Wilfrid Gangbo Department of Mathematics Carnegie Mellon

Wilfrid Gangbo
Department of Mathematics
Carnegie Mellon University
Pittsburgh, PA 15213
Research Report No. 92-NA-035
October 1992
t
v
-'e.;-s:>v
' -* ... -
>'\ U / ]
On the weak lower semicontinuity of energies with
polyconvex integrands
Wilfrid Gangbo*
November 1992
Abstract: Let / : ft x RN x RNxN —> [0, oo) be a Borel measurable function such that
/(x, u, •) is polyconvex in the last variable f for almost every x £ ft and for every u € R v .
It is shown that if / is continuous and F(u) := / n /(x,u(x),Vu{x))dx, u € VF^ft^R*),
then F is weakly lower semicontinuous in WltP, p > N — 1, in the sense that F(xt) <
lim jtof^ti,,) for u^u € Wx'N(ft,RN) such that u» — u in Wl*. On the contrary if /
is only a Caratheodory function then in general F is not weakly lower semicontinuous in
Wl* for N > p > N - 1. Precisely, it is shown that if F(u) := fK \det(Vu(x))\dx where
if is a compact set, then F is weakly lower semicontinuous in W ltP , N > p > N — 1 if
and only if mea$(dK) = 0.
Contents
1 Introduction.
2
2 The case of continuous integrands.
5
3 The case of Caratheodory integrands.
10
Pi.-'Viv.-r.-':
* Carnegie Mellon University, Department of Mathematics, Pittsburgh PA 15213-3890, USA.
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-;--oo
1
Introduction.
Let N > 2 be an integer number, ft C RN be an open bounded set and let / : ft x HN x
R N x N —• [0, oo) be a Borel measurable function. We set
F(u) :=
/R/(X,U(X),
Vu(z))dz, « € W^(ft,R N ) := Wl*.
If one uses the direct method of the calculus of variations to obtain existence of minimum
for F, one needs to show that F is weakly lower semicontinuous in Wl%v. Since Morrey's
works ([Mol], [Mo2]) and later Acerbi-Fusco ([AF]), Marcellini ([Ma2]) and others, it is
well known that if 1 < p < oo and if
0 < / ( x , u , O < a + *>lfP\ V(x,u,0€ ft xRN xR NxN
(1)
then F is weakly lower semicontinuous in Wl*p if only if / is quasiconvex with respect to
the last variable £. We recall that / is said to be quasiconvex if it verifies the following
Jensen's inequality
-- /
Si JQ
Vu(x))dx >
for almost every xo € ft, for every (uo,£o) € R N x R N x N and for every it € W01|OO(0,RN). As
it is very hard to check whether or not a given function is quasiconvex, following Morrey's
work Ball introduced the notion of polyconvex function which is a sufficient condition for
quasiconvexity. A function /(x o ,u o , •) is said to be polyconvex if /(x o ,u o ,-) is a convex
function of all minors of the matrix f. (See Definitions 1.2 and see [Da] for more details
about poly convexity and quasiconvexity). We recall that quasiconvexity does not imply
weakly lower semicontinuous of F in Wl%r> if the growth condition (1) fails. An example
due to Tartar, is given in [BM]. Recently Dacorogna and Marcellini in [DM] proved that
if / ( x , u , f ) = /(£) > 0 is polyconvex, with no particular growth condition, then
for uv,u £ W ^ f t ^ * ) such that uu — tx in W1* provided p > N - 1. Actually in [DM]
the most interesting case to study is N — 1 < p < N. The case p > N is easily proved.
Indeed, using the convexity of / in the last variable ( we can approximate / by a non
decreasing sequence of smooth functions fj such that
(see Lemma 2.3). Then we apply Proposition 1.3 to /j(x,u,£) for j fixed and when j goes
to infinity we obtain
F(u) < lim inf F(uy).
In the case where p < TV — 1 we cannot expect weak lower semi continuity of F. In fact,
Maly proved in [Mai] that for each p < TV — 1 there is a sequence
-»u in
verifying
lim inf / \det{Vun(x))\dx
< I |dei(Vu(x))|<fx,
u{x) = x.
