In connection with approximations for nonlinear evolution equations, it is standard to assume that nonlinear terms are at least locally Lipschitz continuous. However, it is shown here that f = f(X,del sub u(X)) is Lipschitz continuous from the subspace W sup 1, infinity is a subset of L sub 2 into W sup 1,2, and maps W sup 2, infinity into W sup 1, infinity, if and only if f is affine with W sup 1, infinity coefficients. In fact, a local version of this claim is proved

Keeling, Stephen L.

English

NASA Technical Reports Server

-NASA Contractor Report 178263 
ICASE REPORT NO. 87-12 
I C A S E  
ON LIPSCHITZ CONTINUITY OF 
NONLINEAR DIFFERENTIAL OPERATORS 
Stephen L. Keeling 
~hBSA-CP-1782t3) ON L I E S C B I ' I Z  C O N T I N U I T Y  GP B87-2C782 YCBZINEAG D I E E E E E L T I A L  CPBEiAlChS (IASA) 
13 p CSCL 128 
Unclas 
1 G3/64 45350 
Contract No. NAS1-18107 
March 1987 
INSTITUTE FOR COMPUTER APPLICATIONS I N  SCIENCE AND ENGINEERING 
NASA Langley Research Center, Hampton. Virginia 23665 
Operated by the Universities Space Research Association 
https://ntrs.nasa.gov/search.jsp?R=19870011349 2020-03-20T11:53:20+00:00Z
On Lipschitz Continuity of 
Nonlinear Differential Operators 
Stephen L. Keeling* 
Abstract 
In connection with approximations for nonlinear evolution equations, it is standard to assume 
that nonlinear terms are at least locally Lipschitz continuous. However, it is shown here that 
f = f(x,Vu(x)) is Lipschitz continuous from the subspace W1@ c L2 into W-'t2, and maps 
W2@ into W1~O0, if and only if f is affine with W1lm coefficients. In fact, a local version of this 
claim is proved. 
*Supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18107 while 
in residence a t  the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research 
Center, Hampton, VA 23665-5225. 
i 
1 Introduction 
This paper follows efforts to sharply estimate the convergence of some fully discrete approxima- 
tions for semilinear parabolic partial differential equations [3]. At a certain point in the analysis, 
it is tempting to postulate that the semilinearity, viewed as a nonlinear operator, is Lipschitz con- 
tinuous in a sense described below. However, it is proved here that this condition can hold if and 
only if the function in question is actually affine with respect to the argument for which Lipschitz 
continuity is assumed. Hence, while Lipschitz assumptions are standard in proving convergence of 
schemes for nonlinear evolution equations, generalizing them even very weakly to a function space 
setting may amount to linearizing the equation. 
To establish some notation, suppose that 52 is a bounded domain in RN. For 1 5 p 5 00 
and integers m 2 0, let WmJ'(52) represent the well-known Sobolev spaces consisting of functions 
with distributional derivatives of order 2 rn in Lp(52). Also, 1: 8 :]wm,p(nj denotes the usual norm. 
Next, let Cr(52) consist of infinitely differentiable functions with support compactly contained in 
52. Completing the latter with respect to 1 1  - Ilwm,p(n) produces the spaces W,mlp(S2). Then, for 
1 5 p < 00, p-' + q-' = 1 and integers m 2 1, define W-mlq(52) z W,mlp(52)* equipped with the 
norm: " 
Finally, let LF(52) represent a Cartesian product of Lp(52) normed in the natural way, e. g. : 
See A d a m  [l] for more details. 
