Taylor M.E. Partial Differential Equations III: Nonlinear Equations
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308 14. Nonlinear Elliptic Equations
[HW] S. Hildebrandt and K. Widman, Some regularity results for quasilinear systems
of second order, Math. Zeit. 142(1975), 67–80.
[HM1] D. Hoffman and W. Meeks, A complete embedded minimal surface in R
3
with
genus one and three ends, J. Diff. Geom. 21(1985), 109–127.
[HM2] D. Hoffman and W. Meeks, Properties of properly imbedded minimal surfaces
of finite topology, Bull. AMS 17(1987), 296–300.
[HRS] D. Hoffman, H. Rosenberg, and J. Spruck, Boundary value problems for sur-
faces of constant Gauss curvature, CPAM 45(1992), 1051–1062.
[IL] H. Ishii and P. Lions, Viscosity solutions of fully nonlinear second-order elliptic
partial differential equations, J. Diff. Equ. 83(1990), 26–78.
[JS] H. Jenkins and J. Serrin, The Dirichlet problem for the minimal surface equation
in higher dimensions, J. Reine Angew. Math. 229(1968), 170–187.
[Jo] F. John, Partial Differential Equations, Springer, New York, 1975.
[Jos] J. Jost, Conformal mappings and the Plateau-Douglas problem in Riemannian
manifolds, J. Reine Angew. Math. 359(1985), 37–54.
[Kaz] J. Kazdan, Prescribing the Curvature of a Riemannian Manifold, CBMS Reg.
Conf. Ser. Math. #57, AMS, Providence, R. I., 1985.
[KaW] J. Kazdan and F. Warner, Curvature functions for compact 2-manifolds, Ann.
Math. 99(1974), 14–47.
[KS] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities
and Their Applications, Academic, New York, 1980.
[Kry1] N. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations, Math.
USSR Izv. 20(1983), 459–492.
[Kry2] N. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations in a
domain, Math. USSR Izv. 22(1984), 67–97.
[Kry3] N. Krylov, Nonlinear Elliptic and Parabolic Equations of Second Order,
D. Reidel, Boston, 1987.
[KrS] N. Krylov and M. Safonov, An estimate of the probability that a diffusion pro-
cess hits a set of positive measure, Sov. Math. Dokl. 20(1979), 253–255.
[LU] O. Ladyzhenskaya and N. Ural’tseva, Linear and Quasilinear Elliptic Equa-
tions, Academic, New York, 1968.
[Law] H. B. Lawson, Lectures on Minimal Submanifolds, Publish or Perish, Berkeley,
Calif., 1980.
[Law2] H. B. Lawson, Minimal Varieties in Real and Complex Geometry, University of
Montreal Press, 1974.
[LO] H. B. Lawson and R. Osserman, Non-existence, non-uniqueness, and irregular-
ity of solutions to the minimal surface equation, Acta Math. 139(1977), 1–17.
[LS] J. Leray and J. Schauder, Topologie et ´equations fonctionelles, Ann. Sci. Ecole
Norm. Sup. 51(1934), 45–78.
[LM]
J. Lions and E. Magenes, N
on-homogeneous Boundary Problems and Applica-
tions I, II, Springer, New York, 1972.
[LiP1] P. Lions, R´esolution de probl`emes ´elliptiques quasilin´eaires, Arch. Rat. Mech.
Anal. 74(1980), 335–353.
[LiP2] P. Lions, Sur les ´equations de Monge–Ampere, I, Manuscripta Math. 41(1983),
1–43; II, Arch. Rat. Mech. Anal. 89(1985), 93–122.
[MM] U. Massari and M. Miranda, Minimal Surfaces of Codimension One,North-
Holland, Amsterdam, 1984.
[MT] R. Mazzeo and M. Taylor, Curvature and uniformization, Isr. J. Math. 130
(2002), 323–346.
References 309
[MY] W. Meeks and S.-T. Yau, The classical Plateau problem and the topology of
three dimensional manifolds, Topology 4(1982), 409–442.
[Mey] N. Meyers, An L
p
-estimate for the gradient of solutions of second order elliptic
divergence equations, Ann. Sc. Norm. Sup. Pisa 17(1980), 189–206.
