ESI

The Erwin Schr?odinger International Boltzmanngasse 9

Institute for Mathematical Physics A-1090 Wien, Austria

Approximate Solutions to a Linear Dirichlet Problem
with Singular Coefficients

M. Nedeljkov
S. Pilipovi?c

Vienna, Preprint ESI 508 (1997) November 26, 1997

Supported by Federal Ministry of Science and Research, Austria
Available via anonymous ftp or gopher from FTP.ESI.AC.AT
or via WWW, URL: http://www.esi.ac.at

APPROXIMATE SOLUTIONS TO A LINEAR DIRICHLET
PROBLEM WITH SINGULAR COEFFICIENTS

M. Nedeljkov, S. Pilipovi?c

University of Novi Sad, Faculty of Science, Institute for Mathematics
Trg D. Obradovi?ca 4, Novi Sad, Yugoslavia

Abstract. In this paper is constructed an aproximate solution to P (x; D)G = H, Gj@? = F in the space of generaliaed Colombeau functions on ?. Here P is a differential operator with coefficients which are generalized functions (for example singular distributions), ? is a bounded open set, and H and F are generalized functions. Also, solutions to a class of elliptic equations with coefficients in G are obtained in somewhat different way. In the case of smooth coefficients, the consistency of the classical weak solution and the generalized solution is proved. Specially, for a class of second order elliptic equations with bounded coefficients the proposed method of finding generalized solutions produces the approximate solutions to the classical Dirichlet problem.

1. Introduction

Although the main interest in the space of Colombeau's generalized functions and in other spaces of generalized functions (cf. Egorov, [4], Rosinger [11]) comes from non-linear problems, linear problems in this frame are also interesting because equations with coefficients which are singular distributions may be studied. Also, construable approximate solutions G" can be obtained by using Colombeau's spaces.

The following concept of an approximate weak solution is used. Let G" be a family of smooth functions and D be some space of test functions. It is said that G" is a solution to equation P"(x; D)G" = H" in D-s-associated sense if

R
(P"(x; D)G"? H") dx = o("s) as " ! 0 for every 2 D.

The paper is the most important part of our program of re-studying linear problems in Colombeau's algebra G (cf. [6], [7], [8] and [10]). The subject of these papers were partial differential equations with constant coefficients. Some of the results obtained there are now used in the construction of an approximate solution to a partial differential equation with variable coefficients, i. e. coefficients which belong to a space of Colombeau's generalized functions. For example, one can treat stationary Schr?odinger equation with the Dirac ffi-distribution as a potential, ?4u+ ffiu = 0, with an arbitrary Dirichlet condition on a bounded domain. This example illustrates a need for the definition of the product is some new space of

1991 Mathematics Subject Classification. 35D05, 35A35, 35J25, 35J40, 46F10.
Key words and phrases. generalized solutions, Dirichlet problem, elliptic second order linear PDE, singular perturbations.

Typeset by AMS-TEX

2 M. NEDELJKOV, S. PILIPOVI?C

generalized functions which contains embedded distributions, which was the main
motivation for the construction of G (cf. [3] and [9]) and other spaces of generalized
functions (cf. [4] and [11]).

We shall use the simplified version of G which elements are equivalence classes
of nets of smooth functions, elements of EM , with respect to an appropriate ideal
N ae EM . Definitions of EM and N will be given in the next section.
In Theorem 1 is considered a family of partial differential equations

(1) P"(x; D)G" =

jffj<=m
aff;"(x)DffG"(x) = H"(x); x 2 ?;

where aff;" 2 EM (Rn), jffj <= m, H" 2 EM (Rn), " 2 (0; 1), and ? is a bounded open
set in Rn. We assume that P"(x; D) is of the form

(2) P"(x; D) = am;"(x)Dm1 +
n?1

k=0

jff0j<=k
aff;"(x)Dff0Dk1 ; x 2 Rn;

and that there exist C > 0 and ? > 0 such that

(3) jam;"(x)j >= C"N ; x 2 Rn; " 2 (0; ?):

Our goal is to find a family G" 2 EM (Rn) which satisfies

lim
"!0

Z (P"(x; D)G"(x) ? H"(x)) (x)dx = 0; for every 2 Hm(?);

G"j@? = Z";

where H" 2 EM (Rn) and Z" 2 EW;M (@?) are given. The definition of EW;M (@?)
will be given in the next section (cf. [1], [2]).