In this paper we study the general case where / is a non negative function polyconvex in
the last variable and
We prove that if / is continuous in ft x RN x R N x N then F is weakly lower semi continuous
in W lfP , TV — 1 < p < TV in the sense that
Flu) < Urn inf F ( u A
for u,,,!* € iy a ' N (ft,R N ) and uy — u in W 1 *. This generalizes [DM] result to include the
case where / depends on (x,u). Continuity is a necessary condition. Indeed if / is not
continuous but is simply Caratheodory function then in general F is not weakly lower
semicontinuous on W1>p, TV — 1 < p < N. To illustrate this, we show that if K is a
compact subset of ft and TV — 1 < p < TV then
meas{dl<) ^ 0
if and only if
K m j b f ^ \det{Vun{x))\dx < J \det(Vu{x))\dx,
for a suitable u^u € iy 1 ' N (ft,R N ) such that uy — tx in
We give some definitions relevant for this work.
Definition 1.1 Let TV, M > 1 be two integer numbers, ft C RM an open set. A function
f : ft x RN x RNxM — • R t5 said to be a Caratheodory function t//(-,ix,V0 is
measurable for every (tt,V0 € RN x R NxM and /(x,-,-) is continuous for almost every
x € ft.
Definition 1.2 (See [Da] ; .
Let f : HNxM —• R be a Borel measurable function defined on the set of the N x M real
matrices.
• f is said to be convex iff(H+{l-X)r])
and every A € (0,1) .
< A/({)+(l-A)/(if) for every ^
e R VxM
• f is said to be polyconvex if there exists a function h : RTlN*M) —• R convex such
that f(() = h(T(tj) for every te*NxM,wher* T(N, M) = ZI^^SM)
T(£)
= {pdjity
• • •, ad/min(Ar,Af ) £ ) fl^rf <*4?«£ stands
for the m a t r i x of all s x s
( f )( * ) .
m i n o r s of£.
When N = M = 2then T(() =
• / i s said to be quasiconvex if p fa f(£ + W ) > /(£) /or every £ 6 RAxM, /or
every Q CRN open bounded set and for every <f> € W<Jf0° (ft)
the previous inequality for one fixed open bounded ft C R^ )•
, (" it is equivalent to assume
For completeness we state some well known results.
Proposition 1.3 Let N,M > 2 be two integer numbers, $1 C RN an open bounded set
and f : ft x RM x RNxM —• R a continuous function such that /(x,ti,-) is quasiconvex
for each (x,u) 6 fi x R^. Assume further more that f satisfies for 1 < q < p < oc
-o(|u|« + \t\') - f{x) < /(x,ii,0 < a(|u|' + |^|p) + 7(«)
where a > 0, 7 € L1
«;/icrc ^9 > 0 and
where v is a continuous increasing function with i/(0) = 0. Let
F(u) := Jj(xM*),T(Vu(x)))dx, u € W1*
F is weakly lower semicontinuous in WltP.
Proof: For the proof we refer the reader to Theorem 2.4 in [Da].
Lemma 1.4 Let <f> : R —> R 6c a bounded lipschitz function. Let fi C RA' be an open
bounded set and V € C£°(fl,RT). VP > N ~ 1^ */« v ,« € W ^ n . R * ) and »/«„ — u in
Wl* then
lim
Moreover the results stands for p = N — 1, JV = 2. //ere < ; > ts Me 5ca/ar product in
R
T
fN\2
( I
Proof: Lemma 1.4 is obtained as a slight modification of the proof of Lemma 1 in [DM].
4
2
The case of continuous integrands*
Let us first state the main result of this section.
Theorem 2.1 Let N > 2 be an integer number, ft C RN an open bounded set, r(N) =
(
A/A2
] . Let / : fi x RN x Rr(N) —> [0, oo) 6c a continuous function such that
/(x,tx,-) is convex for each (x,u) g f i x HN. Let
F(u) := / f{x,u(x),T(Vu{x)))dx,
u € Wl*(Sl,RN) := Wl*.
Then,
ifu^u
e WlJ*(fl,RN) and uv —F(u)
u in ^limJm^FK),
W1*, p > N - 1. Moreover , if N = 2 ffte re^(2)
ts true even ifp = ^ — 1 = 1.
We recall that T(Vtx) stands for the matrix of all minors of Vu.
Remark 2.2
1.- The case p > N is well known. Indeed, as / is convex in the last variable £, using
Lemma 2.3 we can approximate / by a non decreasing sequence of smooth functions fj
such that
os/ifou.oscifouxi+ier).