[Min] G. Minty, On the solvability of non-linear functional equations of “monotonic”
type, Pacific J. Math. 14(1964), 249–255.
[Mir] C. Miranda, Partial Differential Equations of Elliptic Type, Springer, New York,
1970.
[Morg] F. Morgan, Geometric Measure Theory: A Beginner’s Guide, Academic,
New York, 1988.
[Mor1] C. B. Morrey, The problem of Plateau on a Riemannian manifold, Ann. Math.
49(1948), 807–851.
[Mor2] C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer,
New York, 1966.
[Mor3] C. B. Morrey, Partial regularity results for nonlinear elliptic systems, J. Math.
Mech. 17(1968), 649–670.
[Mo1] J. Moser, A rapidly convergent iteration method and nonlinear partial differen-
tial equations, Ann. Scoula Norm. Sup. Pisa 20(1966), 265–315.
[Mo2] J. Moser, A new proof of DeGiorgi’s theorem concerning the regularity problem
for elliptic differential equations, CPAM 13(1960), 457–468.
[Mo3] J. Moser, On Harnack’s theorem for elliptic differential equations, CPAM 14
(1961), 577–591.
[MW] T. Motzkin and W. Wasow, On the approximation of linear elliptic differen-
tial equations by difference equations with positive coefficients, J. Math. Phys.
31(1952), 253–259.
[Na1] J. Nash, The imbedding problem for Riemannian manifolds, Ann. Math.63
(1956), 20–63.
[Na2] J. Nash, Continuity of solutions of parabolic and elliptic equations, Am. J. Math.
80(1958), 931–954.
[Nec] J. Necas, Example of an irregular solution to a nonlinear elliptic system with
analytic coefficients and conditions for regularity, in Theory of Non Linear Op-
erators, Abh. Akad. der Wissen. der DDR, 1977.
[Ni1] L. Nirenberg, On nonlinear elliptic partial differential equations and H¨older con-
tinuity, CPAM 6(1953), 103–156.
[Ni2] L. Nirenberg, The Weyl and Minkowski problems in differential geometry in
the large, CPAM 6(1953), 337–394.
[Ni3] L. Nirenberg, Estimates and existence of solutions of elliptic equations, CPAM
9(1956), 509–530.
[Ni4] L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Sup. Pisa
13(1959), 116–162.
[Ni5] L. Nirenberg, Lectures on Linear Partial Differential Equations,Reg.Conf.Ser.
in Math., #17, AMS, Providence, R. I., 1972.
[Ni6] L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute Lec-
t
ure Notes, New York, 1974.
[Ni7] L. Nirenberg, Variational and topological methods in nonlinear problems, Bull.
AMS 4(1981), 267–302.
[Nit1] L. Nitsche, Vorlesungen ¨uber Minimalfl¨achen, Springer, Berlin, 1975.
[Nit2] J. Nitsche, Lectures on Minimal Surfaces, Vol. 1, Cambridge University Press,
Cambridge, 1989.
[Oss1] R. Osserman, A Survey of Minimal Surfaces, van Nostrand, New York, 1969.
310 14. Nonlinear Elliptic Equations
[Oss2] R. Osserman, A proof of the regularity everywhere of the classical solution to
Plateau’s problem, Ann. Math. 9(1970), 550–569.
[P] J. Peetre, On the theory of L
p;
spaces, J. Funct. Anal. 4(1969), 71–87.
[Pi] J. Pitts, Existence and Regularity of Minimal Surfaces on Riemannian Mani-
folds, Princeton University Press, Princeton, N. J., 1981.
[Po] A. Pogorelov, On convex surfaces with a regular metric, Dokl. Akad. Nauk SSSR
67(1949), 791–794.
[Po2] A. Pogorelov, Monge–Ampere Equations of Elliptic Type, Noordhoff, Gronin-
gen, 1964.
[PrW] M. Protter and H. Weinberger, Maximum Principles in Differential Equations,
Springer, New York, 1984.
[Rad1] T. Rado, On Plateau’s problem, Ann. Math. 31(1930), 457–469.
[Rad2] T. Rado, On the Problem of Plateau, Springer, New York, 1933.
