if F is smooth, to
n
E ^ w ^ M « " ^
V£eR",
(1.3)
a,ß=l
© 1987 International Congress of Mathematicians 1986
1072
THE PROBLEM OF THE REGULARITY OF MINIMIZERS
1073
while in the vector-valued case TV > 1 it is weaker than convexity and implies,
again if F is smooth, the so-called Legendre-Hadamard condition
N
E
n
E
^ ( ï , « , P ) f ^
> 0
VC G R - , V77 E RN.
(1.4)
itj=la,ß=l
We notice that quasiconvexity appears naturally as "the necessary and sufficient condition" for the sequentially lower semicontinuity of 7 in W 1,m -weak.
Moreover we notice that the use of Sobolev spaces W1,m, m > 1, is by now
natural for the application of direct methods and is related to the need of working
on a class of functions with a sufficiently weak topology, so that minimizing
sequences do converge.
So the regularity problem arises in a natural way; it is the problem of showing,
if possible, continuity or differentiability of the minimum points (Hubert's nineteenth problem). Since we would like to avoid, as far as possible, any complications due to boundary conditions, we shall consider most of the time minimizers
of 7. A minimizer is a function u E Wlìm(tì,RN)
such that
7[u; spt <t>] < 7[u + (j>\ spt <f>] Vty E Whm(Ü, RN)
spt cßmü.
In dealing with the regularity problem again, the Euler-Lagrange equation in
its weak formulation
f [Fpi (x,ti,Du)Da^
+ Fui(x,u,Du)<P\dx = 0
V^C0°°(fi,RN)
(1.5)
Jn
has always been the starting point even in recent times—for instance, in the
classical and celebrated works of E. De Giorgi, J. Nash, J. Moser, O. A. Ladyzhenskaya and N. N. Ural'tseva (see, e.g., [39, 46, 29, 17]).
Actually this is natural and in a sense necessary if we want to study the
continuity of the second order (or higher order) derivatives of a minimizer; a
classical result, obtained after a research span of fifty years, is the following one:
if the integrand F is C°°, or analytic, and
F
,'apf,(x>*>P)(atß*W > C(M)I£I2M2
v
£ e R n , Vrç e RN
for \u\ + \p\ < M, with c(M) > 0, then the points of minimum are C°°, or
analytic, as soon as it is known that they are of class C1. But this approach has
some disadvantages and it appears somehow unnatural when studying the first
two steps in the regularity theory: the Holder-continuity and the differentiability
of the minimizers.
In fact, first of all, even if F is smooth, (1.5) does not hold without assuming
additional conditions on the behavior of Fu at infinity, in such a way that for
u E W1,m, Fu(x,u,Du) lies in L\oc. But, even assuming the so-called "natural
growth conditions," which ensure the local integrability of Fu(x, u, Du), the weak
Euler-Lagrange equation (1.5) would still be of no use in order to prove regularity,
without extra conditions on the minimizers themselves such as boundedness or
even smallness in modulus (see for a discussion [17, 33, 39]). Roughly the point
1074
MARIANO GIAQUINTA
is that (1.5), and therefore this approach, does not distinguish between minimizers
and stationary points. But, in general, minimizers have better properties than
just stationary points.
An interesting geometric example is given by the integral
£[u;D] = f \Du\2/(l + \u\2)2dx,
u = (u1,.
..,un),
JD
which, apart from a constant, represents in local coordinates (choosing stereographic coordinates on the sphere Sn minus a point) the energy of a map from
the disk Dn = #i(0) = {x E Rn:\x\ < 1} into Sn-{point}. S. Hildebrandt,
H. Kaul, K. O. Widman [34] have shown, as a consequence of a more general
result, that the minimizers of £, whose image lies strictly in a hemisphere (in
our system of coordinates this corresponds to \u\ < k < 1) are regular, while
the "equator map u*:x —> x/\x\" which is obviously not strictly contained in
a hemisphere, is a stationary point for £. W. Jager, H. Kaul [35] then showed
that u* is actually a minimizer if n > 7 and it is not even stable, so in particular
it is not a minimizer, for n < 6. In general we have that any energy minimizing
map from the disk Dn into a sphere Sn whose image lies (not necessarily strictly)
into a hemisphere is regular provided n < 7 (see [54, 28]).