Then we apply Proposition 1.3 to fj(x^u^) for j fixed and when j goes to infinity we
obtain
Flu) <lim inf F f u A
In the general case (2) would be obvious if we knew that T(Vuu) —* T(Vu) Ll. However
this is not necessarily true. In [DM] there is an example where u» —* u in WltP, N > p >
N - 1 and JXVu,,) y- T{Vu)Ll (cf abo [BM]). For instance if JV = 2, ft = (0,1) 2 ,1 <
tx,, = iz—^l - y n s i n i / x , c o s i / x ) — (0,0)
in
Wl*.
Then ^ ( V u ^ ) = —i/*(l — y)21'"1 is not bounded in Ll if p < 2, and so it does not
converge in L1 weak to det(Vu) = 0, even if p = 2.
2.- The assumption that u^u € W l f N is important. It can be useful to extend the
definition of F(u) to functions u € W ^ p < JV (cf [Mai]). Also Theorem 2.1 is false if
one omits this assumption (cf [BM]).
3.- If 1 < p < iV - 1, and N > 3, then F is not necessarily weakly lower semicontinuous (cf [Mai]). But if p = JV — l,iV > 3, the question to know whether or not
F is weakly lower semicontinuous is still open. However MaJy proved in [Mai] that if
u,uv € WlfN~* are sense preserving diffeomorphisms such that u» —* tx in W 1 ^" 1 , then
F(u) < lim inf Ffa).
4.- The basic idea to prove Theorem 2.1 is the following: in the first step, we write /
as a sum of functions in form a(u)g(xy T(Vu)), with <j(z, •) > 0 being convex. This can be
done using Weierstrass's Approximation Theorem (see Lemma 2.4). In the second step,
changing variables we write a(u)p(x,T(Vu)) on form h = h(xyT(Vv)). Then, following
the idea of Dacorogna and Marcellini in [Da-Ma] who studied integrands of the form
h = h(T(Vv)), we conclude the theorem.
Lemma 2.3 ( De Giorgi's Lemma. )
Let N, T > 1 be two integer numbers, fi C R^ be an open bounded set, and g : £2 x RT —>
[0,oo) be a continuous function such that g{x,-) is convex for each x € Cl. There exists a
non decreseasing sequence of functions (gk) of class C°° such that:
i)9k>-l]
ii) gk converge uniformly to g in every compact subset of ft x RT;
Hi) gk is convex in the variable T\
v) Drgk(^'iT) is uniformly bounded in k in every compact subset of ft x RT by a constant
which does not depend on x, where Djgk = ( arT^^i *" • * afvP*) •
Proof: For the proof we refer the reader to [Ma2].
Lemma 2,4 ( Weierstrass's Approximation Theorem )
Let f : [0,1] —• R be acontinuous function. Then,
and the convergence is uniform.
Proof: For the proof we refer the reader to [Kl].
Remark 2.5
1.- If / is non negative, then the sequence gn(u) = Y^p<k<n ( L I /(^) wfc (l - u)n~k non
is
decreasing.
2.- One can deduce from Weierstrass's Approximation Theorem that for h > 0. a real
number, N an integer number and / : [— h,h]N —> [0,oo) a continuous function, there
exists a sequence (/n)n of non negative functions of class C°°(RN) such that
/-"at A
kszO
i = 1,2,
Proof of Theorem 2.1 We give the proof of Theorem 2.1 only in the case where
N > p > N — 1 since the case p > N is well known (see Remark 2.2). In the first step of
the proof, we truncate the functions (u,,),, and u to get a new sequence which is uniformly
bounded in L°°. Then we write f as a sum of functions of the form a(u)p(x, JT(VU)), where
9(xr) > 0 a r e convex. In the second step we study the particular case where f has the
form a(u)g(x,T(Vu)). In the last step we study the general case where f satisfies the
hypotheses of Theorem 2.1.