[RT] J. Rauch and B. A. Taylor, The Dirichlet problem for the multidimensional
Monge–Ampere equation, Rocky Mt. J. Math. 7(1977), 345–364.
[Reif] E. Reifenberg, Solution of the Plateau problem for m-dimensional surfaces of
varying topological type, Acta Math. 104(1960), 1–92.
[Riv] T. Riviere, Everywhere discontinuous maps into spheres. C. R. Acad. Sci. Paris
314(1992), 719–723.
[SU] J. Sachs and K. Uhlenbeck, The existence of minimal immersions of 2-spheres,
Ann. Math. 113(1981), 1–24.
[Saf] M. Safonov, Harnack inequalities for elliptic equations and H¨older continuity of
their solutions, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov 96(1980),
272–287.
[Sch] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar
curvature, J. Diff. Geom. 20(1984), 479–495.
[Sch2] R. Schoen, Analytic aspects of the harmonic map problem, pp. 321–358 in
S. S. Chern (ed.), Seminar on Nonlinear Partial Differential Equations,MSRI
Publ. #2, Springer, New York, 1984. .
[ScU] R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff.
Geom. 17(1982), 307–335; 18(1983), 329.
[SY] R. Schoen and S.-T. Yau, Existence of incompressible minimal surfaces and
the topology of three-dimensional manifolds with non-negative scalar curvature,
Ann. Math. 110(1979), 127–142.
[Schw] J. Schwartz, Nonlinear Functional Analysis, Gordon and Breach, New York,
1969.
[Se1] J. Serrin, On a fundamental theorem in the calculus of variations, Acta Math.
102(1959), 1–32.
[Se2] J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations
with many independent variables, Phil. Trans. R. Soc. Lond. Ser. A 264(1969),
413–496.
[Si] L. Simon, Survey lectures on minimal submanifolds, pp. 3–52 in E. Bombieri
(ed.), Seminar on Minimal Submanifolds, Princeton University Press, Princeton,
N. J., 1983.
[Si2] L. Simon, Singularities of geometrical variational problems, pp. 187–256 in
R. Hardt and M. Wolf (eds.), No
nlinear Partial Differential Equations in Dif-
ferential Geometry, IAS/Park City Math. Ser., Vol. 2, AMS, Providence, R. I.,
1995.
[So] S. Sobolev, Partial Differential Equations of Mathematical Physics, Dover,
New York, 1964.
References 311
[Spi] M. Spivak, A Comprehensive Introduction to Differential Geometry, Vols. 1–5,
Publish or Perish Press, Berkeley, Calif., 1979.
[Sto] J. J. Stoker, Differential Geometry, Wiley-Interscience, New York, 1969.
[Str1] M. Struwe, Plateau’s Problem and the Calculus of Variations, Princeton
University Press, Princeton, N. J., 1988.
[Str2] M. Struwe, Variational Methods, Springer, New York, 1990.
[T] M. Taylor, Pseudodifferential Operators and Nonlinear PDE,Birkh¨auser,
Boston, 1991.
[T2] M. Taylor, Microlocal analysis on Morrey spaces, In Singularities and Oscilla-
tions (J. Rauch and M. Taylor, eds.) IMA Vol. 91, Springer-Verlag, New York,
1997, pp. 97–135.
[ToT] F. Tomi and A. Tromba, Existence Theorems for Minimal Surfaces of Non-zero
Genus Spanning a Contour, Memoirs AMS #382, Providence, R. I., 1988.
[Tro] G. Troianiello, Elliptic Differential Equations and Obstacle Problems, Plenum,
New York, 1987.
[Tru1] N. Trudinger, Local estimates for subsolutions and supersolutions of general
second order elliptic quasilinear equations, Invent. Math. 61(1980), 67–79.
[Tru2] N. Trudinger, Elliptic equations in nondivergence form, Proc. Miniconf. on Par-
tial Differential Equations, Canberra, 1981, pp. 1–16.
[Tru3] N. Trudinger, Fully nonlinear, uniformly elliptic equations under natural struc-
ture conditions, Trans. AMS 278(1983), 751–769.
[Tru4] N. Trudinger, H¨older gradient estimates for fully nonlinear elliptic equations,
Proc. Roy. Soc. Edinburgh 108(1988), 57–65.