Another interesting example is due to J. Prehse [12] who pointed out that the
functional
(l + e^llogHI12)-1!^!2^,
/
1
./BtO.e- )
which obviously has 0 as minimum point in WQi2(B(0,e~1)), has among its
stationary points the unbounded function of WQ,2(B(0,C~1))
given by u(x) =
12 log log lai - 1 .
In the last five years there has been a strong attempt to develop a kind of
direct approach to the regularity, working directly with the functional 7 instead
of working with its Euler-Lagrange equation. We mention especially the works
of M. Giaquinta, E. Giusti, in the general situation, and the works of R. Schoen,
K. Uhlenbeck, concerning the regularity theory of harmonic mapping between
Riemannian manifolds. Moreover we mention the earlier works of L. Tonelli [56]
and C. B. Morrey [44] where the idea of a direct approach to the regularity is
present and regularity results for minimizers of nondifferentiable functionals in
dimensions 1 and 2 are proved under extremely weak assumptions.
It is the aim of this lecture to illustrate some of the results obtained in this
direction.
—-In=the^following^it-is^convenientto^distinguishrtwo ievels^of^regolarrtjrr^lT"
regularity of the minimizer u; 2. regularity of the derivatives of the minimizer u.
Of course our results will take a different form in the scalar and in the vectorvalued case, respectively N = 1 and N > 1. In fact, while in the scalar case
it is natural to expect, under suitable hypotheses, regularity everywhere, in the
vector-valued case this is quite rare; minimizers are in general noncontinuous
or have noncontinuous derivatives, as shown by well-known counterexamples of
THE PROBLEM OF THE REGULARITY OF MINIMIZERS
1075
E. De Giorgi, E. Giusti and M. Miranda, V. G. Mazya, J. Neöas, S. Hildebrandt
and K. O. Widman, M. Giaquinta (see, e.g., [17]), and we may only expect
partial regularity, that is, regularity except possibly on a singular closed set.
2. Basic regularity: quasimini ma. In studying the first level of regularity the notion of quasiminima, introduced by M. Giaquinta and E. Giusti
[18, 22], plays an important role for its unifying and clarifying feature (besides,
of course, the fact that we can prove interesting results for quasiminima).
Consider the functional 7 in (1.1) and suppose, for simplicity, that
|p| w < F{xtutp) < A(l + IPP),
rn > 1.
(2.1)
We say that a function u E W^(Ü,RN)
is a Q-minimum, Q > 1, for the
functional 7 if for every open set A m fi and for every v E ^ ^ ( f i j R ^ ) with
v = u outside A we have
7[u;A)<Q7[v,A)
(2.2)
or, equivalently, for any <j> E Wlim(ü,RN)
with spt^ e fi we have
7[u; {x E fi: (ß(x) £ 0}] < Q7[u + <j)\ {x E fi: tß(x) + 0}].
(2.3)
We notice that the comparison in (2.2) is made for all open sets A\ we may
think of choosing special classes of A^s, for example, balls BR. Then we say that
u E W1,m(Q,RN) is a spherical Q-minimum if for any ball BR C fi and for any
</)EWoim(BR,RN) we have
7\u]BR)<Q7[u
+ (t>',BR].
(2.4)
Obviously any Q-minimum is a spherical Q-minimum, but the opposite is not
true; moreover, in the scalar case N = 1, spherical Q-minima may be unbounded
in dimension n > 3 (see [22]).
Of course any minimizer of 7 is a Q-minimum for 7 with Q = 1; but it is
also a Q-minimum for the simpler functional / n ( H - |^w| m ) dx. This shows once
more the special relevance of the "Dirichlet integral." But the class of Q-minima
is much wider [22]:
(a) Weak solutions of elliptic systems with L°° (and not even symmetric)
coefficients
/ A^f(x)DQuiDß^
dx = 0,
il>E W 0 M (fi,R N ),
J Q
\Ag(x)\<L,
Afftf$>\t\a
VURnN
are Q-minima for the Dirichlet integral: In order to see that, it is sufficient to
choose as test function ip = u - (u + (/>), (j) E Wl'2(ü, RN).
