Jost J. Partial Differential Equations

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10.2 L

-Regularity Theory and Applications 277



Du · Dϕ = −



fϕ for all ϕ ∈ C

∞

(Ω). (10.2.2)

Then u ∈ W

2,p

(Ω



) for any Ω



⊂⊂ Ω,and

u

2,p

(Ω



)

≤ const



u

(Ω)

+ f

(Ω)



, (10.2.3)

with a constant depending on p, d, Ω



,andΩ. Also,

Δu = f almost everywhere in Ω. (10.2.4)

We do not provide a complete proof of this result either. This time, how-

ever, we shall present at least a sketch of the proof:

Apart from the fact that (10.1.8) needs to be replaced by the inequality

)

(B(x,R))

≤ const. Δv

(B(x,r))

(10.2.5)

coming from the Calderon–Zygmund inequality (Theorem 10.2.1), we may

ﬁrst proceed as in the proof of Theorem 10.1.2 and obtain the estimate

)

(B(x,R))

≤ const



f

(B(x,R))

(1 − σ)R

Du

(B(x,

1+σ

R))

(1 − σ)

u

(B(x,r))



(10.2.6)

for 0 <σ<1, B(x, R) ⊂ Ω. The second part of the proof, namely the

estimate of Du

, however, is much more diﬃcult for p = 2 than for p =

2. One needs an interpolation argument. For details, we refer to Gilbarg–

Trudinger [9] or Giaquinta [8]. This ends our sketch of the proof.

The reader may now get the impression that the L

-theory is a technically

subtle, but perhaps essentially useless, generalization of the L

-theory. The

-theory becomes necessary, however, for treating many nonlinear PDEs.

We shall now discuss an example of this. We consider the equation

Δu + Γ (u)|Du|

= 0 (10.2.7)

with a smooth Γ . We also require that Γ (u) be bounded. This holds if we

assume that Γ itself is bounded, or if we know already that our (weak)

solution u is bounded.

Equation (10.2.7) occurs as the Euler–Lagrange equation of the varia-

tional problem

I(u):=



g(u(x))|Du(x)|

dx → min, (10.2.8)

with a smooth g that satisﬁes the inequalities

278 10. Strong Solutions

0 <λ≤ g(v) ≤ Λ<∞, |g



(v)|≤k<∞ (10.2.9)



is the derivative of g), with constants λ, Λ, k, for all v.

In order to derive the Euler–Lagrange equation for (10.2.8), as in Sec-

tion 8.4, for ϕ ∈ H

1,2

(Ω), t ∈ R, we consider

I(u + tϕ)=



g(u + tϕ)|D(u + tϕ)|

dx.

In that case,

I(u + tϕ)

|t=0





2g(u)



ϕ + g



(u)|Du|







−2g(u)Δu − 2



g(u)D

u + g



(u) |Du|



ϕdx





−2g(u)Δu − g



(u)|Du|



ϕdx

after integrating by parts and assuming for the moment u ∈ C

The Euler–Lagrange equation stems from requiring that this expression

vanish for all ϕ ∈ H

1,2

(Ω), which is the case, for example, if u minimizes

I(u) with respect to ﬁxed boundary values. Thus, that equation is

Δu +



(u)

2g(u)

|Du|

=0. (10.2.10)

With Γ (u):=



(u)

2g(u)

, we have (10.2.7).

In order to apply the L

-theory, we assume that u is a weak solution of

(10.2.7) with

u ∈ W

1,p

(Ω)forsomep

>d (10.2.11)

(as always, Ω ⊂ R

,andsod is the space dimension).

The assumption (10.2.11) might appear rather arbitrary. It is typical

for nonlinear diﬀerential equations, however, that some such hypothesis is

needed. Although one may show in the present case

that any weak solution

u of class W

1,2

(Ω) is also contained in W

1,p

(Ω) for all p, in structurally simi-

lar cases, for example if u is vector-valued instead of scalar-valued (so that in

place of a single equation, we have a system of—typically coupled—equations

of the type (10.2.7)), there exist examples of solutions of class W

1,2

(Ω)that

are not contained in any of the spaces W

1,p

(Ω)forp>2. In other words,

for nonlinear equations, one typically needs a certain initial regularity of the

solution before the linear theory can be applied.