[TU] N. Trudinger and J. Urbas, On the Dirichlet problem for the prescribed Gauss
curvature equation, Bull. Austral. Math. Soc. 28(1983), 217–231.
[U] K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems, Acta Math.
38(1977), 219–240.
[Wen] H. Wente, Large solutions to the volume constrained Plateau problem, Arch.
Rat. Mech. Anal. 75(1980), 59–77.
[Wid] K. Widman, On the H¨older continuity of solutions of elliptic partial differential
equations in two variables with coefficients in L
1
, Comm. Pure Appl. Math.
22(1969), 669–682.
[Yau1] S.-T. Yau, On the Ricci curvature of a compact K¨ahler manifold and the complex
Monge–Ampere equation I, CPAM 31(1979), 339–411.
[Yau2] S.-T. Yau (ed.), Seminar on Differential Geometry, Princeton University Press,
Princeton, N. J., 1982.
[Yau3] S.-T. Yau, Survey on partial differential equations in differential geometry,
pp. 3–72 in S.-T. Yau (ed.), Seminar on Differential Geometry, Princeton
University Press, Princeton, N. J., 1982.
15
Nonlinear Parabolic Equations
Introduction
We begin this chapter with some general results on the existence and regularity of
solutions to semilinear parabolic PDE, first treating the pure initial-value problem
in 1, for PDE of the form
(0.1)
@u
@t
D Lu C F.t;x;u; ru/; u.0/ D f;
where u is defined on Œ0; T / M ,andM has no boundary. Some of the results
established in 1 will be useful in the next chapter, on nonlinear, hyperbolic equa-
tions. We also give a precursor to results on the global existence of weak solutions,
which will be examined further in Chap. 17, in the context of the Navier–Stokes
equations for fluids.
In 2 we present a useful geometrical application of the theory of semilinear
PDE, to the study of harmonic maps between compact Riemannian manifolds
when the target space has negative curvature.
In 3 we extend some of the results of 1 to the case @M ¤;, when boundary
conditions are placed on u. Section 4 is devoted to the study of reaction-diffusion
equations, of the form
(0.2)
@u
@t
D Lu C X.u/;
where u takes values in R
`
and X is a vector field on R
`
. Such systems arise in
models of chemical reactions and in mathematical biology. One way to analyze
the interplay of diffusion and the reaction due to X.u/ in (0.2) is via a nonlinear
Trotter product formula, discussed in 5.
In 6 we examine a model for the melting of ice. The source of the nonlin-
earity in this problem is different from those considered in 1–5; it is due to the
equations specifying the interface where water meets ice, a “moving boundary.”
M.E. Taylor, Partial Differential Equations III: Nonlinear Equations,
Applied Mathematical Sciences 117, DOI 10.1007/978-1-4419-7049-7
3,
c
Springer Science+Business Media, LLC 1996, 2011
313
314 15. Nonlinear Parabolic Equations
In 7–9 we study quasi-linear parabolic PDE, beginning with fairly
elementary results in 7. The estimates established there need to be strengthened
in order to be useful for global existence results. One stage of such strengthening
is done in 8, using the paradifferential operator calculus developed in 10 of
Chap. 13. We also include here some results on completely nonlinear parabolic
equations and on quasi-linear systems that are “Petrowski-parabolic.”
The next stage of strengthening consists of Nash–Moser estimates, carried out
in 9 and then applied to some global existence results. This theory mainly applies
to scalar equations, but we also point out some ` ` systems to which the Nash–
Moser estimates can be applied, including some systems of reaction-diffusion
equations in which there is nonlinear diffusion as well as nonlinear interaction.
1. Semilinear parabolic equations
In this section we look at equations of the form
(1.1)
@u
@t
D Lu C F.t;x;u; ru/; u.0/ D f;
for u.t; x/, a function on Œ0; T M . We assume M has no boundary; the case
@M ¤;will be treated in 3. Generally, L will be a second-order, negative-
semidefinite, elliptic differential operator (e.g., L D ), where is the Laplace
operator on a complete Riemannian manifold M and is a positive constant. We
suppose F is C
1
in its arguments.
