On the regularity of the positive part of functions

Nonlmear

Analysis,

Theory,

Melhods

& Applications,

Vol. 27. No. 9, pp. 1055-1074, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/96 $15.00+0.00

Pergamon 0362-546X(95)00104-2

ON THE

REGULARITY

OF THE

POSITIVE

GIUSEPPE Institute

di Analisi

Numerica

Key words andphrases: bounded variation.

Truncations,

OF FUNCTIONS

SAVARB

de1 C.N.R.-Via

(Received

PART

Abbiategrasso

for publication

intermediate

25 May

Sobolev

0. INTRODUCTION

AND

The aim of this paper is to study the regularity

209, 27100 Pavia, 1995)

and Besov spaces,

MAIN

functions

with gradient

of

RESULTS

properties of the truncation u

u++u + = max(u, 0) =

Italy

operator?

if u > 0;

(0.1) otherwise i 0 in various Banach spaces of functions defined on a Lipschitz open set Q c lRN. It is a well known result (see [l] for example), that (0, 1) defines a contraction in F(Q) which is bounded in W1,p(sZ), vp E [l, SQ)], that is v 24,u E LP(c2)

Ilu+ - u+ Lqn) -< IIU - 4ILp(O)~

(0.2)

v u E w’~p(Q).

IIu+IIw’qn) 5 1141 W’q2)3

(0.3) On the other hand, u E W2*“(sZ) does not imply the same regularity for u+, since jump discontinuities of the gradient along manifolds of dimension N - l$ may occur. One can ask if u E W”(sZ) * u+ E W”~p(sZ) for some intermediate s E [l, 2[.$ The same remark on the gradient discontinuities carries to exclude the cases L l/p. In [9] it is proved that the range s E [l, 1 + l/p[ can be allowed, that is (0, 1) maps W”*p(sZ) into itself, vs E [l, 1 + l/p[, 1 5 p < 03, with

IIu+I/wyn) 5 cll4 w”qn)

(0.4) the same result holds for the Besov

for a suitable constant C > 0. By (nonlinear) interpolation spaces B&(Q), with q E [l, 001 and 1 5 s < 1 + l/p. 7 In order to fix our ideas, we only consider the positive and can be extended to u - Iu] and so on. $ For W’@(Q)-functions, we have the formula [l] V(u’)

= H(u).

vu,

a.e. in Q, where

part;

W(u)

obviously,

all our statements

1

ifu>O

0

,otherwise

=

hold for the negative

is the Heaviside

one,

function.

We recall that further generalizations are given by [2] (where the positive part is replaced by any Lipschitz continuous function) and by [3] (where a general chain rule is established for BV-functions u). # For the definitions, see [4-61; we shall use the interpolation real functor (. , .)B,p, 0 < 0 < 1, 1 5 p I 00 (see [7, 81 and Section 2 for more details) and we set $.qCQ,

= wm

0 < s < 2, p, q E [l, +m]

~2’pm,,2,q;

with W”q2)

= B;,#2)

ifO
1.

G. SAVARI?

1056

Remark 0.1. In the Hilbertian

framework of the family w”*2(Q) = HS(Q) (see [lo]) we have

u E HS(f2) *

u+ E W(L2),

vs < 3/2.

The limiting case p = 1 has an interesting development which is not covered by the previous result; instead of W2’1(Q) we can consider the space (see [ll]) BH(Q) or, equivalently, lD2&

= lu E W’*‘(Q):

du

is a matrix-valued

bounded measure)

Vu E SV(sZ; RN), that is the total variation = sup

au . n~dlV4iOdx,

$i E C,“(Q

RN), C ld%(x)12 5 1 in Q i

is finite.? We prove the following theorem. THEOREM

1. If u E BH(Q) then U+ E &Y(Q) and we can choose an equivalent norm in BE@)

such that

IIu+IIBIT(Q) 5 Il&M(n,* Remark 0.2. BH(Q) has been minimizing Hessian dependent creasing without fracture and regularity. Another interesting has been studied in [12].

thoroughly studied by Demengel in [ll] as natural domain of functionals with linear growth at infinity. This space allows it is not surprising that the positive part does not modify this function space related to this kind of problem is SBH(Q), which

Remark 0.3. We can also characterize BH(Q as the Gagliardo completion of W2p1(sZ) relative to L’(Q) (see [13] and Section 4); in particular, BH(Q) is an exact interpolation space between the other two. This result allows us to extend some properties proved in [l l] and [12] for C2 domains to any Lipschitz open subset of R N. We only mention the extension theorem, the Poincare type inequality and the continuous inclusion BH(cl) c CO@) ,

N=

2.

In the one-dimensional case we can be more precise: if u is a locally integrable real function defined on the open (and possibly unbounded) interval (a, b), we define the essential variation of v as b

ess-Vab(u) = sup

a

4WC4 dx, 4~E C,“
tBH(L2) is a (dual) Banach space, equipped by the norm II4 BH(0)= II4W’J@, + IWT. Observe that Wz3’(Q) c EN(B) and for a W2-‘(Q function u the total variation of the Hessian becomes

(0.5)

1057

Regularity of the positive part of functions

and we say that u is of essential bounded variation (V E BV(/(a, b)) if ess- V,“(V) < +oc .f For this kind of function it is possible to define two linear trace operators which we call for simplicity u - u-(b).

u - u+w, We have the following

theorem.

