Taylor M.E. Partial Differential Equations III: Nonlinear Equations
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98 13. Function Space and Operator Theory for Nonlinear Analysis
Note that the essential difference between H
.X; Y / and the space we have just
introduced is that we have completely dropped any continuity requirement at
fRe z D 0g. We also do not require continuity from fRe z D 1g to Y . The space
H
b
.X; Y / is a Banach space, with norm
(A.15) kuk
H
b
.X;Y /
D sup
z2
e
ku.z/k
X
C sup
y
ku.1 C iy/k
Y
:
Now, for 2 .0; 1/,weset
(A.16) ŒX; Y
b
Dfu./ W u 2 H
b
.X; Y /g;
with the same sort of Banach space structure as arose in (A.4) and (A.12). We
have continuous injections
(A.17) ŒX; Y
,! ŒX; Y
IX
,! ŒX; Y
b
:
Our next task is to extend the standard result on operator interpolation from the
setting of ŒX; Y
to that of ŒX; Y
IV
and ŒX; Y
b
.
Proposition A.1. Let .X
j
;Y
j
;V
j
/ be compatible triples, j D 1; 2. Assume that
T W V
1
! V
2
is continuous and that
(A.18) T W X
1
! X
2
;TW Y
1
! Y
2
;
continuously. (Continuity is automatic, by the closed graph theorem.) Then, for
each 2 .0; 1/,
(A.19) T W ŒX
1
;Y
1
IV
1
! ŒX
2
;Y
2
IV
2
:
Furthermore, if Y
j
X
j
(continuously) and T is a continuous linear map satis-
fying (A.18), then for each 2 .0; 1/,
(A.20) T W ŒX
1
;Y
1
b
! ŒX
2
;Y
2
b
:
Proof. Given f 2 ŒX
1
;Y
2
IV
,picku 2 H
.X
1
;Y
1
;V
1
/ such that f D u./.
Then we have
(A.21) T W H
.X
1
;Y
1
;V
1
/ ! H
.X
2
;Y
2
;V
2
/; .T u/.z/ D T u.z/;
and hence
(A.22) Tf D .T u/. / 2 ŒX
2
;Y
2
IV
2
:
This proves (A.19). The proof of (A.20) is similar.
A. Variations on complex interpolation 99
Remark: In case V D X C Y , with the weak topology, ŒX; Y
IV
is what is
denoted .X; Y /
w
in [JJ], and called the weak complex interpolation space.
Alternatives to (A.6) for a family X
s
of Banach spaces include
(A.23) ŒX
s
;X
t
IV
D X
.1/sCt
and
(A.24) ŒX
s
;X
t
b
D X
.1/sCt
:
Here, as before, we take 2 .0; 1/. It is an exercise, using results of 6,toshow
that both (A.23)and(A.24), as well as (A.6), hold when X
s
D H
s;p
.M /,given
p 2 .1; 1/,whereM can be R
n
or a compact Riemannian manifold. We now
discuss the situation for Zygmund spaces.
We start with Zygmund spaces on the torus T
n
. We recall from 8 that the
Zygmund space C
r
.T
n
/ is defined for r 2 R, as follows. Take ' 2 C
1
0
.R
n
/,
radial, satisfying './ D 1 for jj1.Set'
k
./ D '.2
k
/.Thenset
0
D ';
k
D '
k
'
k1
for k 2 N,sof
k
W k 0g is a Littlewood–Paley partition of
unity. We define C
r
.T
n
/ to consist of f 2 D
0
.T
n
/ such that
(A.25) kf k
C
r
D sup
k0
2
kr
k
k
.D/f k
L
1
< 1:
With ƒ D .I /
1=2
and s; t 2 R,wehave
(A.26) ƒ
sCit
W C
r
.T
n
/ ! C
rs
.T
n
/:
By material developed in 8,
(A.27) r 2 R
C
n Z
C
H) C
r
.T
n
/ D C
r
.T
n
/;
where, if r D k C ˛ with k 2 Z
C
and 0<˛<1, C
r
.T
n
/ consists of functions
whose derivatives of order k are H¨older continuous of exponent ˛.
We aim to show the following.