We will begin with very general considerations, which often apply to an even
more general class of linear operators L. For short, we suppress .t; x/-variables
and set
ˆ.u/ D F.u; ru/:
We convert (1.1) to the integral equation
(1.2) u.t/ D e
tL
f C
Z
t
0
e
.ts/L
ˆ.u.s// ds D ‰u.t/:
We want to set up a Banach space C.Œ0; T ; X / preserved by the map ‰ and es-
tablish that (1.2) has a solution via the contraction mapping principle. We assume
that f 2 X, a Banach space of functions, and that there is another Banach space
Y such that the following four conditions hold:
e
tL
W X ! X is a strongly continuous semigroup, for t 0;(1.3)
ˆ W X ! Y is Lipschitz, uniformly on bounded sets;(1.4)
e
tL
W Y ! X; for t>0;(1.5)
and, for some <1,
1. Semilinear parabolic equations 315
(1.6) ke
tL
k
L.Y;X /
Ct
; for t 2 .0; 1:
We will give a variety of examples later. Given these conditions, it is easy to see
that ‰ acts on C.Œ0; T ; X /, for each T>0.Fix˛>0,andset
(1.7) Z Dfu 2 C.Œ0; T ; X / W u.0/ D f; ku.t/ f k
X
˛g:
We want to pick T small enough that ‰ W Z ! Z is a contraction. By (1.3), we
can choose T
1
so that ke
tL
f f k
X
˛=2 for t 2 Œ0; T
1
.Now,ifu 2 Z, then,
by (1.4), we have a bound kˆ.u.s//k
Y
K
1
,fors 2 Œ0; T
1
, so, using (1.6), we
have
(1.8)
Z
t
0
e
.ts/L
ˆ
u.s/
ds
X
C
t
1
K
1
:
If we pick T
2
T
1
small enough, this will be ˛=2 for t 2 Œ0; T
2
; hence
‰ W Z ! Z, provided T T
2
.
To arrange that ‰ be a contraction, we again use (1.4) to obtain
kˆ
u.s/
ˆ
v.s/
k
Y
Kku.s/ v.s/k
X
;
for u;v 2 Z. Hence, for t 2 Œ0; T
2
,
(1.9)
k‰.u/.t/ ‰.v/.t /k
X
D
Z
t
0
e
tL
ˆ.u.s// ˆ.v.s//
ds
X
C
t
1
K sup ku.s/ v.s/k
X
I
and now if T T
2
is chosen small enough, we have C
T
1
K<1,making‰ a
contraction mapping on Z. Thus ‰ has a unique fixed point u in Z, solving (1.2).
We have proved the following:
Proposition 1.1. If X and Y are Banach spaces for which (1.3)–(1.6) hold, then
the parabolic equation (1.1), with initial data f 2 X, has a unique solution
u 2 C.Œ0; T ; X /,whereT>0is estimable from below in terms of kf k
X
.
As an example, let M be a compact Riemannian manifold, and consider
(1.10) X D C
1
.M /; Y D C.M/:
In this case we have the conditions (1.3)–(1.6)ifL D . In particular,
(1.11) ke
t
k
L.C;C
1
/
Ct
1=2
; for t 2 .0; 1:
Thus we have short-time solutions to (1.1) with f 2 C
1
.M /.
It will be useful to weaken the hypothesis (1.3) a bit. Consider a pair of Banach
spaces X and Z of functions, or distributions, on M , such that there are continu-
316 15. Nonlinear Parabolic Equations
ous inclusions
C
1
0
.M / X Z D
0
.M /:
We will say that a function u.t/ taking values in X,fort 2 I , some interval in R,
belongs to C.I; X / provided u.t/ is locally bounded in X,andu 2 C.I;Z/.More
generally, this defines C.I; X/ for any locally compact Hausdorff space I .Then
we say e
tL
is an almost continuous semigroup on X provided e
tL
is uniformly
bounded on X for t 2 Œ0; T ,givenT<1, e
.sCt/L
u D e
sL
e
tL
u, for each
u 2 X; s; t 2 Œ0; 1/,and
u 2 X H) e
tL
u 2 C.Œ0; 1/; X/:
Examples include e
t
on L
1
.M / andonH¨older spaces C
r
.M /; r 2 R
C
n Z
C
,
when M is compact. The space C.I; X / may depend on the choice of Z,but
we omit reference to Z in the notation. For example, when we consider e
t
on
L
1
.M /, with M compact, we might fix p<1 and take Z D L
p
.M /.