2. Let u be a locally absolutely continuous function defined on (a, 6) and let u be the positive part of u; if ess-V$(u’) < +oo then we have ess-Kb(U’) < +oo with

THEOREM

ess-Kb(u’) + lu:(a)l + Id(

In particular,

5 ess-cb(u’) + lu:(a)l + IuL(b)l.

(0.6)

if u:(a) = u’(b) = 0 then ess-V,“(u’) 5 ess-Vab(u’).

(0.7)

Remark 0.4. In general we cannot control ess-VUb(U’) just by ess-VUb(u’): consider for example

the function u(x) = x,

x E (-1, 1).

We have ess-V!,(u’) = 1 and ess-V!l(u’) = 0. However, (0.7) holds if u;(a) I 0 or u(a) > 0, and u’(b) L 0 or u(b) > 0. When (a, b) = Rand u E L!‘(R) or u’ E Lp(R) for somep < co, (0.7) is always verified. Remark 0.5. We denoted by BV(a, b) the space of functions of essential bounded

variation;

BV(a, 6) becomes a Banach space with the norm Ilull BV(a,b) = Iu+(a)I

+ [u-(b)1

+ ess-cb(u)

and we easily control that BV(a, b) c ,?(a, 6), with continuous inclusion. Observe that the term lu+(a)] + lu-(b)( is usually substituted by the L’(a, 6)-norm; if (a, b) is bounded these norms are equivalent whereas in the unbounded cases the L’-norm raises a smaller space. Our choice allows a slightly bigger generality and is justified by (0.6). Let us now consider the two extreme cases with respect to the summability power p and to the corresponding regularity, that is u E W”“(a, b) and u E BH(a, b); since u c* U+ is bounded in both spaces, it would be interesting to know if it is bounded in the family of the real interpolation spaces obtained from them. In other words, let (+, *)e,, be the real interpolation functor introduced by the K-method of Peetre (see [7, 81 and Section 2) and let us define (following [ 16]), for 0 < 8 < 1, 1 5 p c: co ZO~%, b) = V%

b), BW,

Z l+osp(a, b) = (w E W”“(a,

m,,,

b): w’ E z”‘p(a, 6)).

W3)

t We know (see [14, 151) that we can find a real function w: (a, b) - m such that w = u a.e. on (a, b) and V,“(w) = sup

f

t ,=I

Iw(t,) - w(t,-,)I, a < to < t, < ... < tm< b

In Section 1 we use the analogous definition of ess-Eb(v) by using the “approximate”

= ess-V$(u). I

notions of limit and continuity.

1058

G.SAVAR@

Nonlinear interpolation theory (see [17]) cannot be applied, as it requires a Lipschitz or Holder hypothesis for the mapping on some of the spaces; however, we can prove the following theorem. THEOREM

3. Let u E ACroc(a, b) with V’ E ZesP(a, b); if u = u+ then we have 24’ E zeqa,

In particular,

u E Z’+B,P(a, b) *

b),

II4l zSqa,b) 5 Ib’IIZ~qa,b)~

(0.9)

U+ E Z1+e,P(a, b).

Remark 0.6. The family Ze,p has been introduced

in [6] to study the regularity properties of the solutions of scalar conservation laws. In this framework the good function spaces have to contain discontinuous functions and their behaviour with respect to truncations plays an important role. At this point it is natural to ask if the spaces ZoPP(a,b) are related to those of the more familiar Besov class; we can prove a conjecture stated in [ 161.

THEOREM

4. Choosing in (0.8) p = l/8, the space 0<8<1

Z’+‘(a, b) = W’,P(a, 6) fl Z’+e,P(a, b),

is of class (1 + 13)/2 between LP(a, b) and WzTP(a, 6), that is, Bj,+Ie(a, 6) C Z’+‘(a, b) C @,?(a, b).

This last property says that Z’,e(a, b), for 0 < 0 < 1, is an “upper” limiting bound for the family Bi,,(a, b), 1 5 p, q < +co, 1 I s < 1 + l/p; then it is not surprising that the result of [9] can be recovered by (nonlinear) interpolation. In this way we can also give further information on the constant C of (0.4). THEOREM 5. (0.1) maps B;.4(Q

an equivalent

into itself, vs E [l, 1 + l/p[, norm on these spaces such that

IIu+IIB;,,(n)5 II4 lb;:,(%I) *

1 I p, q I cc),and we can choose (0.10)

Remark 0.7. In [9] is given an example of a BP,l+l’P(lR)-function

v whose positive part does not belong to the same space; Theorems 3 and 4 show, however, that this regularity can be recovered as soon as u is a little more regular, that is v E BP,‘+““(lR). r Forp = 2 this fact has a rather surprising counterpart in the results of [ 181 for evolution variational inequalities. On the other hand, our technique can also be applied to the evolution problems of the type studied in [19], extending the results of [18] and [20]. In a forthcoming article we exploit this feature. The outline of the paper is the following: in the next two sections we deal with BH(a, 6) and Z l+evP(a, b), respectively; we prove Theorems 2 and 3 for piecewise linear functions and we extend them to the whole spaces by an approximation technique. The last two theorems will be proved in Section 3; the proof of Theorem 1 with Remark 0.3 will be given in the last section, but it depends only on the results of Section 1.