Proposition A.2. If r<s<tand 0<<1,then
(A.28) ŒC
s
.T
n
/; C
t
.T
n
/
IC
r
.T
n
/
D C
.1/sCt
.T
n
/;
and
(A.29) ŒC
s
.T
n
/; C
t
.T
n
/
b
D C
.1/sCt
.T
n
/:
Proof. First, suppose f 2 ŒC
s
;C
t
IC
r
,sof D u./ for some u 2 H
.C
s
;
C
t
;C
r
/. Then consider
(A.30) v.z/ D e
z
2
ƒ
.ts/z
ƒ
s
u.z/:
100 13. Function Space and Operator Theory for Nonlinear Analysis
Bounds of the type (A.8)onu, together with (8.13) in the torus setting, yield
(A.31) kv.iy/k
C
0
; kv.1 C iy/k
C
0
C;
with C independent of y 2 R.Inotherwords,
(A.32) k
k
.D/v.z/k
L
1
C; Re z D 0; 1;
with C independent of Im z and k. Also, for each k 2 Z
C
,
k
.D/v W !
L
1
.T
n
/ continuously, so the maximum principle implies
(A.33) k
k
.D/ƒ
.ts/
ƒ
s
f k
L
1
C;
independent of k 2 Z
C
.Thisgivesƒ
.1/sCt
f 2 C
0
, hence f 2
C
.1/sCt
.T
n
/.
Second, suppose f 2 C
.1/sCt
.T
n
/.Set
(A.34) u.z/ D e
z
2
ƒ
.z/.t s/
f:
Then u./ D e
2
f . We claim that
(A.35) u 2 H
.C
s
;C
t
;C
r
/;
as long as r<s<t. Once we establish this, we will have the reverse containment
in (A.28). Bounds of the form
(A.36) ku.z/k
C
s
C; ku.1 C iy/k
C
t
C
follow from (8.13), and are more than adequate versions of (A.8). It remains to
establish that
(A.37) u W
! C
r
.T
n
/; continuously.
Indeed, we know u W
! C
s
.T
n
/ is bounded. It is readily verified that
(A.38) u W
! D
0
.T
n
/; continuously,
and that
(A.39) r<sH) C
s
.T
n
/,! C
r
.T
n
/ is compact.
The result (A.37) follows from these observations. Thus the proof of (A.28)is
complete.
A. Variations on complex interpolation 101
We turn to the proof of (A.29). If u 2 H
b
.C
s
;C
t
/,formv.z/ as in (A.30), and
for " 2 .0; 1 set
(A.40) v
"
.z/ D e
"ƒ
v.z/; v
"
W
e
! C
0
.T
n
/ bounded and continuous
(with bound that might depend on "). We have
(A.41)
k
.D/v
"
." C iy/ D e
."Ciy/
2
k
.D/e
"ƒ
ƒ
.ts/"
ƒ
i.ts/y
ƒ
s
u.z/:
Now fƒ
s
u.z/ W z 2
e
g is bounded in C
0
.T
n
/, and the operator norm of ƒ
i.ts/y
on C
0
.T
n
/ is exponentially bounded in jyj.Wehave
(A.42) fe
"ƒ
ƒ
".ts/
W 0<" 1g bounded in OPS
0
1;0
.T
n
/;
hence bounded in operator norm on C
0
.T
n
/. We deduce that
(A.43) k
k
.D/v
"
." C iy/k
L
1
C;
independent of y 2 R and " 2 .0; 1. The hypothesis on u also implies
(A.44) k
k
.D/v
"
.1 C iy/k
L
1
C;
independent of y 2 R and " 2 .0; 1. Now the maximum principle applies. Given
2 .0; 1/,
(A.45) k
k
.D/e
"ƒ
v./k
L
1
C;
independent of ".Taking" & 0 yields v./ 2 C
0
.T
n
/, hence u."/ 2
C
.1/sCt
.T
n
/.
This proves one inclusion in (A.29). The proof of the reverse inclusion is sim-
ilar to that for (A.28). Given f 2 C
.1/sCt
.T
n
/,takeu.z/ as in (A.34). The
claim is that u 2 H
b
.C
s
;C
t
/. We already have (A.36), and the only thing that
remains is to check that
(A.46) u W
e
! C
s
.T
n
/ continuously;
and this is straightforward. (What fails is continuity of u W
! C
s
.T
n
/ at the
left boundary of
.)