The proof of Proposition 1.1 readily extends to the following variant:
Proposition 1.1A. Let X and Y be Banach spaces for which (1.4)–(1.6) hold. In
place of (1.3), we assume e
tL
is an almost continuous semigroup on X.Also,we
augment (1.4) with the condition that ˆ W C.I; X/ ! C.I; Y /. Then the initial-
value problem (1.1), given f 2 X, has a unique solution u 2 C.Œ0; T ; X/,where
T>0is estimable from below in terms of kf k
X
.
As examples, we can consider
(1.12) X D C
rC1
.M /; Y D C
r
.M /;
r 0.Ifr is not an integer, these are H¨older spaces. We have, for any s>0,
(1.13) ke
t
k
L.C
r
;C
rCs
/
C
s
t
s=2
;0<t 1:
It follows from (1.2)thatiff 2 C
rC1
and one has a solution u in the space
C.Œ0; T ; C
rC1
/, then actually, for each t>0;u.t/ 2 C
rCs
for every s<2.
We can iterate this argument repeatedly, and also, via the PDE (1.1), obtain the
regularity of t-derivatives of u, proving:
Proposition 1.2. Given f 2 C
1
.M /; L D , the equation (1.1) has, for some
T>0, a unique solution
(1.14) u 2 C
Œ0; T ; C
1
.M /
\ C
1
.0; T M
:
A number of different pairs X and Y can be constructed; it is particularly
of interest to have results for cases other than X D C
1
.M /; Y D C.M/,as
these are often useful for establishing the existence of global solutions. When
1. Semilinear parabolic equations 317
(1.4) holds depends on the nature of the nonlinearity in (1.1). We list here some
estimates that bear on when (1.6) holds, in case L D . The bound in the right
column is on the operator norm over 0<t 1. In the cases listed here, we
assume that p q,ands r.
(1.15) YXbound on ke
t
k
L.Y;X/
L
q
.M / L
p
.M / C t
.n=2/.1=q1=p/
I
H
r;p
.M / H
s;p
.M / C t
.1=2/.sr/
I
H
r;q
.M / H
s;p
.M / C t
.n=2/.1=q1=p/.1=2/.sr/
I
We now take a look at the case F.u; ru/ D
P
j
@
j
F
j
.u/ of (1.1), with L D
;thatis,
(1.16)
@u
@t
D u C
X
j
@
j
F
j
.u/; u.0/ D f:
For simplicity, we take M D T
n
. The limiting case D 0 of this, which we
will consider in 5 of the next chapter, includes important cases of quasi-linear,
hyperbolic equations. We will assume each F
j
is smooth in its arguments (u can
take values in R
K
) and satisfies estimates
(1.17) jF
j
.u/jC hui
p
; jrF
j
.u/jC hui
p1
;
for some p 2 Œ1; 1/. We will show that the Banach spaces
(1.18) X D L
q
.M /; Y D H
1;q=p
.M /
satisfy the conditions (1.3)–(1.6) for a certain range of q. First, we need q p,
so q=p 1 in (1.18). Only (1.4)and(1.6) need to be investigated. For (1.4)we
need F
j
W L
q
! L
q=p
to be locally Lipschitz. To get this, write
(1.19)
F
j
.u/ F
j
.v/ D G
j
.u;v/.u v/;
G
j
.u;v/ D
Z
1
0
F
0
j
su C.1 s/v
ds:
By (1.17), we have an estimate on kG
j
.u;v/k
L
q=.p1/
, and, by the generalized
H¨older inequality,
(1.20) kF
j
.u/ F
j
.v/k
L
q=p
kG
j
k
L
q=.p1/
ku vk
L
q
;
so we have (1.4). To check (1.6), we use the third estimate in (1.15), to get
(1.21) ke
t
k
L.H
1;q=p
;L
q
/
Ct
.n=2/.p=q1=q/1=2
;