See Ladyzhenskya and Ural’tseva [17] or the remarks in Section 12.3 below.

10.2 L

-Regularity Theory and Applications 279

In order to apply the L

-theory to our solution u of (10.2.7), we put

f(x):=−Γ (u(x))|Du(x)|

. (10.2.12)

Because of (10.2.11) and the boundedness of Γ (u), then

f ∈ L

(Ω), (10.2.13)

and u satisﬁes

Δu = f in Ω. (10.2.14)

By Theorem 10.2.2,

u ∈ W

2,p

(Ω



) for any Ω



⊂⊂ Ω. (10.2.15)

By the Sobolev embedding theorem (Corollary 9.1.1, Corollary 9.1.3, and

Exercise 2 of Chapter 9),

u ∈ W

1,p

(Ω



) for any Ω



⊂⊂ Ω, (10.2.16)

with

d −

because of p

>d. (10.2.17)

Thus,

f ∈ L

(Ω



) for all Ω



⊂⊂ Ω, (10.2.18)

and we can apply Theorem 10.2.2 and the Sobolev embedding theorem once

more, to obtain

u ∈ W

∩ W

1,p

(Ω



)withp

d −

(10.2.19)

for all Ω



⊂⊂ Ω



. Iterating this procedure, we ﬁnally obtain

u ∈ W

2,q

(Ω



) for all q. (10.2.20)

We now diﬀerentiate (10.2.7), in order to obtain an equation for D

u, i =

1,...,d:

ΔD

u + Γ



(u)D

u|Du|

+2Γ (u)



u =0. (10.2.21)

This time, we put

f := −Γ



(u)D

u|Du|

− 2Γ (u)



u. (10.2.22)

280 10. Strong Solutions

Then

|f|≤ const (|Du|

+ |Du||D

u|),

and because of (10.2.20) thus

f ∈ L

(Ω



) for all p.

This means that v := D

u satisﬁes

Δv = f with f ∈ L

(Ω



) for all p. (10.2.23)

By Theorem 10.2.2, we infer

v ∈ W

2,p

(Ω



) for all p,

i.e.,

u ∈ W

3,p

(Ω



) for all p. (10.2.24)

We diﬀerentiate the equation again, to obtain equations for D

u (i, j =

1,...,d), apply Theorem 10.2.2, conclude that u ∈ W

4,p

(Ω



), etc. Iterating

the procedure again (this time with higher-order derivatives instead of higher

exponents) and applying the Sobolev embedding theorem (Corollary 9.1.2),

we obtain the following result:

Theorem 10.2.3: Let u ∈ W

1,p

(Ω),forp

>d(Ω ⊂ R

),beaweak

solution of

Δu + Γ (u)|Du|

where Γ is smooth and Γ (u) is bounded. Then

u ∈ C

∞

(Ω).



The principle of the preceding iteration process is to use the information

about the solution u derived in one step as structural information about

the equation satisﬁed by u in the next step, in order to obtain improved

information about u. In the example discussed here, we use this information

in the right-hand side of the equation, but in Chapter 12 we shall see other

instances. Such iteration processes are typical and essential tools in the study

of nonlinear PDEs. Usually, to get the iteration started, one needs to know

some initial regularity of the solution, however.

10.2 L

-Regularity Theory and Applications 281

Summary

A function u from the Sobolev space W

2,2

(Ω) is called a strong solution of

Δu = f

if that equation holds for almost all x in Ω.

In this chapter we show that weak solutions of the Poisson equation are

strong solutions as well. This makes an alternative approach to regularity

theory possible.

