# A remark on the regularity at the boundary for solutions of elliptic equations

## Full text

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e

o

### 4 (1961), p. 355-370

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(2)

### by

M. K. VENKATESHA MURTHY

1.

The

### object

of this note is to prove the

### following

result.

Let A be a linear

order

with

### infinitely

diffe-

rentiable coefficients in a

a smooth

### boundary

in a

euclidean space and let

be differential

with

### infinitely

differentiable coefficients on

and

are functions

### in

certain classes of

### infinitely

differentiable functions

### (referred

to as Friedman classes in the

then

any function u

in Sd and

in

in

### aS2,

is itself in a Friedman class in

### S~

when the coefficients of A

of

### By

and the functions which define the

are

in certain Friedman classes.

When

### f, gj,

and the coefficients of A and of

are real

### analytic

functions of their

the real

### analyticity

of the solution u

the

the

has been

a method of

and

and

that

is

an

and

is

### analytic.

Our result includes that of Ma-

and

### Stampacchia.

In the case of the Dirichlet

this result

wa.s

### proved

in the case of real

functions

and Niren-

### berg 

and in the case of functions in Friedman classes

Friedman in

The

### notation,

necessary norms, and other

are introduced

2.

3 two

### L2-estimates

for u and its tan-

### gential

derivatives of all orders and normal derivatives of order

are

In

4

### L2-estimates

for derivatives of all orders of u are obta- ined and

an

### application

of Sobolev’s lemma

### yields

the result.

The author wishes to

B. V.

### Singbal

for his valuable sugge- stions and

(3)

2.

### Notation and Preliminaries.

Let Sl denote a bounded domain in a v-dimensional Euclidean space and let

be a coordinate

### system

in D. First we define certain

classes of

differentiable

### (C°°)

fanctions on Let be a se-

quence of

numbers

the

### following

condition: there exists

a

constant

n, such that

Then

is any

we denote

### by C (In p ; S

the

class of C°°-functions

on S

the

condition: . .

### (QA)

For every closed subdomaiu of S~ there exist two constants

and

on

and on euch

### that,

for any x E we have

where k =-

... ,

### ky) and k ~ = k! +... + k" .

We call a class of the

C a cla88.

### Siinilal·ly

we define the classes C

### Q)

when the condition

is satisfied in

It is clear that

/ r

with a

constant

of n

fact we can

=

### G .

Now we make some remarks on the Friedman classes C

### (Mn ; Q)

which

will be of use in the

### (i) If f is

a function in the class C

such that

0 for

r E Q

### then 7 1 . f

is itself in the class

let

be a coor-

dinate

in Q and let

a

of order one.

differentiation

### operator

of order k can be written in the ...

Then

(4)

where

is a

of

### degree n

in the set of ar-

and where the

### degree of dit ...

is taken to be r.

monomial of

in these

is

### majorized

oii any closed subset

Hence one can

see that

with a suitable

constant c

of n.

we see that

on

### S~a

which establishes

If

are in C

then their

is itself in In

### fact,

we have for an a =

... ,

Leibniz

### formule,

where

Then

suitable constant C’ &#x3E; 0.

The remarks

and

the

If

to 0

in Q then

### f/g

is itself in the class C

### (2M,, ~ Of.

Let s denote a fixed

### positive

real number. Let H8 denote the space 01

,

all

### tempered

distributions 99 such that its h’ourier

### transform g2

satisfies the condition that

is square

### integra~ble

and we define the scalar

in H 8

and the

norm

### (see ).

(5)

In view of the local nature of the

it is

### enough

to consider

the solution of the

iu a

with the

conditions

defined on the

of the

### Throughont,

the function u is assumed to be

### infinitely

differentiable in the

with the

of the

This is so, for

in the

### following

cases:

Let the coefficients of A and be

### infinitely

differentiable functions of their

### arguments.

Then any solution u of

is

### infinitely

differentiable either when

is an

in the

sense defined.

J. Peetre

or when the

operators

the

condition of

and

with

to the

### elliptic operator

A.

Next we introduce the -differential

### operators.

Let o),, denote the hemi-

the

of the

of CÏJr.

denote the

### (v -1)-dimensional subspace

and let x’ denote either

or

### (xl , ... , xy_1) inadvert- ently

in the context. We

the

notation

then ap

### (x)

denotes a function

### (x)

and DP =

Similar notation is used in non also with x’ iu

of x and

in

### place

of DP.

All our functions are (defined in cvxo

with ’tbe

of the

of

where

a fixed

number. Let

be au

linear

differential

### operator

of order 2w on wRo with the coefficients c~~

### (x)

6’°° in mm and let

be differential

where

are

.functions

on al CORO.

Let o

### (t) .be

a real valued C°° function of the variable t

oo t

such that

(6)

then for any

of

### positive

numbers r aud h with define

Then

### clearly

we have

and farther for any we have

where

is a

constant.

### depending

on v, p and the bounds for the derivatives of Lo.

DEFINITION. The

### system (A, (Bj))

is said to be an admissible

if A and the

the

### following

condition : there exists a constant

### 03

such that for aiiy C°° function u anu for any r with

we have

where if

is cousidered as

its

containd in

then

### BJ ((P,,h 11)

is extended to the whole of no

it to be

to

zero 11 ~co -

Here

defined

REMARK,

When

the

of

a solution u of

the

### boundary (the

coefficients of A and of

real

### analytic

functions of their

was

and

The

### iiieqnality (2)

has been obtained

J. Peetre when

is an

### elliptic system

in the sense defined in

When A is

and

the

condition with

to A an

ine-

has been

and

The

is the

### precise

statement of onr theorem.