permit a stronger convergence theorem in [3] is that for some u E W2*@'(52), and p > 0: 
Now given a function f : RaN -+ R, the generalized local Lipschitz postulate which would 
3c, > 0 such that VU', U 2  E W1@(52) satisfying max IlVU, - Vull~z(n) I p 
m=1,2 
(1.1.i) 
However, with the additional assumption: 
( 1.1 .ii) 
f(X,VU(X)) E Wl*@'(n) 
it is shown in section 3 that (1.1.i) and (1.1.ii) are actually equivalent to the following: 
1 
This equivalence is established in Theorem 3.1 using techniques found in Dacorogna [2], where for 
example, Theorem 2.1 is proved. There are various aspects of the latter which impede its adaptation 
for the question at hand. However, most important is the fact that the set {VU : U E W1lrn(D)) is 
not dense in L z ( D )  for N 2 2, as demonstrated in Lemma 2.2. In spite of this, results in Chapter 
4 of Morrey [4] can be distilled to obtain Theorem 2.2. For the significance of the arbitrariness of 
D, note that Morrey's proof requires sequential weak * continuity of G(u, 0) for vanishingly small 
hypercubes. On the other hand, (1.l.i) and (1.l.ii) are equivalent to (1.2) for a fixed, but arbitrarily 
bounded domain 0. Finally, for (31, it is important not to append regularity assumptions to (1.l.i) 
and (l.l.ii), since for example, finite element approximation subspaces consisting of continuous 
piecewise linear functions are only in W1lm (n). Nevertheless, Example 3 below shows that assuming 
additional regularity widens the class of functions for which the generalized local Lipschitz inequality 
holds. 
2 Examples and Related Results 
In this section, a few examples are offered to capture the spirit of claims made in the Intro- 
duction. The first two are intended to demonstrate the restrictive character of (1.l.i). 
Example 1. Let N = 1, 0 = ( O , l ) ,  and f(p) = p 2 .  Now, for arbitrary p > 0, a sequence 
{Un}z l  c W ' J ~ ( ~ )  is constructed in such a way that for a certain u E W2lrn(,): 
= P21 r,' 4(z)dzl v4 E w,'I2(n), I1411w192(n) = 1. 
The plan is to construct a sequence of saw-toothed functions which converge to zero as f remains 
constant. First define the characteristic function for [0, k]: 
I 
O<ZS+ {: + < Z I l .  x ( 4  = 
Now, let U ( z )  be given by: 
U ( z )  = P X ( 4  + 41- 5)P - x(41 
and extend this function by periodicity to R to obtain D(z). Similarly, let X(z) be the periodic 
extension of x(z). Next, set: 
Un(z) E n-lD(nz) and xn(z) = ii(n.1 2 E [o, 11 
I 
2 
so that: 
DzVn(z) 2 PXn(.) - P[1- xn(z)]. 
Finally, since: 
f(DzUn(5)) = P2 a. e. 
the claim above follows with u(z)  G 0. 
In spite of the simplicity of Example 1, it may not be sufficiently satisfying because . is not 
monotone, or fails to meet some other favorite condition. So, Example 2 aspires for complete 
satisfaction but a t  a small cost. I t  requires the following Lemma which is also used in the next 
section. The proof of a special case is given here for completeness. (See Dacorogna [2].) 
Lemma 2.1 Let Q be a hypercube in  RN and suppose that x E L,(Q). Eztend x b y  periodicity 
to RN to  obtain x and define xn(x) x(nx). Then the following holds: 
Proof: Only the case N = 1, and Q = [0,1] is considered here. Since the simple functions are dense 
in Ll(Q), it suffices to show for example, that: 
VCY E [o, 11. 
This follows after taking the limit in: 
where [.I represents the greatest integer function. 
Example 2. Except for the form of f, let every element of Example 1 be transported for use here. 
Now assume that f ( p )  is any function which satisfies: 
rn n+oo Hence, the left side cannot be made to vanish as I(Unll~2cn) - 0. 
following might be useful in proving Theorem 3.1. (See Dacorogna [2].) 
These examples also suggest that the method of characteristic functions used to prove the 
3 
Theorem 2.1 Let g : RN t R be continuous and define: 
G(U,D)  = g(U(x))dx U E  L z ( D ) ,  D c RN. 
D 
Then G(U,D) is sequentially weak * continuous for  every D c RN if and only i f  g i s  af ine,  i. e., 
for every D c RN: 
n-co  
G(Un,D) G(U,D)  
whenever: 
Un(x). @(x)~x  n+m ----.t I ,  U(x) * @(x)dx V@ E L Y ( D )  I ,  
if and only if: 
g(Xa+  (1 - X)b) = Xg(a) + (1 - X)g(b) VX E [O, 11, Va, b E RN. 
Now, the next Lemma is presented to demonstrate the limits of Theorem 2.1 in connection with 
weak * convergence in W11"(!2). 
L e m m a  2.2 Let N 2 2 and suppose D i s  any domain in RN. Then {VU : U E W'@(D)}  is  not 
dense in L g ( D ) .  