Remark: In contrast to (A.28)–(A.29), one has
(A.47) ŒC
s
.T
n
/; C
t
.T
n
/
D closure of C
1
.T
n
/ in C
.1/sCt
.T
n
/:
Related results are given in [Tri].
102 13. Function Space and Operator Theory for Nonlinear Analysis
If OPS
m
1;0
.T
n
/ denotes the class of pseudodifferential operators on T
n
with
symbols in S
m
1;0
, then for all s; m 2 R,
(A.48) P 2 OPS
m
1;0
.T
n
/ H) P W C
s
.T
n
/ ! C
sm
.T
n
/:
Cf. Proposition 8.6. Using coordinate invariance of OP S
m
1;0
and of C
r
.T
n
/ for
r 2 R
C
n Z
C
, we deduce invariance of C
s
.T
n
/ under diffeomorphisms, for all
s 2 R.
From here, we can develop the spaces C
s
.M / on a compact Riemannian man-
ifold M and the spaces C
s
.M/ on a compact manifold with boundary. These
developments are done in 8.
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14
Nonlinear Elliptic Equations
Introduction
Methods of the calculus of variations applied to problems in geometry and
classical continuum mechanics often lead to elliptic PDE that are not linear. We
discuss a number of examples and some of the developments that have arisen to
treat such problems.
The simplest nonlinear elliptic problems are the semilinear ones, of the form
Lu D f.x;D
m1
u/,whereL is a linear elliptic operator of order m and the
nonlinear term f.x;D
m1
u/ involves derivatives of u of order m 1.In 1 we
look at semilinear equations of the form
(0.1) u D f.x;u/;
on a compact, Riemannian manifold M , with or without boundary. The Dirichlet
problem for (0.1) is solvable provided @
u
f.x;u/ 0 if each connected com-
ponent of M has a nonempty boundary. If M has no boundary, this condition
does not always imply the solvability of (0.1), but one can solve this equation if
one requires f.x;u/ to be positive for u >a
1
and negative for u <a
0
.Weuse
three approaches to (0.1): a variational approach, minimizing a function defined
on a certain function space, the “method of continuity,” solving a one-parameter
family of equations of the type (0.1), and a variant of the method of continuity
that involves a Leray–Schauder fixed-point theorem. This fixed-point theorem is
established in Appendix B, at the end of this chapter.
A particular example of (0.1)is
(0.2) u D k.x/ K.x/e
2u
;
which arises when one has a 2-manifold with Gauss curvature k.x/ and wants
to multiply the metric tensor by the conformal factor e
2u
and obtain K.x/ as the
Gauss curvature. The condition @
u
f.x;u/ 0 implies that K.x/ 0 in (0.2).
In 2 we study (0.2) on a compact, Riemannian 2-fold without boundary, given
K.x/ < 0. The Gauss–Bonnet formula implies that .M / < 0 is a necessary
M.E. Taylor, Partial Differential Equations III: Nonlinear Equations,
Applied Mathematical Sciences 117, DOI 10.1007/978-1-4419-7049-7
2,
c
Springer Science+Business Media, LLC 1996, 2011
105
106 14. Nonlinear Elliptic Equations
condition for solvability in this case; the main result of 2 is that this is also a
sufficient condition. When you take K 1, this establishes the uniformization
theorem for compact Riemann surfaces of negative Euler characteristic. When
.M / D 0, one takes K D 0 and (0.2) is linear. The remaining case of this
uniformization theorem, .M / D 2, is treated in Chap. 10, 9.
The next topic is local solvability of nonlinear elliptic PDE. We establish this
via the inverse function theorem for C
1
-maps on a Banach space. We treat un-
derdetermined as well as determined elliptic equations. We obtain solutions in 3
with a high but finite degree of regularity. In some cases such solutions are actu-
ally C
1
.In 4 we establish higher regularity for solutions to elliptic PDE that are
already known to have a reasonably high degree of smoothness. This result suf-
fices for applications made in 3, though PDE encountered further on will require
much more powerful regularity results.