THEOREM. Let

with A

such

that the

### following

conditions are satisfied :

### (i)

the coefficients ap

of A are in C

and

the coefficients

of

are hi C

### (Mn; Ôt

(7)

Then any function u, C°° in wR0 and

the

where

is

C

in C

### at roBot I respectively,

is

a function in C mm

### U at COR.)-

In the course of the

### proof

of the theorem we need the

norms

in

### ):

and

We make the covention that

and

and introduce the

notation

### (in analogy

with that introduced in

and

(8)

3. In this

we

two lemmas

t.o the

### proof

of

the main theorem stated in the

In

we obtain

an

for the

### derivatives, upto

order 2m in the transverse direction and of all orders in the

for a function

the

To

with we have the

due to

and

p.

### 331).

If u is any C°° function and if

### system

then

there exists a constant,

y

of

such that for

we have

Now we observe

for any

we have

It follows from

for k &#x3E;

### 0,

2m we have

On the otherhand we also have

### Taking

i can now be written in the form

LEMMA 3.1. If u is any C°° fnnetion and if

### system

then there exists a

constant

indendent

R and of

(9)

such that for

&#x3E; 0 the

### following inequality

holds:

PROOF. Consider any one of the

derivatives with

and

in the form

B and

We obtain

### Using

Leibniz formula for the derivation of a

### product

of two functions and the fact that

are

such that q,

qv = k and

...

### -E-

8" = p we have the

aiid

over

= k in

the

(10)

the constants

and

use of

we obtain

where

is a

constant

### independent

of R. Moreover we

with

### C7 independent

of v. From

this remark it is clear that the last term of the second member of the abo-

ve

can be

### majorized by

the last but one term. Hence

where

is a

constant

lc.

the

### inequality (4)

to the last term of the second member of

we obtain

### Multiplying

both sides of estimates :

we have the

(11)

Further since

because

we have

But

and

it follows thaf,

Hence we have:

Then the

### inequality (11)

becomes

Since it follows that there

(12)

exists a constant

such that

### Taking It +

I = i in the last term of the second member and the sup-

remum for

### R/2 _

t* R of the first member we obtain

and this

the

of the

LEMMA 3.2. Let

### system

and u be any 000 function

the

with

in the classes C

and C

### (

Then there exist two

### positive

constants M and such that

PROOF. We can suppose, if necessary after soine modification that the constants

and

### R,

are the same as before and are such that

and

### Let fJ

denote the volume of the unit ball in the y dimensional Euclidean space. Then for R

we have

I

(13)

of

we obtain

where

are

constants

of 2X and k~

Then the

for any k and R

Now

as in the

of

and

with the

M ?

and A _

we obtain

after

an induction

on k. This

the

of lem-

ma 3.2.

4. We

the

### proof

of the main theorem

in this pa-

### ragraph.

For this purpose it is

### necessary

to obtain estimates of the

for all

as well as

of u.

obtain

and

in

We introduce the

norms

to those

2.

For

2~n define

to

we have

(14)

This

that

We now prove the

### following

extension of the estimation

if R is smaller

than or

### equal

to a fixed number

on the

differential

y

,

with

_ 1 fixed

and 9

on the

,_ P g

The

### following

is a sketch of the derivation of the estimate

us

### denote 3)" by

y for convenience.

### By assumption

y = 0 is not a charecteristic surface for the

### given equatiori Au = f.

Hence, one can eolve for

### derivative D;tJ’

u of u in terms of the derivatives

### involving

normal deriva- tives of u of orders less than 2m :

where in view of the remarks on the classes C

and

are functions

to the class C

This

### implies

that both

for suitable constants

and

### 80 ~ 1.

We can assume these constants to be the same as before

### by

suitable choice. Then we have from

Hence

It is clear from

and

that

follows for

and all

p &#x3E; 0

that R

chosen

and

(15)

We prove

for q &#x3E;

p &#x3E; 0

### by

induction on q. Let us assume that

### (15)

holds for all values

### of q

less than a certain

which we

denote

q.

both sides of

and

### integrating

over wr we obtain

### Multiplying

both sides of this

for R

### taking

the supremum over all r witl

r R and

the induction

### assumption

we obtain

where g is a suitable constant.

But

we have the

Then the

### inequality (20)

becomes

(16)

Here all the terms in the summation over

are less than

### unity. Taking

and the second member does not exceed which is

less than

if

and

that

Thus we have

holds for

2w and for R

### R1

if we take

As we have

said in the introduction the result is deduced

### by applying

Sobolev’s lemma to the

### L2-norms

of the derivatives of u. For this

we need estimates for the square

of the

These are

obtained from

as follows:

Hence

Thus we obtain

Now we

### apply

Sobolev’s lemma in the form used in

### , namely,

for

(17)

Since R - r _ r and R 1 we obtain the

after

a

constant

### independent of p).

This proves the fact that u E C

thus

the

of

the theorem.

BIBLIOGRAPHY

### 

S. AGMON, A. DOUGLIS and L. NIRENBRG, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm.

Pare. Appl. Math., 12 (1959), pp. 623-727.

### 

A. FRIEDMAN, On the regularity of the solutions of non-linear elliptic and parabolic systems of partial differential equations, Journal of Mathematics and Mechanics, 7 (1958), pp.

43-59.

### 

E. MAGENES and G. STAMPACCHIA, I problemi al contorno per le equazioni differenziali

di tipo ellittico, Scuola Normale Snperiore, Pisa, 12 (1958), pp. 247-257.

### 

C. B. MORREY and L. NIRENBERG, On the analyticity of the solutions of linear elliptic systems of partial differential equations, Comm. Pure. Appl. Math., 10 (1957), pp.

271-290.

### 

J. PEETRE, Théorèmes de régularité pour quelques classes

### d’opérateurs

différentiels, Commu-

nioations dn Séminaire Mathématique de l’université de Lund, Tome 16 (1959).

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