(b) In general "solutions" (here the word "solution" has to be understood in
the right sense; depending on the situations, they have to be bounded or even
small (see [22, 17])) of nonlinear elliptic systems DaAf(x, u, Du) = B(x,u,Du)
under "natural hypotheses" are Q-minima for suitable functionals.
(c) Minimizers of functionals in constrained classes are Q-minima of free functionals.
1076
MARIANO GIAQUINTA
(d) Quasiconformal maps and, more generally, quasiregular maps, i.e., u:ü C
Rn -> Rn, u E W^n(ü,Rn)
such that \Du(x)\n < cdetDu a.e. in fi, are Qn
minima for fn \Du\ dx.
We have [18, 22]
THEOREM 2.2. (i) In the scalar case N = 1: let u E V^' c m (fi,R) be a
Q-minimum for 7; then u is locally Holder-continuous in fi.
(ii) In the vector-valued case N > 1: let u E W^(Q,RN)
be a spherical Qminimum for 7\ then there exists an exponent r > m such thatu E
W^(Ü,RN)',
moreover the gradient of u satisfies the following reverse Holder inequality with
increasing supports,
a
(l + \Du\r)dx)
<c(jf
(l + \Du\«)dx\
\
for all BR e fi and all q, 0 < q < m. In particular if m — n, then u is locally
Holder-continuous.
The proof of (i) uses De Giorgi's results and their extensions due to Ladyzhenskaya and Ural'tseva on what we now call De Giorgi's classes [7, 39], while the
proof of (ii) uses a result of M. Giaquinta and G. Modica [24] on reverse Holder
inequalities related to a previous result of W. Gehring [15]. Both results rely on
a Caccioppoli type inequality (see, e.g., [17]).
If we read Theorem 2.2 for minimizers, it states the basic and optimal regularity properties of minimum points. We notice that no hypothesis of ellipticity has
been made, so the basic regularity follows from the minimality and the growth
condition 2.1. It is worth noticing that the results of Theorem 2.2 are not true
for stationary points of 7, even under ellipticity conditions.
On the other hand, Theorem 2.2 permits to recover essentially all the results of
Holder-continuity and higher integrability of the gradient known for "solutions"
(under inverted commas!) of "nonlinear elliptic systems" as consequences of the
minimality condition 2.2.
As mentioned, well-known counterexamples show that in the vector-valued
case minimizers (and therefore Q-minima) are in general noncontinuous. J.
SouCek [55] has shown that solutions of linear elliptic systems with L°° coefficients may be discontinuous on a dense set. Therefore, because of (b), Q-minima
of the Dirichlet integral, for N > 1, may be discontinuous in a dense set; so it is
quite surprising that the gradient which lies in Lm is in fact p-summable with
some p larger than m.
^Finally^wcrwoukHike tor remark^thatrthe higher idegreenjMnfcegrabilitjr
(ii) clearly appears as a consequence of a comparison on BR with "harmonic
functions" with the same boundary value as the Q-minimum u, so it can be
considered as a result of a linear perturbation. On the contrary, the result (i) of
Theorem 2.2 does not hold for spherical Q-minima [22], so it is to be considered,
in a sense, as a purely nonlinear result: this, in particular, is true for De GiorgiNash theorem.
THE PROBLEM OF THE REGULARITY OF MINIMIZERS
1077
Most of the known properties of solutions of elliptic equations have been shown
to hold for scalar Q-minima, as the weak maximum principle, Liouville's type
theorems, and so on. In particular we mention the very interesting paper by
E. Di Benedetto and N. S. Trudinger [8] where they prove, using some ideas of
De Giorgi and N. V. Krylov-M, V. Safanov, that the classical result of J. Moser
[48] on Harnack's inequality holds for nonnegative quasi(-super-)minima and the
paper of W. P. Ziemer [58] where Wiener type conditions for the regularity of
Q-minima at boundary points are given. We also mention that the notion of
Q-minimum has proved to be very useful in various contexts (see, e.g., [17, 42,
49]) and finally that Theorem 2.2 is an essential step in proving "regularity" of
the derivatives of a minimizer.