First step.
a) Fix h > 0 and 6(h) «
1. Truncate u and uv by considering ^(u), where <j> is given by
and t/> G C°°(R,R) is defined in the following way
{
-h
si f < -ft - 6(h)
t
h
if \t\ < h
if t> h + 6(h),
0 < rl>'{t) < 1 for every t € R and xf>'(t) = 0 if and only if |t| > h + 6(h).
b) By Lemma 2.4 and Remark 2.5 we get
n
,u,T) = n—*oo
Urn S X t O / ^ T )
V(«, u,r) 6 f l x [-2A,2*]W x RT,
with
and
a;€C°°(R,R), a } > 0 , i =
(4)
Using explicitly Weierstrass's Approximation formula it is easy to see that fj(x*£) is in
the form bj / ( x , ^ , (). Therefore /j(x, •) is convex in Rr for every j = 1,2, • • •.
Second step. We show that for all j
Um^/^^W^K^Cx^CVK))) > J^aj(u)<f>\u)fj{x,T(V(u))). (5)
- If lim^inf/ aj(u i/ )^(u i/ )/ j (x,r(V(u i/ ))) = oo then (5) is trivial.
- Assume that lim inf / ai(tX|/)^'(u1/)/j(x,r(V(ul/))) < oo. By Lemma 2.3 we can approximate fj by a sequence of non decreasing functions (/*)* which converge uniformly
to fj in every compact set of fi x RT and such that {fj)k verifies all properties given in
Lemma 2.3. Without loss of generality we may assume that fj verifies
/i€C~(ftxir,[(>,oo)),
fj{xr)
is convex,
/j(x,') = 0 if dist(x,d£l) < — for a suitable k ,
Drfj{x,T)
is uniformly bounded in every compact set of Q x RT independly on T. (6)
We can also assume that it € C°°(ft,RN). If this wasn't the case then it would suffice to
replace u by ut € C°°(ft,RN) such that \\uc — u||vyi.* < £ following the proof with the
suitable modifications.The convexity of fj implies that
Urn inf / C j ^ j /
>
lim inf / o i ( « l ( ) /
+
Urn inf / ajM+'M
+
limmlJ^iuM'M < W,r(%)));r(VK)) - T(V(ti)) >,
< DrfsfaTiVtuWiTWu,)) - T(V(u)) >
where we used Fatou's Lemma and the fact that
) —> aj{u)<f> (u) a.e.
For T € RT, we set T = ( f , t ) , t € R. For fixed i 6 fi , let Df/j(x,-) denote the matrix
of the partial derivatives of fj(xy') with respect to the r — 1 first variables in RT. Let #
be the functional defined on 0, x RN x R N x N by
));f (0 -
It is easy to see that H and — H are quasiconvex in the last variable. Using the fact that
u € C°°(ft,R N ), (6) and the fact that \4>(uy)\ < 1, we get that H and -H verify the
assumptions of Proposition 1.3. We deduce that
lim inf / ajMt'M
< Cf/j(i,r(V(«)));f(VW) - f(V(«)) >= 0.
On the other hand setting
vl . AJ(iKtii)), v* = A}(tf(u')), where Aj(t) = / ' ^ a} o r l (*)&,
then we obtain
t;1 a.e.,
and
By Lemma 1.4 we obtain
= Km m^
^jf iffaTViuHfaWM)
- det(V(v)))
) - o^tt)*'^)) ^/,(x, rv(«))«to(V(ti)) j = o
which together with (7), yields (5).
Third step.
Let n € N be fixed. By (4), (5) and the definition of \j> we deduce that
lim jj^jf/(*,«.
>
lim inf^ / (E«iK)^(u 1 ,)/ i (x,r(VK
jasO
When n goes to infinity, Lebesgue's Monotone Theorem and (4) give
limjjn^ jf/(^ii^rCVu^)) >Jj'(u)f(x,u,T(V(u))).
Letting h go to infinity in the previous inequality we obtain (2).
(7)
3
The case of Caratheodory integrands.
We state the main result of this section.
Theorem 3.1 Let N > 2 be an integer number, N -1 < p < N, ft C RN and open
bounded set, and K C ft a compact set. The two following assertions are equivalent:
meas{dK) ^ 0,
(8)
lim n inf/ \det(Vun(x))\dx < j \det{Vu{x))\dx
(9)
for a suitable u^u € W a ' N (ft,R N ) such that uv — u in
Before proving Theorem 3.1 we begin with some remarks.
Remark 3.2
Let us recall that if F(u) = fK \det(Vu(x))\dx and K is a compact set then, for p > A\
F is weakly lower semicontinuous on WliP even if meas(dK) ^ 0 (see Proposition 1.3).