1059

Regularity of the positive part of functions 1. PROOF

OF THEOREM

2

We begin by recalling the main properties of functions of bounded variation in (a, b). In order to work with a.e. defined functions we must consider the “approximate” notions of limit and continuity (see [14, 15, 211); for a measurable set E, (E 1 is its Lebesgue measure. Definition that

1.1. Let U: (a, b) H R be a Lebesgue measurable function

and x0 E [a, b]. We say

ap lim u(x) = 1 x-x0 if lim

IbE

R:

14x1

-

r+o+ Consequently,

II

>

Eln~,(x,)l

=

o

2r

x0 is a point of approximate

continuity

VE > 0.

,

for u if

ap lim U(X) = u(xo). x+x0 Left and right approximate limits are analogously defined.7 We denote by e(u) the set of approximate continuity points of u; recall that I@, b)\G(u)] =O. It is possible to give an equivalent

characterization

of BP’ functions.

1.2 [14, 15, 211. For a function u E L&&r, 6) we have

THEOREM

ess-Kb(u) = SUP

i

i Moreover,

lU(tj)

-

a < to < t, <

U(tj_l)l,

..’

<

t,

<

b,

tj

E

e(u)

j=l

if u E BV(a, b) then,

,

1

VX,

E (a,

b)

3 ap lim U(X) = lim U(X) = 2.-(x0); 3 ap lim u(x) = lim U(X) = u+(x,) x-x; r-x, x-x, x+x; x Ee(u) x Ec?(u) and these values are equal outside a countable set. Analogously, 3 ap lim U(X) = lim U(X) = u+(a); 3 ap lim U(X) = x-a+ x+a+ x-bx E c?(u)

we have lim

u(x) = u-(b).

x+bx E c?(U)

Remark 1.3. Thanks to (0.5) the ess-Eb functional is lower semicontinuous with respect to the B’
ap lim u(l/x) = I x-o+

G. SAVARI?

1060

Remark 1.4. We can study the ess-Vt functional

with respect to the extrema CY,/3 E [a, b];

setting

bml) = lo+ the following

- GGJL

vx,~(a,b)

formula hold for each function of BV(a, b) and for a < cx < /3 < b ess-Kb(u) = ess-Ka(u) + es-V,P(u) + ess-Vpb(u)+ [u](a) + [u](p)

lim ess-K@(u) = ess-K’(U);

lim ess-Q(u)

O-b-

= 0.

pa+

Now we want to study some approximation properties in BV(a, b). We consider a finite subdivision a < to < t, < . -. < t, < b and we call S the associated family of intervals (a, toI, (to, t,>, . . . , L1

, &A CL, b).

(1.1)

We denote by P’(S) the vector space of the piecewise constant function u such that VIES.

4I = CI, P’(S) will be the space of continuous a function u E BV(a, b) we setf

piecewise linear functions u such that U’ E F”(S). Given ‘j

1 tj

ii = rI;u E PO(S): ii(x) =

ii

=

II:24

tj-l

0)

1 tj-,

ifxE(fj-l,tj),j=

dx

u+(a)

if x E (a, to);

\ u-(b)

if x E (t,, b).

i

If u E At&(a,

-

l,...,

m;

6) and U’ E BV(a, b) we set E

Pi(S):

ii

=

U(tj),

j

=

0, 1, . .., m;

u’(x) =

u:(a)

if x E (a, to);

u:(b)

if x E (f,, b).

The following theorem summarizes the results we shall use. THEOREM

1.5. Let S be a subdivision

of (a, b) as in (1. l), and u E BV(a, b); we have

II4 L”(a,b) ess-Vab(ii) 5 ess-Vab(u),

and, if u E AC&a,

5

(1.2)

~~“l~Lm(a,b)

lIdIBV(a,b)

s

hh’(a,b)

(1.3)

b) and U’ E BV(a, b), we have

llfillP(t(),t,) 5 IIhyt,,t,, (r&l)

= rgu’.

t We cannot choosethe mean value on the intervals (a, t,) and (t,,,, b) since they may be unbounded.

(1.4) (1.5)

Regularity

of the positive

Moreover, there exists a sequence S, of subdivision corresponding projection of a function u 2.4E BV(a, b) * u E ~Gx@,

1061

part of functions

lim I/u - ii&,~~~,~) = 0, n4m

of (a, 6) such that, setting ii,, , a,, the VU < a < j? < b, vp E [l, $00)

u’ EBVa, b) * IIU - iinIIL”(a,p)+ 0, VU ess- V,“(u) = lim ess-V,b(ii,) IlullBV(a,b) = nli_mmhzliBV(o,b)* n-m b),

< a
(1.6)

< b (1.7) (1.8)

Proof. The first three formulae are well known and (1.5) follows by the above definitions. A good choice of S, requires that lim R, = 6,

lim L, = a,

n-m

lim An = 0,

n-m

(1.9)

n-cc

where L,, R, are the left and right extrema of the internal intervals of S, , respectively, and A,, is the maximum amplitude of the internal intervals of S,. In this case, (1.6) reduces to standard approximation results on piecewise constant functions and (1.7) is a consequence of the uniform continuity of u on every bounded interval. Finally, (1.8) follows from (1.3) and from the 1.s.c. of the ess-Kb functional and of the BV-norm. n We begin now the proof of Theorem 2 by considering the special case of piecewise linear functions. If u is such a function, so is u = U+ and we can find a common subdivision S such that U, u E P’(S). Moreover, u is related to u by u(x) z 0; We focus our attention Definition that

on this relation and we fix the following

. . . . t,].