In 5 we establish the theorem of J. Nash, on isometric imbeddings of com-
pact Riemannian manifolds in Euclidean space, largely following the ingenious
simplification of M. G¨unther [Gu1], allowing one to apply the inverse function
theorem for C
1
-maps on a Banach space. Again, the regularity result of 4 ap-
plies, allowing one to obtain a C
1
-isometric imbedding.
In 6 we introduce the venerable problem of describing minimal surfaces. We
establish a number of classical results, in particular the solution to the Plateau
problem, producing a (generalized) minimal surface, as the image of the unit disc
under a harmonic and essentially conformal map, taking the boundary of the disc
homeomorphically onto a given simple closed curve.
In 7 we begin to study the quasi-linear elliptic PDE satisfied by a function
whose graph is a minimal surface. We use results of 6 to establish some re-
sults on the Dirichlet problem for the minimal surface equation, and we note
several questions about this Dirichlet problem whose solutions are not simple
consequences of the results of 6, such as boundary regularity. These questions
serve as guides to the results of boundary problems for quasi-linear elliptic PDE
derived in the next three sections.
In 8 we apply the paradifferential operator calculus developed in Chap. 13,
10, to obtain regularity results for nonlinear elliptic boundary problems. We
concentrate on second-order PDE (possibly systems) on a compact manifold with
boundary
M and obtain higher regularity for a solution u, assumed a priori to be-
long to C
2Cr
.M/,forsomer>0, for a completely nonlinear elliptic PDE, or to
C
1Cr
.M/, in the quasi-linear case. To check how much these results accomplish,
we recall the minimal surface equation and note a gap between the regularity of
a solution needed to apply the main result (Theorem 8.4) and the regularity a
solution is known to possess as a consequence of results in 7.
Section 9 is devoted to filling that gap, in the scalar case, by the famous
DeGiorgi–Nash–Moser theory. We follow mainly J. Moser [Mo2], together with
complementary results of C. B. Morrey on nonhomogeneous equations. Morrey’s
results use spaces now known as Morrey spaces, which are discussed in Appendix
A at the end of this chapter.
1. A class of semilinear equations 107
With the regularity results of 8 and 9 under our belt, we resume the study of
the Dirichlet problem for quasi-linear elliptic PDE in the scalar case, in 10, with
particular attention to the minimal surface equation. We note that the Dirichlet
problem for general boundary data is not solvable unless there is a restriction on
the domain on which a solution u is sought. This has to do with the fact that the
minimal surface equation is not “uniformly elliptic.” We give examples of some
uniformly elliptic PDE, modeling stretched membranes, for which the Dirichlet
problem has a solution for general smooth data, on a general, smooth, bounded
domain. We do not treat the most general scalar, second-order, quasi-linear elliptic
PDE, though our treatment does include cases of major importance. More material
can be found in [GT] and [LU].
In 11 we return to the variational method, introduced in 1, and prove that a
variety of functionals
(0.3) I.u/ D
Z
F.x;u; ru/dV.x/
possess minima in sets
(0.4) V Dfu 2 H
1
./ W u D g on @g:
The analysis includes cases both of real-valued u and of u taking values in R
N
.
The latter case gives rise to N N elliptic systems, and some regularity results
for quasi-linear elliptic systems are established in 12. Sometimes solutions are
not smooth everywhere, but we can show that they are smooth on the complement
of a closed set † of Hausdorff dimension <n 2 (n D dim ). Results of
this nature are called “partial regularity” results.
In 13 we establish results on linear elliptic equations in nondivergence form,
due to N. Krylov and M. Safonov, which take the place of DeGiorgi–Nash–Moser
estimates in the study of certain fully nonlinear equations, done in 14.In 15
we apply this to equations of the Monge–Ampere type.
In 16 we obtain some results for nonlinear elliptic equations for functions of
two variables that are stronger than results available for functions of more vari-
ables.
One attack on second-order, scalar, nonlinear elliptic PDE that has been very
active recently is the “viscosity method.” We do not discuss this method here; one
can consult the review article [CIL] for material on this.
1. A class of semilinear equations
In this section we look at equations of the form
(1.1) u D f.x;u/ on M;