Proof: First, fix xo E D and let Q c D be a hypercube centered at xo. Then, note that since 
W'iM(Q) c+ Co(Q) [l], the set: 
Wo {U E W'l"(Q) : U(3) = 0)  
is a well-defined closed linear subspace of W1l"(Q). Also, when applied to gradients, 11 - ~ [ L N  (8) is 
actually a norm on Wo equivalent to 1 1  - Ilw1,m(s). If it were not so, there would exist a sequence 
IIVnllw1-(s) = 1 Vn 
IlvvnIILC(Q) + O. 
Since the imbedding W'*"(Q) L, L,(Q) is compact [l], there is a subsequence which converges 
in L,(Q) and hence in W'@(Q). Further, the limit V E Wo must be constant and satisfy 
~ ~ V l l w l , m ~ ~ )  = 1. However, this leads to a contradiction since V E V(xo) = 0. 
m 
{Vn}r=1 C Wo such that: 
while: 
n+oo 
Now, choose a smooth V E L g ( D )  for which: 
a Z J . l ( x )  # az,v2(x) X E Q  
IIVfin - VIIL~(D) - 0. 
VUn(x) = Vfin(x) X E Q  
and suppose there exists a sequence {Vfin}r=l c {VU : U E W'@(D)}  such that: 
n+m 
Then for n 2 1, select Un E Wo to satisfy: 
4 
I so that: 
! 
I 
n+co 
I Ilvun - vllLZ(Q) - O. 
Hence, V must be the gradient of some smooth U E Wo. However, since af,,,U = d:,,,U cannot 
hold, the contradiction completes the proof. rn 
In spite of this Lemma, there is the following generalization of Theorem 2.1 [4]. 
Theorem 2.2 Let g : R2N+1 -, R be continuous and define: 
G(u, D) = I ,  g(x, U(X),VU(X))dX u E W1@(D), D c RN. 
Then G(u, D )  is sequentially weak * continuous for  every D c RN i f  and only i f  g(x, u, p) i s  afine 
with respect to p, i. e., for every D c RN: 
n+oo 
G(un,D) - G(u,D) 
whenever: ID un(x)4(x)dx n--*M ID u(x)4(x)dx v4 E h ( D )  
Vun(x) * @(x)dx ==T ID Vu(x) @(x)dx V@ E LIy(D) 
ID 
and: 
if and only if: 
g(x,u,Xa+ (1 - X)b) = Xg(x,u,a) + (1 - X)g(x,u,b) 
VA E [0,1], V(x,u) E RN+l, Va,b E RN. 
Actually, with W1@(D) viewed as a closed linear subspace W c LZ+'(D), its predual is the 
quotient space Lf+'(D)/WI.  Nevertheless, the above is an equivalent formulation of sequential 
Finally, the next example addresses the question of whether the class of functions satisfying 
I weak *. continuity on W'@'(D). 
(1.l.i) and (1.l.ii) can be widened by appending regularity assumptions. 
Example 3. As in Example 1, let N = 1, h2 
u E W2+"'(h2) and consider whether it is possible to show that for some cp > 0: 
I 
i 
I m=1,2 
j 
(0, l), and f (p) p2. Now choose an arbitrary 
 VU^, ~2 E w2tco(o) satisfying max ~ / D , U ~  - ,uJJwi,=(n) L p 
{ I l f  (DZU2) - f(DzUl)llw-'~2(n) L 4u2  - UIIILa(n). 
That this holds with cp = 2p can be seen from the following calculation: 
8 
3 Demonstration of the Theorem 
In this section, the equivalence advertised in the Introduction is established in the following. 
Theorem 3.1 Let n be a bounded domain in RN. Also, suppose f : RaN + R, u E W2~O0(R), 
and p > 0. Then (1.2) i s  necessary and suficient for (1.1.;) and (1.l.ii). 
Proof: First, sufficiency is established. Condition (1.l.ii) follows immediately from (1.2). Now, let 
U1 and U2 satisfy: 
max I l V ~ m  -VUIILg(n) I P. 
m=1,2 
Using (1.2), condition (1.l.i) is obtained as follows: 
For necessity, t is first shown that: 
VX E [O, 11, Vx E R, Va,b E RN such that rnax{llallL, llbllL} 5 p 
f(x, Vu(x) + [Xa + (1 - X)b]) = Xf (x, Vu(x) + a) + (1 - A)f (x, Vu(x) + b) (3.1) 
where ( 1  + Il!, represents the usual norm on RN. Now fix X E (0, l), a, b E RN satisfying: 
(3.2) max{llallL, IlbllL,) 5 P 
and define: 
c E (1 - X)(a - b) d E Xa + (1 - X)b. 