(1.10)

definition.

1.6. For a given function w E P’(S) we denote by IK[w] the family of u E P’(S) such u z 0;

By (l.lO),

vx E (a, b)\&,

u’(x) = ewm4x)),t

(1.11)

v x E (a, b)\ (to, . . . , t,).

u’(x) = e4m4x)),

we have u = u+ *

u E IK[u']

and we shall see that this is sufficient to obtain the bound

llu’llBV(a,b)

s

iI u’iIBV(a,b)-

The idea is the following: we divide the zero-set of u into the disjoint union of maximal intervals I i , . . . , Z, and we observe that U’ can be obtained by multiplying u’ by the functions Xj(X)

=

0

if x E

1

ifx$Ij,

Ij,

j=l

, **., k.

Our thesis follows if we show that each step does not increase the BV-norm

of the function.

t Observe that H@(x)) interval of F?.

function

= H(u(x)).

All our results hold even if His

replaced

by the characteristic

of any open

1062

G.SAVARl?

Definition 1.7. We say that an interval Z = (a, p) C (a, b) belongs to Z(U), the family of the maximal zero intervals of u E P’(S), if UII = 0, z c I’ = (cY’)/Y), f& 3 0 *

z = I’.

(1.12)

For an interval Z = (CY,/3), 3z1(x) will be the function which takes the value 0 on Z and 1 outside in,

0 1

= (1 - XI(x)> =

if x 15I; if x $ I.

We can state now our first result. THEOREM

1.8. If u E IK[w] then IIU’II

BV(a,b)

s

iiwil

Proof. We observe that, if Z E Z(U) is a maximal u and X,w satisfy (1.11)

BV(a,b)*

zero interval of u and u E lK[w] then also

u’(x) = ~.,wwm(x)), We call I, , Z, , . . . . Zk the disjoint wJ(x)

By induction,

=

(1.13)

u E lK[X,w].

intervals of Z(U) (which is obviously finite) and we define

w(x);

wj(x)

=

j=l

~Ij(x)wj-l(X),

, .--9 k.

we have u E IK[Wj],

j=O,l

9 . ..> k

and u’(x) = Wk(X) since the zeros of u outside I,, . . . , Zk are isolated. So, our thesis follows by induction following basic lemma. 1.9. Let u E IK[w] be as in the statement of Theorem a maximal zero interval of U; then

LEMMA

llm4l

BV(a,b)

5

from the

1.8, and let Z = (01,/3) E Z(U) be

IIWiIBV(a,b)*

(1.14)

Proof. Call ii, = 3z,w; since u is piecewise linear and nonnegative, Z has the following nice property CY# a * ul(a) < 0; (1.15) P # b * UP) > 0, which holds for w, ti too, by (1.1 l), ul(cu) = w-(CY) = k(oc);

UP)

= w+(P) = C+(P).

(1.16)

Regularity of the positive part of functions

1063

Except the trivial case I = (a, b), we distinguish three other cases: (I) CY= a and /I < b. As i%+(a) = 0 and ii-(b) = w-(b), we have to prove that ess-Vab(G) 5 ess-Vab(w) By Remark

+ Iw+(a)l.

1.4, we have

ess-Vab(G) = ess-Vao(iG) + ess-Vi(k)

+ [ii](p)

= ess-Vpb(w) + i++(p) = es-Vpb(w) + w+(p)

since 4(,,,)

= 0,

+i,,,b)

=

w*

However,

w+(P) = w+(P) - w+(a) + w+(a) 5 [WI(P) + es+I/,P(w) + w+(a) so that, by 1.4, the result follows. (II) a < 01 and p = b. We use the same argument. (III) a < Q! < p < b. In this case we have ess-Vab(iG) = ess-Vna(G) + ess-Vi(C) + [l%](a) + [%](/3) = ess-I/,(w)

+ ess-Vi(w) + Iti-(cY)~ + Ii?+(p)\

= ess-Vab(w) + ess-Vi(w) + w+(p) - W-((Y), thanks to (1.15) and (1.16). As in previous calculations,

we obtain

w+(p) - w-(a) 5 [w](p) + ess-V,P(w) + [w](a). The result follows since i?+(a) = w+(a)

and C-(b) = w-(b).

n

Before concluding the proof of Theorem 2, let us note that the previous proof can be adapted to a more general situation. Let 3 be an operator defined on AC,,,(a, b) with the following properties (which are obviously satisfied by u y u’): 3p E [l, co[: U, -+ u in W$$(a, b) -

3[u,] -+ 3[u] in LD’(a,b)

3 maps piecewise linear functions into piecewise linear ones r 3[u] E IK[u'],

v u E P(S),

(1.17)

v s.

We have the following theorem. THEOREM

1.10. Let u E A&,Ja,

b), u’ E BV(a,

II4

BV(a,b)

b) and u = 3[v]; then U’ E BV(a, s

h%V(o,b)

b) and

*

Proof. Let v, E P’(S,) be a sequence of piecewise linear functions which approximate

Theorem

1.5 and set u,, = 3[u,].