For p < N — 1 then F is not weakly lower semicontinuous on W l t P even if rneas{dK) = 0
(see [Mai]).
We state and prove the following lemma that will be used to prove that (8) implies (9).
Lemma 3.3
Let N, r > 2 be two integer numbers, ft C RN an open bounded set and K C ft a
compact set such that meas(dK) > 0. Let p < N be a real number. Then there exists a
sequence uk G W ^ f t , ! * * ) such that
i)
u)
Hi)
Proof:
= id in W1'p(ft,RAr) with id{x) := x,
\det(Vuk(x))\ < 1 on K,
Uk->u
meas{xedK:
det(Vuk{x))
^ 0} < ^ - .
we divide the proof into five steps. We assume without loss of generality that
ft = (0,1)*.
First step. We construct the sequence uk. Let k G N be fixed. Using Vitali's Covering
Theorem we find two sequences (xt*)t C dK, (/?*), C (0, ^r) such that
10
/ xr \ ^ meas(dK)
'
meas (Nk) < — ^ + 1
(11)
tacl
for i = l,
£(x,/?) stands for the open ball in RN with center x and radius fi and
open set. Since A" is a compact set we have
dKcNk\J[ U *(*?,#)),
is an
(12)
where T(k) is a constant depending on k. Now we want to change the centers xf by
other centers which belong to the complementary of K. Using (10), (11), (12) and the
fact that xf € dK, we deduce that there exist an open set Nk and two sequences of G
B(xl ft) \ K, 0 < cf < ft, such that
A=T(k)
dK cNk\J[
U B(o?,e?)),
- mea» (dK))
Since U \ K is an open set and a* €
(13)
<
) \ K, there exists Sj- > 0 such that
(16)
and
(17)
We define
is easy to see that uk is a diffeomorphism from I?(a*,6*) into £(a*, ef) and u* maps
(a?,e?)\ £(«?,#) into dB(a$,ef).
11
. As
Second step. In this step we show that uk £
and
is continuous on J9(a*,e*),
we have
(18)
and since
(19)
= x on
we conclude that
(20)
Using the definition of u* on ft \ (U't=i
^( a ?> c ?)) it is obvious that
J9(a?fef)),
(21)
f=:l
which together with (18) and (20) yields
(22)
Third step. We show that, up to a subsequence, tx* —* u = id in W lfP (fl,R N ). Using the
definition of n* on $7 we obtain
|ufc(x) - x| < ^ - for every x € ft,
(23)
and
€ *(«?,«?
where Is is the identity matrix in R N x N . For a, b € RN, a ® b denote the N x N matrix
with component at-6j and \a\ = yja\ +
V a%. Cleary, there exits a constant C = C( Ar)
such that
k
12
Thus by (15) and (16) we have
L
where wN = measB(O,1). Recalling that B(af,cf) does not intersect £(a£,e*) for t
and B(ak, cf) C ft = (0, l ) w we conclude that
(24)
Therefore (ujt)* is bounded in WltJ> and by (23) we deduce that, up to a subsequence,
uk^u = id in
Fourth step. We show that \det(Vuk(x))\ < 1 a.e. on A'. Indeed (E) implies that
.=T(fc)
dc*(Vu*(*)) = 1 a.e. x € f t \ ( |J £(a?,e*)).
(25)
We know that uk € Cl[B{aki,eki) \ B(ak,6k),RN) and
|u*(«) - of | = ek Vx € 5(«f, ef) \ J5(af, 5,fc).
As u* is the identity on dB(a^ cf) we obtain
uk(B(alek) \B(ak,6k)) = a£(a,fc,ef).
Therefore uk is not invertible at any point x € B(ak,ek) \ B(ak,6k). We conclude that
det(Vuk(x)) = 0 a.e. x 6 £(o?f«?) \ B(a?,^fc),
(26)
•which, together with (17) and (25) implies that
0 < <fet(Vut(x)) < 1 a.e. x € K.
Fifth step. We claim that meas{x € dK : det(Vuk(x)) ^ 0} <
(25) and (26) we have
: det(Vuk(x)) ? 0} C Nk
13
(27)
mea aA
^ ">. By (13). (17),
(28)
and the result follows now from (14).
Proof of Theorem 3.1.