Then, let Q be a hypercube containing R, with two of its (N - 1)-dimensional faces F1 and F 2  
orthogonal to  c: 
Fi E {xE Q : C - X =  ai} i =  1,2. 
Also, define: 
(3.3) F A - { x E Q :  C * X = C Y A }  ax (1 - X ) a 1 +  X a 2  
and let the convex hull of {Fl, FA} be represented by: 
Qx 3 {X E Q : X =  txl+ (1 - t)xA, t E [0,1], XI E PI, XA E FA}. 
6 
Before proceeding, it is shown that: 
(3.4) P ( Q ~  = Xlc(Q). 
Let ql E FI and q2 E F2 be vertices forming an edge of Q, chosen so that 1142 - qlllE = p(&). 
Then, qx 9 1  + X(q2 - qi) E FA since: 
qx * c = a1 + X(Q2 - a1) = ax, 
Thus: 
P ( Q ~  = I h x  - qilIt2Ilq2 - qiII:-' = Xllqz - qillz = WQ) 
and (3.4) is obtained. Now on Q, define the characteristic function of Qx: 
X E Q A  {: x E Q\Qx. x(x) = 
Let x be the periodic extension of x to RN, and define: 
By (3.4) and Lemma 2.1: 
Now on Q, define the hypertent function: 
Vo(x) = (c - x - a1)x(x) - X ( 1 -  X)-'(c - x - a 2 ) [ l  - x(x)] X E  Q. 
By (3.3): 
1 lim 
QA 3 x - + F ~  
v~(x) =ax - a1 = X ( a 2  - a1) = -X(I  - A)- (ax - a 2 )  = lim VO(X). 
Q \ O A 3x+ FA 
Hence, Vo E Co(Q) and: 
VVo(x) c{x(x) - X ( 1 -  X)- ' [ l -  x(x)]} x E Q .  
Therefore, Vo E W'@(Q). Further, since VO is constant on hyperplanes parallel to FA, and zero on 
F1 and F2, it can be extended periodically to RN to obtain vo E W1jm(RN). Now on Q, define: 
Vn(X) = n-'Vo(nx) X E Q  
7 
Then on 0, define: 
Un(x) U(X) + d * x + Vn(x) XES1 U(X) G U(X) + d x and 
so that: 
n+a, 
( 3 4  IIU - Unllk(n) - 0. 
Also, note that by (3.2): 
or: 
[f(x, Vu(x) + Xa + (1 - X)b) - f(x, Vu(x) + d + vVn(x)]p(x)dx 
- 0  vp E w,',"n). n+w 
(3.9) 
So, once it is established that: 
i [ ~ f ( x ,  vu(x) + a> + (1 - ~ ) f ( x ,  vu(x) + b) - f(x, vu(x> + d + vvn(x)lp(x)dx 
vp E w;I2 (n) 
(3.10) 
n--rgo - 0  
the claim (3.1) follows from (3.9) and (3.10). For (3.10), note that: 
f(x, VU(X) + d + VVn(X)) 
Hence, with (3.5): 
J,f(x, VU(4 + d + VVn(X))P(X)dX 
= L{Pa(x)xn(x) + P'b(X)[l- xn(x)])dx 
= &X,VU(X) + 4 + (1 - Nf(X,VU(X)  +b))P(X)dX 
/O{Pa(X)A + P'b(X)[l- 4 ) d x  
and (3.10) is obtained. 
Condition (1.2) is now extracted from (1.l.ii) and (3.1). First, select any v E RN for which: 
Ilvllr, 5 P. 
I.jl = &&j 1 5 4 3 -  ' < N  
Then with E+ denoting the Kronecker delta, let a basis {z i )E1 c RN, and an 
arbitrarily but satisfying: 
in addition to: 
Now, fix x E s1 and for convenience, take: 
> 0 be chosen 
(Iv + SZ' + tZj11L 5 p vs, t  E [0,1] 15 ; , j  I N. 