(1.18) v as in

G. SAVARl?

1064

From (1.17) we deduce that

lim IIGIIBV~a,b) = II~‘IIBvw) IIdw(a,b) 5 lIG3v@,b)~ n-m and by Theorem

1.5, V” + v

in w(k$@, b)

withp given by (1.17); it follows that u; -+ U’ in ‘%~‘(a, b) and by the lower semicontinuity BP-norm we obtain

2. PROOF

We first recall the definition

OF THEOREM

of interpolation

of the

3

spaces by the K-method

Definition 2.1. Let E0 > E, be two Banach spaces (the inclusion v E E, let us define

of Peetre (see [7, SD.

being continuous)

mt, v>= .‘$ [Ilx - UIIEo+ MEJ.

and for (2.1)

1

Then for 0 < 19< 1 and 1 I p I 03, (E,, E1)B,p denotes the Banach spaces of v E E0 such that t-‘K(t,

u) E Lp,(O, ~0) = P(O, 00; dt/t)t

with norm

II4 (%~dB,p= Ilt-‘K(t, v)II~:co,q. Our purpose is to extend Theorem 1.10 to the interpolation spaces BH(a, 6) and W’,“(a, b). We have the following theorem. THEOREM2.2. Let v E AC&(a,

family obtained from the Banach

b) with v’ E ZBVP(a, b); if u = z[v], with 3 satisfying (1.17), we

have U’ E ZBSP(a, b) with

IId Z%?,b)5 Ib’IIZ~qa,b)~ We divide the proof into steps; first we study the approximation LEMMA

2.3.Let S be a subdivision

(2.2)

by IPO-functions.

of (a, b); the associated linear map

@: u E BV(a, b) u ~2E lP’(S)

is a contraction with respect to the L”(a, b)-norm and it can be extended to a linear contraction on each ZosP(a,6). t That is the L!‘-space on (0, m) with respect to the Haar measure dt/t;

whenp = 00 we are dealing with the usual one.

1065

Regularity of the positive part of functions

Proof. It is an easy consequence of (1.2) and (1.3). Let us denote by X(a, b) the closure of

BP’@, b) in L”(a, 6); by (1.2) II: can be extended by density to a linear contraction of X(a, b). On the other hand, (1.3) says that II: is a contraction also of BP’@, 6) and, by interpolation, of (X(0, b), B&z, b))B,p for any admissible choice of 0 andp. However, we know that (see [IS]) CEO5m,p

= EL”,

m9,p 5

with equal norms

so that Ze’p(a, 6) = (X(a, b), BV(a, b&J,.

We study now the semicontinuity

properties of the interpolated

n

norms.

PROPOSITION 2.4. Assume that E, is the dual of a separable space; then the interpolated

II-II (-%.Eh,p are sequentially

norms

1.s.c. with respect to the weak* topology of E,,.

Proof. Let u,, E (E,, E1)B,p with

u, -*

u

in EO.

Since K(t, *) is an equivalent norm for E,, (see [17]) we have, for all t > 0 K(t, u) I lim inf K(t, u,) n-co and by Fatou’s lemma?

COROLLARY

12, = II&u;

2.5. Let S, be a sequence of subdivisions we have

for any u E Ze,P(a, b); if u E AC&z,

Now we study the K-functional LEMMA

2.6. Let S be a subdivision

satisfying (1.9) and set ii, = IIi”u,

b) and U’ E ZePP(a, b), then

of a piecewise constant function. of (a, b) and w E P’(S); we have

K(t, WI= xy&llX

- 4P(a,b) + ~ll&V@,b)1.

(2.3)

Proof. We shall show that, for any x E BV(a, b), X E iP’(S) realizes a better choice for the

minimizing

functional,

i.e.

lb - WIIP@,b) + f IIa?v(a,b)5 IIX - WIILy7,b)+ t II&v(qb, * t The case p = 00 is immediate, too.

G.SAVAl&

In fact, by Theorem

1.5, we know that IM EV(U,b)5 IIxIIBv@,b,~

on the other hand, we have w = w, w being in P’(S), so that II2 - WIIL”(a,b) = Ilacx

- W)IIL”(a,b)

5 IIX - 4L”@,b)*

COROLLARY 2.7. For any w E P’(S) there exists a family Mt,

Proof. Immediate,

WI = IIW - W’IILya,b)

.

w’ E P’(S) such that (2.4)

+ WIIBV(a,b).

since P’(S) is finite dimensional.

n

The second step is to consider the situation of Lemma

1.9 and to bound K(t, ti) by K(t, w).

LEMMA 2.8. Let w c P’(S) and u E P’(S), with u E lK[w], as in (1.11). If Z = (a, j3) E Z(U) is a maximal zero interval of u and ii, = Xl w (see Definition 1.7), then zqt, i+) I zqt, w),

vt>o.

(2.5)

Proof. We choose a family w’ E IP’(S) as in the previous corollary and we want to exhibit a corresponding 6’ such that lIti - iit’ll Lm(u,b) 5 Iiw - WfI&z,b), (2.6) IlJ+‘ll

BV(a,b)

s

(2.7)

h%“(n,b).

The cases 01 = a or p = b are easy to check if we choose ti’ = X1w’; (2.6) is immediate 11%

-

gViliL”(a,b)

=

II%(w

-

W%=(a,b)

5

/Iw

-

since

w’llLm(a,b)

and for the other one, we have (in the case 01 = a) II@‘II B@,b)

= Iw:(p)I

+ ess-I/Bb(w’)

5 lw:(a)I

+ j w:(p)

5 IIW%iV(a,b)

+ Iw?(b)I

- w:(a)1

+ ess-Vi(w’)

+ I wf.(b)I

*

The case a < 01,p < b is more delicate. We denote by I- the interval of S preceding Z and by I+ the following one; we know that U’I~- = i?ll_ = WII- = w-(a) < 0, We distinguish

#‘lr+ = ccl,+ = W/f* = w+(P) > 0.

two situations again, according to the sign of U’ on I- and I+.