We prove that (8) implies (9). Assume that meas(dK) ^ 0. By Lemma 3.3 there exists
a sequence uk € W^llN(n,RN) such that:
»)
it)
Hi)
t*it — u in W1*(n,RAr), u(x):=x,
\det(Vuk(x))\ < 1 a.e on K,
(29)
{xedK:
(30)
det(Vuk(x))?0}<±;.
(29) and (30) imply that
f \det(Vuk(x))\dx = f \det(Vuk(x))\dx + f
<
p
\det(Vuk(x))\dx
+ meas(K \ oK)
and so
lim inf f \det(Vuk(x))\dx <
meas(K\dK)
*-*<» JK
< meas{K) = / \det(Vu(x))\dx,
and we conclude (9).
In order to prove that (9) implies (8), we assume that meas(dK) = 0. It is easy to
construct a sequence an € C°(Sl,RN) such that (see [Ga])
) a.e x €
ft,
0 < on(x) < a n+1 (x) < lK(x) a.e x € fl
(31)
(32)
Let uk,u € H ^ - ^ ^ R ^ ) be such that uk — u W^1>J)(ft,RN). Theorem 2.1 implies that
/ an(x)|<fet(Vu(x))|<fx
< Urn inf /
an{x)\det(Vuk(x))\dx
< lim inf / \det{Vuk(x))\dx,
for each fixed n. Using (31), (32) and Fatou's Lemma we conclude that .
/ \det(Vu(x))\dx < limmf^J
I
14
\det(Vuk{x))\dx.
Acknowledgements
This work was supported by the Army Research office and the National Foundation
through the Center for Nonlinear Analysis at Carnegie Mellon University. I would like to
thank Stefan Muller for the helpfull discussion we had while he visited Carnegie Mellon
University, I would like also to thank Irene Fonseca for her comments on the original
manuscript.
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91-NA-006
[1
Gurtin, M.E. and Voorhees, PM Two-phase continuum mechanics
with mass transport and stress, August 1991
91-NA-007
[]
Fried, E., Non-monotonic transformation kinetics and the
morphological stability of phase boundaries in thermoelastic
materials, September 1991
91-NA-008
[]
Gurtin, ME., Evolving phase boundaries in deformable
continua, September 1991
91-NA-009
[]
Di Carlo, A., Gurtin, M.E., and Podio-Guidugli, P., A regularized
equation for anisotropic motion-by-curvature, September 1991
91-NA-010
[]
Kinderlehrer, D. and Ou, B., Second variation of liquid crystal
energy at x/|x|, August 1991
91-NA-011
[]
Baughman, L.A. and Walkington, N., Co-volume methods for
degenerate parabolic problems, August 1991
91-NA-012
[]
James, R.D. and Kinderlehrer, D., Frustration and
microstructure: An example in magnetostriction, November
1991
91-NA-013
[ ]
Angenent, S.B. and Gurtin, M.E., Anisotropic motion of a phase
interface, November 1991
92-NA-001
[ ]
Nicolaides, R.A. and Walkington, NJ., Computation of
microstructure utilizing Young measure representations,
January 1992
92-NA-002
[ ]
Tartar, L., On mathematical tools for studying partial differential
equations of continuum physics: H-measures and Young
measures, January 1992
92-NA-003
[ ]
Bronsard, L. and Hilhorst, D., On the slow dynamics for the CahnHilliard equation in one space dimension, February 1992
92-NA-004
[ ]
Gurtin, M.E., Thermodynamics and the supercritical Stefan
equations with nucleations, March 1992
92-NA-005
[ ]
Antonic, N., Memory effects in homogenisation linear second order
equation, February 1992
92-NA-006
[ ]
Gurtin, M.E. and Voorhees, P.W., The continuum mechanics of
coherent two-phase elastic solids with mass transport, March 1992
92-NA-007
[ ]
Kinderlehrer, D. and Pedregal, P., Remarks about gradient Young
measures generated by sequences in Sobolev spaces, March 1992
92-NA-008
[ ]
Workshop on Shear Bands, March 23-25,1992 (Abstracts), March
1992
92-NA-009
[ ]
Armstrong, R. W., Microstructural/Dislocation Mechanics Aspects
of Shear Banding in Polycrystals, March 1992
92-NA-010
[ ]