In terms of Colombeau generalized functions the above means that for given
H 2 G(Rn) and Z 2 GW;M (@?) we solve a Dirichlet problem

P (x; D)G =

jffj<=m
affDffG Hm(?)
ss H; in ?; Gj@? = Z; (G 2 G(Rn)):

Construction of a solution is the same for all types of differential operators.
Note that we have used Theorem 1 of [1] in order to solve the above problem with
homogeneous boundary conditions.

We shall examine elliptic equations with generalized functions as coefficients in
order to present relations of our results with the classical ones.

A class of families of elliptic equations of order 2m is studied in Theorem 3. By
using the classical Lax-Milgram lemma (cf. [12]) we construct a family of solutions
in the H2m
0 (?)-s-associated sense. We show that our general method given in

Theorem 2 for an arbitrary equation gives a solution which is C1
0 (?)-associated to
the solution constructed in Theorem 3.

Since the space of Schwartz distributions is naturally embedded into the space of
generalized functions G, the above approach can be used when the coefficients aff in

APPROXIMATED SOLUTIONS TO A DIRICHLET PROBLEM 3

(1) are distributions. Recall that the multiplication of distributions is not defined in D0(?) for all elements but when the distributions are embedded into G(?) their product exist in G(?) (cf. [3]).

In order to analyze the generalized solutions to Dirichlet's problems and the classical ones, in Theorems 4 and 5 we study a class of elliptic second order linear partial differential equation with coefficients belonging to L1(?). In this case, a generalized solution G is C1(?) \ H10(?)-associated to the classical solution g 2 H10(?).

2. Basic spaces

The space EM(?) consists of families of smooth functions G" in ?, " 2 (0; 1) such that for every K aeae ? and ff 2 Nn0 there exists N > 0 such that

sup
x2K
jDffG"(x)j = O("?N )

and N(?) consists of families G" 2 EM (?) such that for every K aeae ?, ff 2 Nn0
and r 2 R
sup
x2K
jDffG"(x)j = O("r):

The space of Colombeau's generalized functions, G(?), is defined by EM (?)=N (?).

If in previous definitions we have families of complex numbers instead of smooth functions, then we obtain the spaces EM;0, N0 and C = EM;0=N0, respectively. C
is called the space of Colombeau complex numbers.

Generalized functions and generalized numbers are denoted by capital letters and their representatives have a subscript ". The restrictions of generalized functions are defined by the restrictions of appropriate representatives. The support of G 2 G(?) is the complement of the largest open subset of ? where G is the zero generalized function. The space of all compactly supported generalized functions is denoted by Gc(?). It is naturally embedded into G(?).

We define Gm;t(?), m 2 N[f1g, t 2 R, to be the space of all elements G 2 G(Rn) which have representatives G" such that supjfij<=m

x2?

jDfiG"(x)j = O("?t). Let us note

that G1;t(?) is a subspace of Oberguggenberger's space G1(?) (cf. [9]).
If g 2 D0(?), then G"(x) = hg(?); "?nOE((? ? x)=")i; x 2 ?, where OE 2 C1
0 , R
OE(x)dx = 1, is a representative of the corresponding element in G.
Let D be some space of test functions, for example Hm0 (?), Hm(?), C1(?)

or Gm;t(?). We say that G1; G2 2 G(?) are equal in D-sense, G1 D= G2, if hG1; 'i = hG2; 'i in C for every ' 2 D. Similarly we define equality in D-sense

for a generalized function G1 and g 2 D0 by G1 D= g, if hG1; 'i = hg; 'i for every

' 2 D. Recall, Hm (Hm0 ) is the Sobolev space also denoted by W m;2 (W m;2
0 ).
The s-association (sss), s >= 0 of generalized complex numbers is defined as follows. For G 2 C , G sss 0 means that G has a representative G" such that

G" = o("s) as " ! 0. If G1; G2 are in G(?) then G1
Dsss G2 if hG1 ? G2; 'i sss 0 for

every ' 2 D. If s = 0, then instead of : : : -0-association we use the notation : : : - association. As in the previous case one can define association between a generalized function and a function in D0.

4 M. NEDELJKOV, S. PILIPOVI?C

In order to give a meaning to a Dirichlet problem in EM and thus in G we recall the definition of the space of generalized functions on a closed set (cf. [1]).