3. Regularity of t h e first derivatives of a minimizer. When studying
the regularity of the first derivatives of a minimizer, "ellipticity" will clearly play
an important role; but "growth conditions," as explained later, are important
too, at least for the methods of proof. In any case it is not necessary to assume
that our functional be Gateaux differentiable, and this is not the case under our
assumptions on the integrand F(x, u,p):Qx RN x RnN —• R in (1.1) which will
be the following:
HYPOTHESIS 1. Growth conditions on F: for m>2
A we have
and a positive constant
ipr<i',(!B,ti)p)<A(i+ipn
(3.1)
where ra > 2. Actually it is sufficient that (3.1) holds in the integrated form on
small balls.
HYPOTHESIS 2. F is twice continuously differentiable with respect to p and
\FPP(x,u>P)\ < c i(M 2 + | p | a ) { m _ a ) / 2 i
(3-2)
in particular,
\Fp(x,u,p)\<co(p2
where
^\p\2)^-1)'2
(3.3)
p>0.
HYPOTHESIS 3. The function (1 + \p\2)~m/2F(x,u,p)
in (x, u) uniformly with respect to p.
is Holder-continuous
HYPOTHESIS 4. Strict uniform quasiconvexity: for all XQ E fi, uo E
Po E RnN and for any (j) E CQ(Q,RN)
we have
f [F(x0,u0,po + D(l>)-F(xoiuo,po)}dx>
Jn
f \V(p0 + D<ß) - V(p0)\2dx
Jn
where V(p) is the vector-valued function defined by V(p) = (p? +
RN,
(3.4)
\p\2)^m~2^4p.
1078
MARIANO GIAQUINTA
We notice that in the scalar case N = 1, Hypothesis 4 is equivalent to
HYPOTHESIS 4'. Strict uniform ellipticity: for some positive v
*W*o,«o,Po)&fc > * V + |Po| 2 ) (m - 2)/2 |£| 2
V£ e R",
(3.5)
soifp = 0we have degeneration at the points where the gradient of our minimizer
is zero.
We now have the following partial regularity result,
THEOREM 3.1. Suppose Hypotheses 1-4 hold with p > 0, N > 1. Let
u E Wfo™(ft,RN) be a minimizer of (1.1). Then there exists an open set fio C
fi where the first derivatives of u are locally Holder-continuous. Moreover the
Lebesgue measure of the singular set fi —fiois zero.
Theorem 3.1 was first proved by C. B. Morrey, E. Giusti and M. Miranda, and
E. Guisti essentially in the case F = F(p) and under the stronger assumption of
uniform ellipticity
F
Pi4lx>u>d&$
* & + N 2 ) (m " 2)/2 I£I 2
VZ e RnN
(3.6)
by means of an indirect argument. Still under condition (3.6) it was then proved,
by a direct argument, in the general case F = F(x,u,p) by M. Giaquinta and
E. Guisti for m = 2 and M. Giaquinta and P. A. Ivert for m > 2 in 1983-1984
(see, e.g., [17, 19, 23]). Under the strict quasiconvexity assumption, Theorem
3.1 was proved by L. C. Evans [10] (see also [11]) in the case F = F(p), again
by an indirect argument (see also [51]); in the form given above it was proved
in 1986 independently by M. Giaquinta and G. Modica [25], and N. Fusco and
J. Hutchinson [13].
I shall not insist on Theorem 3.1 and I refer to the talk of L. C. Evans at
the 1986 International Congress of Mathematicians. I would only like to remark
that no result seems to be available, in this generality, when 1 < m < 2 and in
the degenerate case p = 0, and that the result of Theorem 3.1 uses in a strong
way the minimality property (see also [27, 51]). It is worth recalling that our
functional 7 need not be differentiable.