Soner, H. M. and Souganidis, P. E., Singularities and Uniqueness of
Cylindrically Symmetric Surfaces Moving by Mean Curvature,
April 1992
92-NA-011
[ ]
Owen, David R., Schaeffer, Jack, andWang, Kerning, A Gronwall
Inequality for Weakly Lipschitzian Mappings, April 1992
92-NA-012
[ ]
Alama, Stanley and Li, Yan Yan, On "Multibump" Bound States
for Certain Semilinear Elliptic Equations, April 1992
92-NA-013
[ ]
Olmstead, W. R , Nemat-Nasser, S., and Li, L., Shear Bands as
Discontinuities, April 1992
92-NA-014
[ ]
Antonic, N., H-Measures Applied to Symmetric Systems, April
1992
92-NA-015
[ ]
Barroso, Ana Cristina and Fonseca, Irene, Anisotropic Singular
Perturbations - The Vectorial Case, April 1992
92-NA-016
[ ]
Pedregal, Pablo, Jensen's Inequality in the Calculus of Variations,
May 1992
92-NA-017
[ ]
Fonseca, Irene and Muller, Stefan, Relaxation of Quasiconvex
Functional in BV(Q,SRP) for Integrands f(x,u,Vu), May 1992
92-NA-018
[ ]
Alama, Stanley and Tarantello, Gabriella, On Semilinear Elliptic
Equations with Indefinite Nonlinearities, May 1992
92-NA-019
[ ]
Owen, David R., Deformations and Stresses With and Without
Microslip, June 1992
92-NA-020
[ ]
Barles, G., Soner, H. M., Souganidis, P. E., Front Propagation and
Phase Field Theory, June 1992
92-NA-021
[ ]
Bruno, Oscar P. and Reitich, Fernando, Approximation of Analytic
Functions: A Method of Enhanced Convergence, July 1992
92-NA-022
[ ]
Bronsani, Lia and Reitich, Fernando, On Three-Phase Boundary
Motion and the Singular Limit of a Vector-Valued GinzburgLandau Equation, July 1992
92-NA-023
[ ]
Cannarsa, Piermarco, Gozzi, Fausto and Soner, H.M., A Dynamic
Programming Approach to Nonlinear Boundary Control
Problems of Parabolic Type, July 1992
92-NA-024
[ ]
Fried, Eliot and Gurtin, Morton, Continuum Phase Transitions With
An Order Parameter; Accretion and Heat Conductor, August
1992
92-NA-025
[ ]
Swart, Pieter J. and Homes, Philip JM Energy Minimization and the
Formation of Microstructure in Dynamic Anti-Plane Shear,
August 1992
92-NA-026
[ ]
Ambrosio, I., Cannarsa, P. and Soner, H.M., On the Propagation of
Singularities of Semi-Convex Functions, August 1992
92-NA-027
[ ]
Nicolaides, R.A. and Walkington, Noel J., Strong Convergence of
Numerical Solutions to Degenerate Varational Problems, August
1992
92-NA-028
[ ]
Tarantello, Gabriella, Multiplicity Results for an Inhomogenous
Neumann Problem with Critical Exponent, August 1992
92-NA-029
[ ]
Noll, Walter, The Geometry of Contact, Separation, and
Reformation of Continous Bodies, August 1992
92-NA-030
[ ]
Brandon, Deborah and Rogers, Robert C , Nonlocal
Superconductivity, July 1992
92-NA-031
[ ]
Yang, Yisong, An Equivalence Theorem for String Solutions of the
Einstein-Matter-Gauge Equations, September 1992
92-NA-032
[ ]
Spruck, Joel and Yang,Yisong, Cosmic String Solutions of the
Einstein-Matter-Gauge Equations, September 1992
92-NA-033
[ ]
Workshop on Computational Methods in Materials Science
(Abstracts), September 16-18,1992.
October 1992
92-NA-035
[ ]
Gangbo, Wilfrid, On the weak lower semicontinuity of energies
with polyconvex integrands, October 1992
Stochastic Analysis Series
91-SA-001
[ ]
Soner, H.M., Singular perturbations in manufacturing, November
1991
91-SA-002
[ ]
Bridge, D.S. and Shreve, SJE., Multi-dimensional finite-fuel singular
stochastic control, November 1991
92-SA-001
[ ]
Shreve, S. E. and Soner, H. M., Optimal Investment and
Consumption with Transaction Costs, September 1992
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