F(Y) = f(x, Wx)  i- Y) IlYllt- 5 P- 
With h, s, t E (0,1], the following is obtained from repeated applications of (3.1): 
Aij(h, s,t)F(v) E h-l{s-l[F(v + szi + h z j )  - F(v + hzj ) ]  - t-'[F(v + tz') - F(v)]) 
= h-l{s-l[(l - S)F(V + hzJ)  + sF(z' + v + hzj)  - F(v + hzj ) ]  
-t-l[(1 - t ) F ( v )  + tF(Z' + v) - F(v)]) 
= h- l { (h  - 1)F(v) - hF(zj + v) + (1 - h)F(z' + v) + hF(zj + zi + v) + F(v) - F(Z' + v)} 
= 2 { F ( 3 [ z j  + zi + VI + 3.) - F ( i [ Z j  + v] + $[zi + VI)} = 0. 
Hence: 
NOW all that remains for (1.2) is establishing the regularity of the coefficients. For this, define: 
a:i"jf(x, Vu(x) + v) = 0 vx E 0, 
V k ( x )  U(X) + PXk l 5 k l N  
IlVllL I P, 1 5 ; , j  I N. 
so that: 
Then according to  (1.l.ii): 
I l v V k  - Vu\lLg(R) I P- 
1 I k I N .  Pfk(x) = f(X,VVk(X)) - f (x ,vu(x) )  E wl'm(n) 
Also: 
Thus, (1.2) is obtained. 
fo(x) = f ( x , V u ( x ) )  - f(x) * VU(X) E wllyn). 
9 
References 
[l] Adams, R. A., Soboleu Spaces, Academic Press, New York, London, Toronto, Sydney, San 
Francisco, 1975. 
[2] Dacorogna, B., ‘Weak Continuity and Weak Lower Semicontinuity of Non-Linear Functionals,” 
Lecture Notes in Mathematics, v. 922,  Springer-Verlag, Berlin, Heidelberg, New York, 1982. 
[3] Keeling, S. L., “Galerkin/Runge-Kutta Discretizations for Semilinear Parabolic Equations,” 
ICASE Report No. 87-13, NASA Langley Research Center, Hampton, VA, 1987 (NASA CR-178264). 
[4] Morrey, C. B., Multiple Integral8 in the Calculus of  Variations, v.130, Springer-Verlag, Berlin, 
Heidelberg, New York, 1966. 
Standard Bibliographic Page 
.. Report No. NASA CR-178263 12. Government Accession No. 13. Recipient's Catalog No. 
ICASE Report No. 87-12 
L. Title and Subtitle 5. Report Date 
6. Performing Organization Code 
ON LIPSCHITZ CONTINUITY OF NONLINEAR DIFFERENTIAL 
OPERATORS 
'. Author(s) 
Stephen L. Keeling 
fnsEF%PFYor Computer fppTications in Science 
Mail Stop 132C, NASA Langley Research Center 
Hampton, VA 23665-5225 
#. P rfor 'zatio Name and A dre 
and Engineering 
L2. Sponsoring Agency Name and Address 
National Aeronautics and Space Administration 
Washington, D. C. 20546 
87-12 
- 13. Type of Report and Period Covered 
Contractor Report 
14. Sponsoring Agency Code 
505-90-21-01 
L5. Supplementary Notes 
Langley Technical Monitor: 
J. C. South 
Submitted to SIAM J. Numer. Anal. 
Final Report 
16. Abstract 
In connection with approximations for nonlinear evolution equations, it is standard to assume 
that nonlinear terms are at least locally Lipschitz continuous. However, it is shown here that 
f = f(x,Vu(x)) is Lipschitz continuous from the subspace W1pm c L2 into W-lt2, and maps 
W2im into W1ioo, if and only if f is affine with Witm coefficients. In fact, a local version of this 
claim is proved. 
17. Key Words (Suggested by Authors(s)) 
Lipschitz continuity, affine, 
nonlinear equations 
18. Distribution Statement 
6 4  - Numerical Analysis 
Unclassified - unlimited 
19. Securit Classif. of this report) 20. Security Classif.(of this page) 21. No. of Pages 22. Price 
Uncfassiffed 1 Unclassified 1 12 I A02 


On Lipschitz continuity of nonlinear differential operators

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