(2.8)

1067

Regularity of the positive part of functions

(I) JV~\,_* wtll+ 5 0. We set ii’ = 3t1w’ and we immediately we have ess-Q(i%‘) = ess-v,“(iG’) + )X(a)1

have (2.6); on the other hand,

+ Ii%~(p)I + ess-Vj(ti*)

= ess-V,“(w*) + Iwi.(a)( + Iw:(/?)I + ess-Vi(w*) = ess-V,*(w’) + /w!.(a) - w:(p)1 + ess-V$w’) 5 ess- Vob(w’) since we assumed that

Iw’(a)I + iw:m = IWt((-y> - wml. Consequently (2.7) holds, too. (II) w’lI- * wilr+ > 0. In order to fix ideas, suppose that they are both positive; then we set Iv(x) =

w’(a)

if x E I;

W*

otherwise.

Then tit - ti differs from w* - w only on I, where we have 1%’ - cil = i%’ = w’(a) I W’lf- - WI,- 4 IIW, - WIILya,b) thanks to (2.8). The variation

of tit can be bounded by

ess-Vab(iY) = ess-V,cY(iif) + [iV](/l)

+ ess-T/pb(tit)

= ess-I/6Y(w*) + (w:(B) - wt(a)I + ess-P$(w*) 5 ess- Vab(wt) obtaining (2.7) again. If wtll- and wtll+ are both negative, we set analogously iv(x) =

As we argued in Theorem

w:m

if x E 1;

Wf

otherwise.

1.8, we obtain the following corollary.

COROLLARY2.9. Let u E IK[w] as in Theorem

By our usual approximation

n

argument,

1.8. Then

we conclude the proof of Theorem 2.2.

G. SAVARl? 3. PROOF

OF

THEOREMS

4 AND

5

We start with a definition. Definition 3.1. For 0 < 8 < 1, Z’(a, b) will be the space? ~?‘(a, b) Il ZBsP(a, b), with p = l/0 and Z’+‘(a, b) = ( u E W1*P(a, b): u’ E Z’(a, b)] with

= IIUIIW~qa,b) + IbllZQ(a,b) * II4 ZI+B(a,b) The particular

choice of 13,p of this definition

THEOREM 3.2. Z’(a, b) is of interpolation $,l(a,

b) = (Wa,

is justified

by the following theorem.

class 0 between LP(a, 6) and W1,p(a, b), that is

b), WltP(a, b))O,l C Z’(a,

6) C (LP(a, b), W’rP(a,

b))e,, = Bi,,(a,

Proof. For the sake of simplicity we suppose (a, b) = lR. The right inclusion [16]; in order to prove the other one, it is sufficient to obtain the estimate

b).

is showed in

p = i/e

l14Z@qa,b) 5 CII~%@,&II&&)~

for some constant C > 0 and every nonnegative function u E W1,P(R). We use a consequence of the “power theorem” (see [7]) which is related to the original representation of (E,, E,)O,p as “espaces de moyennes” given by Lions and Peetre in [22]: for a pair E0 > E, of Banach spaces let u be a given element of E, and u’ E E, , t > 0 be a strongly measurable family such that lItl-%‘llL~(E,)

=

O”t’-ellufllE, I;0

$ < +a

and

IIt-% - fJ’)llL:(&) = esst>osup t-“l/u - u’(IE, < +m. Then u E CEO,4)e,p withp = l/6’ and there exists a constant c > 0 independent of u such that

Ilull(-%EI)~.~ 5 Clltl-w,~(E,) * II t-Tu - ~m&o). In our case, we choose u ‘(x) = (u(x) - tq+ so that

t-qu - u’I(L”(m) % 1, t When

(a, 6) is bounded,

Z’ = ZBsp, p = l/O.

vt>O

(3.1)

1069

Regularity of the positive part of functions

and U’ E BV(R) as U’ E W1,P(R) and is 0 outside a set of finite measure. We have ~\(-a) u?(+co) = 0 and m

(I’-“ess-V(d)):

=

=

dt f

Iw~lx[u>t”](x)

0

By applying (3.1) and using p = l/0, the proof is complete. By a simple application corollary.

of the reiteration

theorem

COROLLARY 3.3. For any s E (0, 1) and q E [l, +a]

n

(see [7, 221) we find the following

we have p=

(P(a, b), Z’+e(a, b)),,, = B;;;e’s(a, b), with equivalent

1/e

norms.

By the results of the previous section, we have u E Z’+e(a, 6) Since u y U+ is a contraction u E We,

lb+ IIz’+@(a,b) s lI~IIZ’+Q7,b)~

u+ E Z1+e(a, b), of P(R),

by nonlinear

@, zl+e(G b)),,,

*

interpolation

(3.2)

(see [17]) we deduce

u+ E (Eya, b), Z’+e(a, b)),,,

with

lIu+II(rP(a,b),zlfe(u,b))s,, --= II4 (LP(a,b),Z’+B(a,b)),,~ and we conclude the proof of Theorem 5 in the one-dimensional case. Now we study the multidimensional case, by using a slicing argument. vector of IR”; we set n, = (x E RN: (x, v) = oj 24,(x; t> = u(x + tv),

We denote by XN-’ the (IV - 1)-dimensional canonical basis of RN.