As already stated, in the vector-valued case N > 1 the singular set is in
general nonempty; on the contrary, we expect that it will be empty and the
minimizers regular everywhere in the scalar case. This is actually true, even in
the degenerate case p = 0, and we have
THEOREM 3.2. In the scalar case N = 1, suppose that Hypotheses 1-3 and
Hypothesis 4' hold with p > 0. ThenjanyLminimizer_tLJEJVfo^(Q
Holder-continuousfirstderivatives in fi.
Theorem 3.2, in case F = F(p) and m = 2, is a consequence of the celebrated
De Giorgi-Nash Theorem [7] and, for m > 2, p > 0, F = F(p), it was proved in
a very interesting paper by K. Uhlenbeck [57]. For F = F(x,u,p) it was proved
for m = 2 by M. Giaquinta and E. Giusti [19] and, in the general case m > 2,
ß > 0, by M. Giaquinta and G. Modica
THE PROBLEM OF THE REGULARITY OF MINIMIZERS
1079
It is worth noticing again that under the assumptions of Theorems 3.1 and
3.2, the functional 7 in (1.1) is not differentiable, and that the result of Theorem
3.2 is not true for "stationary points." Finally we notice that in the degenerate
case p — 0, minimizers have not, in general, continuous second derivatives even
in the simple situation F = |p| m , m > 2 (see, e.g., [26, 40]); and in the general
situation F = F(x,u,p), it seems that there is an upper bound for the Holder
exponent which is strictly less than the Holder exponent of the function F(-, -,p)
(see, e.g., [20, 50]).
4. Further contributions. Ever since the first results of partial regularity
of minimizers were proved, many questions have been raised; and most of them
still have no answer. In this final section I shall state a few of these questions
and discuss some contributions.
1. Under which conditions are vector-valued minimizers regular everywhere?
In this direction surely the most interesting result is due to K. Uhlenbeck [57]
who showed that under the assumptions of Theorem 3.1, if moreover we assume
that F(x,u,p) = G(|p|2) with G smooth, then minimizers are regular everywhere.
Everywhere regularity can also be proved if the functional 7 "is not far" from
a quadratic functional or more generally from a functional whose minimizers are
regular (see, e.g., [38]).
2. Nothing is known on the structure of the singular set nor even on its
stability or instability with respect to perturbations of the data. More simply,
we may ask whether we can improve the estimate of the dimension of the singular
set. "Optimal" results have been proved for quadratic functionals.
Consider the variational integral
A[u;Q]=
f <A?f(x, uiDavfDßU? dx
Jn
(4.1)
where A®? are bounded and smooth functions satisfying the ellipticity condition
^ f f t f > |f|a
VURnN-
(4.2)
We notice that A is not differentiable. M. Giaquinta and E. Giusti [18] showed
that ifuE W^ioc (fijR^) is a minimizer of (4.1), and (4.2) holds, then the first
derivatives of u are Holder-continuous except possibly on a closed singular set
whose Hausdorff dimension is strictly less than n — 2.
Furthermore we have (see [26] and also [14]): under the assumptions of Theorem 3.1, suppose moreover that
F(x,u,p) = G(x,u,a^(x)gij(u)piJß).
(4.3)
Then any minimizer has Holder-continuousfirstderivatives in an open set fio
and the singular set fi —fiohas Hausdorff dimension strictly less than n-m.
A "special" case of both (4.1) and (4.3) is given by the variational integral
£>;fi] = / aaß(x)gij(u)DauiDßu3\/a~(x)dx
./fi
(4.4)
1080
MARIANO GIAQUINTA
where (aa@) and (gy) are smooth symmetric positive definite matrices and
a(x) = det(aaß)i (aaß) = ( a a / 3 ) - 1 , £ in (4.4) represents in local coordinates
the energy of a map between two Riemannian manifolds Mn —• MN with metric tensors respectively aaß and gij. In this situation we have (see [21, 52]):
bounded minimizers of (4.4) can have at most isolated interior singularities in
dimension n = 3 and, in general, the singular set has Hausdorff dimension not
larger than n — 3; while (see [36, 53]) no singularity can occur at the boundary,
provided, of course, the boundary datum is smooth. We notice that it seems
instead reasonable to expect singularities at the boundary for stationary points
and even minimizers of (4.1) (see [16]).