More generally, for a weak solution u ∈ W

1,1

(Ω)of

Δu = f,

where f ∈ L

(Ω), one may utilize the Calderon–Zygmund inequality to get

the L

-estimate for all Ω ⊂⊂ Ω,

u

2,p

(Ω



)

≤ const (u

(Ω)

+ f

(Ω)

This is valid for all 1 <p<∞ (but not for p =1orp = ∞).

This estimate is useful for iteration methods for the regularity of solutions

of nonlinear elliptic equations. For example, any solution u of

Δu + Γ (u)|Du|

with regular Γ is of class C

∞

(Ω), provided that it satisﬁes the initial regu-

larity

u ∈ W

1,p

(Ω)forsomep>d(= space dimension).

Exercises

10.1 Using the theorems discussed in Section 10.2, derive the following result:

Let u ∈ W

1,2

(Ω) be a weak solution of

Δu = f,

with f ∈ W

k,p

(Ω)forsomek ≥ 2andsome1<p<∞.Thenu ∈

k+2,p

(Ω

) for all Ω

⊂⊂ Ω,and

u

k+2,p

(Ω

)

≤ const (u

(Ω)

+ u

k,p

(Ω)

20.2 Consider the map

u : B(0, 1)(⊂ R

) → R

x →

|x|

282 10. Strong Solutions

Show that for d ≥ 3, u ∈ W

1,2

(B(0, 1), R

) (this means that all compo-

nents of u are of class W

1,2

). Show, moreover, that u is a weak solution

of the following system of PDEs:

Δu

+ u



i,β=1

=0 forα =1,...,d.

Since u is not continuous, we see that solutions of systems of semilinear

elliptic equations need not be regular.

11. The Regularity Theory of Schauder and

the Continuity Method

(Existence Techniques IV)

11.1 C

-Regularity Theory for the Poisson Equation

In this chapter we shall need the fundamental concept of H¨older continuity,

which we now recall from Section 9.1:

Deﬁnition 11.1.1: Let f : Ω → R, x

∈ Ω, 0 <α<1. The function f is

cal led H¨older continuous at x

with exponent α if

sup

x∈Ω

|f(x) − f(x

|x − x

< ∞. (11.1.1)

Moreover, f is cal led H¨older continuous in Ω if it is H¨older continuous at

each x

∈ Ω (with exponent α); we write f ∈ C

(Ω). If (11.1.1) holds for

α =1, then f is called Lipschitz continuous at x

. Similarly, C

k,α

(Ω) is the

space of those f ∈ C

(Ω) whose kth derivative is H¨older continuous with

exponent α.

We deﬁne a seminorm by

|f|

(Ω)

:= sup

x,y∈Ω

|f(x) − f(y)|

|x − y|

. (11.1.2)

We deﬁne

f

(Ω)

= f

(Ω)

+ |f|

(Ω)

and

f

k,α

(Ω)

as the sum of f

(Ω)

and the H¨older seminorms of all kth partial derivatives

of f. As in Deﬁnition 11.1.1, in place of C

0,α

, we usually write C

.The

following result is elementary:

Lemma 11.1.1: If f

∈ C

(G) on G ⊂ R

, then f

∈ C

(G),and

(G)

≤



sup



(G)



sup



(G)

284 11. Existence Techniques IV

Proof:

(x)f

(x) − f

(y)f

(y)|

|x − y|

≤

(x) − f

(y)|

|x − y|

(x)| +

(x) − f

(y)|

|x − y|

(x)|,

which directly implies the claim. 

Theorem 11.1.1: As always, let Ω ⊂ R

be open and bounded,

u(x):=



Γ (x, y)f (y)dy, (11.1.3)

where Γ is the fundamental solution deﬁned in Section 1.1.

(a) If f ∈ L

∞

(Ω) (i.e., sup

x∈Ω

|f(x)| < ∞),

then u ∈ C

1,α

(Ω),and

u

1,α

(Ω)

≤ c

sup |f | for α ∈ (0, 1). (11.1.4)

(b) If f ∈ C

(Ω), then u ∈ C

2,α

(Ω),and

u

2,α

(Ω)

≤ c

f

(Ω)

for 0 <α<1. (11.1.5)

The constants in (11.1.4) and (11.1.5) depend on α, d,and|Ω|.