VXE7c,,

Hausdorff

Let v be a unitary

telR.

measure and by e, , . . . , eN the

1070

Fubini’s

G. SAVARG

theorem ensures that u E Lp(fRN) if and only if

f-qx; *I ELP(Q An analogous characterization

for EN-‘-a.e.

j=l

, ..a, N

holds for w”,P(RN) (see [5, Theorem 5.6.21) z&(x; *) E w”~p(lR)

u E W”,p(RN)

x E 7cej,

H ”I G,

for XN-‘-a.e.

Ilgx; %-qR,

d3CN-‘(x) < +m,

x E rce,, j=l

9 -.., N

and we can choose as norm of VJ’“,~(R~) the following

With the aid of this powerful result, the proof of the corollary since for a fixed u E W”~p(RN), s < 1 + l/p, we have (u’),(x;

1) = k)‘(x;

for RN is immediately

given,

0

and we already know the one-dimensional result. The case of a general Besov space follows now easily by interpolation. Finally, if Q is a given Lipschitz subset of RN, we know that every Bi,q(sZ) function admits an extension to RN with uniformly bounded norm; so we can choose as norm in Bi,4(sZ)

Since we conclude.

n

4.

PROOF

OF

THEOREM

1

We begin by considering the case Q = RN; for BH(Q) an analogous characterization previous section does not hold, so we have to use a different argument. With the vectors ej,j = 1, . . . . N of the canonical basis of RN we define eii = ei, For C,“(RN) functions we have (i # j)

eij

1 = ~(e;

+

ej),

i#j.

of the

1071

Regularity of the positive part of functions

so that (4.1) is an equivalent norm for W2*‘(RN), thanks to the density of C,“(RN)-functions. We want to extend (4.1) to BH(RN) by using the essential variation along a fixed direction. For a unitary vector v of RN and a function u E L’(RN) such that (see (3.3)) t y z&(x; t) E BV(rn), we set

for EN-l-a.e.

x E 7ry,

1 ess-V,(u) =

ess-V(u,(x; *)) d%?-‘(x) I *”

and we know that u E BV(lRN) Moreover,

if u E W”‘(RN)

-3 ess-y(u) < 0;) Vv E RN, Iv/ = 1.

(4.2)

then n

ess- V,(u) = i

a" dx. RN av I

(4.3)

I

It is not surprising that a good substitute of (4.1) can be obtained by taking the variation partial derivative along the same direction. PROPOSITION

4.1. A function

u E W’*‘(F?)

[ul BH(lR”) Moreover,

Ilull

w~~I(P)

+

[~IBH(R

C i,j

=

belongs to BH(lRN) ess-V(g)

<

of any

iff (4.4)

+co.

lJ

N) is an equivalent norm for BH(P).

Proof. (4.4) is well defined and continuous on BH(@) thanks to (4.2). Let u be a W”‘(F?) function such that [u] BH(R~j is finite; we shall prove that u E BH(RN)

by showing that there exists a constant c > 0 independent

llUEllBH(R”) where uc is the usual approximation

5

ct II W’,‘(RN)

+

with a symmetric mollifier

q&(x) = &-Nrf(x/&).

It is well known that

IIUEIIw’s’(R”)5 II4W’+?~) and we want to check the same bound for the seminorm

(4.5)

[dBH(R”)~,

of u by a convolution

UC = u * 1;1&;

of u and E such that

[*lBH(n~).

t For example see [15]; (4.5) is equivalent to our thesis by the lower semicontinuity of the BY-norm.

rf’

G. SAVARI?

1072

Letting Uij = au/&,,

we have

I

m,~IUij(X

+

&j)

-

VU(X)1

dX I

* h;

eSS-K/,,(Uij)

By the usual properties of the mollifiers

so that IR,~lUb(X

+

heij)

-

$(X)1

d.X I

! RN

, I I

v IR.~,

1. ess-

+

I[Uij(X

I uij(X

+

heij)

Ve,j(V,)

-

he,)

-

*T/&l dX

VU(X)/

vij(x)I

dX

’ h.

Since vi belong to C” we deduce that 3 5 ess- VeiJ(v,..) IIde, IIL~(R,~)

Since we already pointed out that for P-functions and [~EIBH~R~~~ 5 bd~~~~~~~~.

we have

5 dIIav~+R~) + mmf(R”)l II&?H(R.~j= II4 W~.‘(l!P) with c > 0 independent COROLLARY

H

of u&, we conclude our proof.

4.2. If u E BH(lRN) then also U+ E BH(lRN) with (4.6)

Proof.

For any unitary vector v we have

(u+), = W’;

g (x; t) = ; 24,(x;t) 0 Y

so that, for HNW1 a.e. x E rr, we have ess-F$(g).>

5 ess-f$(($)).

Choosing v = eij and summing up we obtain (4.6).

H

In order to prove the same result for any Lipschitz open set a, we have to refine the extension theorem of [l l] for BH functions, which requires a boundary of class C2. First we show that BH(S2) is an exact interpolation space with respect to L’(Q) and I@‘(sZ).

1073

Regularity of the positive part of functions

4.3. Let Q be an open subset of RN; BH(Q) is the Gagliardo completion of W2’1(i2) relative to L’(a); in particular it is an exact interpolation space between W2*1(t2) and L’(Q).