The previous results apply to energy minimizing maps between two Riemannian manifolds U: Mn —> MN only if we know a priori that the image of U lies
on a coordinate chart of MN. This is of course a strong restriction. A general
regularity theory for energy minimizing harmonic maps between Riemannian
manifolds, which gives analogous results, has been developed by R. Schoen and
K. Uhlenbeck [52, 53]. These results have been extended to the case of target manifolds MN with boundary in [9]. In this context we also mention the
recent work of H. Brezis, J. M. Coron, G. H. Lieb [5] where singular energy
minimizing maps from a domain of Rn into S n _ 1 are studied, and the work of
R. Hardt, D. Kinderlehrer, F. H. Lin [32] in connection with the theory of liquid
crystals (we refer to the talk of R. Hardt at the 1986 International Congress of
Mathematicians).
3. Are growth conditions really necessary? A theorem of L. Tonelli [56], in
dimension n = 1, states: if u is a minimizer of the functional (1.1) on an interval
I where the integrand F is a C°° function satisfying FVP(x,u,p) > 0, then there
exists a closed set E with measE = 0 such that u E C°°(I — E). A more recent
result of F. H. Clarke and R. B. Vinter [6], still in dimension n = 1, says that in
the autonomous case, i.e. F = F(u,u), if F is convex in u and F(u,u) > (ß(\u\)
where <f>(r)/r —> -f-oo as r —> +oo, then any minimizer is Lipschitz-continuous
everywhere (we refer to [4, 6] for more information). How far can we go in
this direction in more than one variable? Not much is known even in terms of
examples and counterexamples.
We shall therefore confine ourselves to a few remarks on Theorem 3.1. Hypotheses 1-3 are surely quite reasonable if we assume the uniform ellipticity in
(3.6). But they are strong under the uniform strict quasiconvexity (3.4). Already L. C. Evans [10] pointed out that the estimate from below in (3.1) is not
necessary if F does not depend explicitly on x and u. Recently Hong Min-Chin
[43] has shown that, in the general situation, (371) can be substituted by
F°(p)<F(x,u,p)<M\p\m
+l
where F°(p) is a strictly quasiconvex function with |Fp p (p)| < 1 -j- | p | m ~ 2 , so
that functionals of the type
{A(x, u) DuDu + L det Du} dx,
/<
n = N = 2,
THE PROBLEM OF THE REGULARITY OF MINIMIZERS
1081
are included. But still the control in (3.2) on the second derivatives of F is too
strong as shown, for instance, by the functional
!{\Du\2 + y/l + (detDu)2} dx.
E. Acerbi and N. Fusco [2] have proved that (3.2) is not necessary for the
partial regularity of the minimizers and that in fact it is sufficient to assume
that (3.3) holds. So that we may state that the conclusion of Theorem 3.1
holds under the weaker assumptions that (i) F is of class C2 with respect to p,
(1 + \p\2)m~2F(x,u,p) be Holder-continuous in (x,u) uniformly with respect to
p, (ii) \F(x,u,p)\ < c(\ + \p\m), (iii) F(x,u,p) be strictly quasiconvex, and finally that there exists a strictly quasiconvex function F°(p) such thatF(x,u,p) >
F°(p) (see [2]).
REFERENCES
1. E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations,
Arch. Rational Mech. Anal. 86 (1984), 125-145.
2.
, A regularity theorem for minimizers of quasiconvex integrals, preprint, 1986.
3. J. M. Ball, Convexity conditions and existence theorems in nonlinear
elasticity,
Arch. Rational Mech. Anal. 6 3 (1977), 337-403.
4. J. M. Ball and V. J. Mizel, One-dimensional variational problems whose minimizers
do not satisfy the Euler-Lagrange equation, preprint, 1985.
5. H. Brezis, J. M. Coron, and E. H. Lieb, Harmonic maps with defects, preprint, 1986.
6. F. H. Clarke and R. B. Vinter, Regularity properties to the basic problem in the
calculus of variations, TYans. Amer. Math. Soc. 289 (1985), 73-98.
7. E. De Giorgi, Sulla differenziabilità e Vanaliticità delle estremali degli integrali
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