Proof: (a) Up to a constant factor, the ﬁrst derivatives of u are given by

(x):=



− y

|x − y|

f(y)dy (i =1,...,d).

From this formula,



) − v

)



≤ sup

|f|·





− y

− y|

−

− y

− y|



dy. (11.1.6)

By the intermediate value theorem, on the line from x

to x

there exists

some x

with



− y

− y|

−

− y

− y|



≤

− x

− y|

. (11.1.7)

We put δ := 2 |x

− x

|.SinceΩ is bounded, we can ﬁnd R>0with

Ω ⊂ B(x

,R), and we replace the integral on Ω in (11.1.6) by the integral

on B(x

,R), and we decompose the latter as



B(x

,R)



B(x

,δ)



B(x

,R)\B(x

,δ)

= I

+ I

, (11.1.8)

“sup” here is the essential supremum, as explained in Appendix 12.3.

11.1 C

-Regularity Theory for the Poisson Equation 285

where without loss of generality, we may take δ<R.Wehave

≤ 2



B(x

,δ)

− y|

d−1

dy =2ω

δ (11.1.9)

and by (11.1.7)

≤ c

δ(log R − log δ), (11.1.10)

and hence

+ I

≤ c

− x

for any α ∈ (0, 1).

This proves (a), because obviously, we also have



(x)



≤ c

sup

|f|. (11.1.11)

(b) Up to a constant factor, the second derivatives of u are given by

(x)=





|x − y|

− d



− y



− y





|x − y|

d+2

f(y) dy;

however, we still need to show that this integral is ﬁnite if our assumption

f ∈ C

(Ω) holds. This will also follow from our subsequent considerations.

We ﬁrst put f(x)=0forx ∈ R

\ Ω; this does not aﬀect the H¨older

continuity of f.Wewrite

K(x − y):=



|x − y|

− d



− y



− y





|x − y|

d+2

∂

∂x



− y

|x − y|



We have



<|y|<R

K(y)dy =



|y|=R

|y|

−



|y|=R

|y|

(11.1.12)

=0,

since

|y|

is homogeneous of degree 1 − d.Thusalso



K(y)dy =0. (11.1.13)

We now write

(x)=



K(x − y)f(y)dy (11.1.14)



(f(y) − f(x)) K(x − y)dy

286 11. Existence Techniques IV

by (11.1.13). As before, on the line from x

to x

there is some x

with

|K(x

− y) − K(x

− y)|≤

− x

− y|

d+1

. (11.1.15)

We again put

δ := 2 |x

− x

and write (cf. (11.1.14))

) − w

)



{(f(y) − f(x

)) K(x

− y) − (f(y) − f(x

)) K(x

− y)} dy

= I

+ I

(11.1.16)

where I

denotes the integral on B(x

,δ), and I

that on R

\B(x

,δ). Since

|f(y) − f(x)|≤f

·|x − y|

, it follows that

|≤f



B(x

,δ)

{K(x

− y) |x

− y|

− K(x

− y) |x

− y|

} dy

≤c

f

· δ

(11.1.17)

Moreover,



\B(x

,δ)

(f(x

) − f(x

)) K(x

− y) dy



\B(x

,δ)

(f(y) − f(x

)) (K(x

− y) − K(x

− y)) dy, (11.1.18)

and the ﬁrst integral vanishes because of (11.1.12). Employing (11.1.15), and

since for y ∈ R

\ B(x

,δ),

− y|

d+1

≤

− y|

d+1

it follows that

|≤c

δ f



\B(x

,δ)

− y|

α−d−1

≤ c

f

. (11.1.19)

Inequality (11.1.5) then follows from (11.1.16), (11.1.17), (11.1.19). 

Theorem 11.1.2: As always, let Ω ⊂ R

be open and bounded, and Ω

⊂⊂

Ω.Letu be a weak solution of Δu = f in Ω.