THEOREM

Proof. Let us recall (see [13]) that for a given pair E, > E, of Banach spaces, the Gagliardo

completion

of E, relative to E0 is defined by E.%’ = 1

v E E,: 3 u, E E, with I]u,]~~~ bounded and lim /Iu, - UllEo = 0 n-m 1

(4.7)

with IItQ0,c

= inf supl]un]lE,, u, satisfying (4.7) n 1

and it is an interpolation space between EO and El. In our case let us denote by X the Gagliardo completion of W2,‘(sZ) in L’(a); by the lower semicontinuity of the norm of BH(Q) with respect to the L’(Q) convergence, we easily have

II41EH(fl)

x c m&2);

s

IbdX,

(4.8)

VUEX.

On the other hand, given a function u E H?(Q) we know (see [ll, that there exists a sequence u, of C”(Q) functions such that

Since for C”(Q) functions the norms of W2,‘(L?) and BH(Q) inclusion and inequality of (4.8). H

231 and the remark of [12])

coincide, we obtain the reverse

COROLLARY 4.4. If Sz is a strongly Lipschitz open subset of lRN, there exist a constant M = M(Q) > 0 and a linear continuous operator 0’: BH(fi) - BH(RN) such that

ll@ull&Y(P)5 m4Mf(n,

(@u)ln = u,

v u EBH(Q),

6( W2J(Q))c W2,‘(lRN). Proof. Let 6 be the “universal” in particular, we know that

linear extension operator of [24, Chapter VI, Theorem 51;

6: Ll(sz) - L’(P)

W2J(c2) -

and

W2,‘(lRN)

with bounded norm. Thanks to the previous result we conclude by interpolation. As we argued in the previous section,

II4 BH(b2) is an equivalent norm on BH(Q

=

II”IiW’s’(n)

+

inf([ulBH(RN)9

for which Theorem

1 holds.

ul~

=

d

(4.9) n

1074

G. SAVARE REFERENCES

1. STAMPACCHIA G., Le probltme de Dirichlet pour les equations elliptiques du second ordre a coefficients discontinus, Ann. Inst. Fourier Univ. Grenoble 15, 189-258 (1965). 2. MARCUS M. & MIZEL V. J., Absolute continuity on tracks and mappings of Sobolev spaces, Archs ration. Mech. Analysis 45, 294-320 (1972). 3. AMBROSIO L. & DAL MASO G., A general chain rule for distributional derivatives, Proc. Am. math. Sot. 108, 691-702 (1990). 4. ADAMS R., Sobolev Spaces. Academic Press, New York (1975). 5. NIKOLSKII S. M., Approximation of Functions of Several Variables and Imbedding Theorems. Springer, Berlin (1975). 6. TRIEBEL H., Theory of Function Spaces, II. Birkhauser, Boston (1992). 7. BERGH P. L. & LGFSTRGM J.. Interpolation Spaces. Sorinaer. Berlin (1976). 8. BUTZER P. L. & BERENS H., Semi-groups of Operators and Approximation. Springer, Berlin (1967). 9. BOURDAUD G. & MEYER Y., Fonctions qui operent sur les espaces de Sobolev, J. funct. Analysis 97, 351-360 (1991). 10. LIONS J.-L. & MAGENES E., Non Homogeneous Boundary Value Problems andApplications I. Springer, Berlin (1972). 11. DEMENGEL F., Fonctions a Hessian Borne, Ann. Inst. Fourier Univ. Grenoble 34, 155-190 (1984). 12. CARRIER0 M., LEACI A. & TOMARELLI F., Special bounded hessian and elastic-plastic plate, R. Accad. Naz. Sri. dei XL, Mem. Mat. XVI, 223-258 (1992). 13. GAGLIARDO E., A unified structure in various families of function spaces. Compactness and closure theorems, Proc. Znt. Symp. Linear Spaces, Jerusalem, pp. 237-241. Jerusalem Academic Press (1960). 14. FEDERER H., Geometric Measure Theory. Springer, Berlin (1969). 15. GIUSTI E., Minimal Surfaces and Functions of Bounded Variation. Birkhluser, Boston (1984). 16. TARTAR L., Remarks on some interpolation spaces, in Boundary Value Problems for PDEs and Applications (Edited by C. BAIOCCHI and J.-L. LIONS), pp. 229-252. Masson, Paris (1993). 17. TARTAR L., Interpolation non lineaire et regularite, J. funct. Analysis 9, 468-489 (1972). 18. SAVARE G., Weak solutions and maximal regularity for abstract inequalities, Adv. Math. Sci. Appl. (to appear). 19. VISINTIN A., Stefan problem with phase relaxation, I.M.A. J. Appl. Math. 34, 225-245 (1985). 20. SAVARE G., Approximation and regularity of evolution variational inequalities, Rc. Act. Naz. Sci. dei XL, Mem. Mat. XVII, 83-l 11 (1993). 21. ZIEMER W. P., Weakly Differentiable Functions. Springer, New York (1988). 22. LIONS J.-L. & PEETRE J., Sur une classe d’espaces d’interpolation, Inst. Hautes Etudes Sci. Publ. Math. 19, 5-68 (1964). 23. TEMAM R., Problemes mathematiques en plasticite. Dunod, Paris (1983). 24. STEIN E. M., Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, N.J. (1970).