Let X be a non-void subset of Rn and fGff" ; ff 2 Nn0 g be a family of mappings Gff" : (0; 1)?X ! C . Denote by EW;M (X) the vector space of families fGff" ; ff 2 Nn0 g which satisfy the following three conditions:

(a) fGff" ; ff 2 Nn0 g has a locally moderate growth when " ! 0. This means that for every ff 2 Nn0 and x0 2 X there exist a neighbourhood V of x0, N 2 R, C > 0 and ? > 0 such that

jGff" (x)j <= C"?N ; for every x 2 V \ X and " 2 (0; ?):

(b) There exists ? > 0 such that the family

fX 3 x 7! Gff" (x); " < ?; ff 2 Nn0 g

satisfies requirements defining Whitney's C1-function on X, that is for every m 2 N, ff 2 Nn0 , jffj <= m and x0 2 X there exist a neighbourhood V
of x0 and c" > 0 such that

(4)

fifififififi

Gff" (x) ?

jfij<=m?jffj

(x ? x0)fiGff+fi
" (x0)

fi!

fifififififi
<= c"jx ? x0jm?jffj?1;

for every x; x0 2 V , " 2 (0; ?).
(c) Constants c" are locally bounded above by c"?N as " ! 0. More precisely, for every m 2 N, ff 2 Nn0 , jffj <= m and x0 2 X there exist a neighbourhood V of x0, N 2 R, C > 0 and ? > 0 such that (4) holds with c" = C"?N .

The ideal NW (X) of EW;M (X) is the set of those fGff" ; ff 2 Nn0 g which satisfy:
For every ff 2 Nn0 , and x0 2 X there exist a neighbourhood V of x0 such that for every q > 0 there exist C > 0 and ? > 0 such that

jGff" (x)j <= C"q; for every x 2 V \ X and " 2 (0; ?):

Put GW (X) = EW;M (X)=NW (X). Clearly, if G 2 G(?), where ? is an open set containing X, then fDffG"jX; ff 2 Nn0 g 2 EW;M defines the restriction GjX 2 GW .

Theorem 1. ( [1]) Let X be a closed subset of Rn. Then the restriction map G(Rn) ! GW (X) is surjective. In particular, for given fGff" ; ff 2 Nn0 g 2 EW;M (X) there exists F" 2 EM (Rn) such that fDffF"jX ? Gff" ; ff 2 Nn0 g 2 NW (X)

3. The generalized Dirichlet problem

A differential operator of the form P (x; D) = P
jffj<=m aff(x)Dff, where aff 2

G(Rn), is called a generalized differential operator. Thus a representative of P (x; D) is given by P"(x; D) = P
jffj<=m aff;"Dff, where aff;" is a representative of aff, jffj <= m.

Note that if ~aff;" is another representative of aff, jffj <= m, then P
jffj<=m aff;"DffG" ? P
jffj<=m ~aff;"DffG" 2 N (Rn) for every G" 2 EM (Rn). Also, if G" 2 N (Rn), then

P"(x; D)G" 2 N (Rn).

APPROXIMATED SOLUTIONS TO A DIRICHLET PROBLEM 5

Let ? be a bounded open set in Rn and H 2 G(Rn). In Theorem 2 we shall suppose that the operator P (x; D) can be written in the form (2) such that (3) holds.

Let F 2 GW (@?) be defined by a family fFff" ; ff 2 Nn0 g. We consider the boundary value problem

(5) P (x;D)G Hm(?)
ss H; in ?; Gj@? = F

that is, in terms of representatives,

lim
"!0

Z (P"(x; D)G"(x) ? H"(x)) (x)dx = 0; 2 Hm(?);

fDffG"j@? ? Fff" ; ff 2 Nn0 g 2 NW (@?):

Theorem 1 implies that there exists ~F 2 G(Rn) such that ~F j@? = F . Let V = P (x; D) ~F and U be a solution to the problem

P (x; D)U Hm(?)
ss H ? V in ?; U j@? = 0:

Then G = U + ~F is a solution to (5).
So, in the sequel we shall consider the following problem

(6) P (x;D)G Hm(?)
ss H in ?; Gj@? = 0;

in terms of representatives,

(6')

lim
"!0

Z (P"(x; D)G"(x) ? H"(x)) (x)dx = 0; 2 Hm(?)

fDffG"j@?; ff 2 Nn0 g 2 NW (@?);

where P" satisfies (3).

Theorem 2. With the above assumptions, there exists a solution G 2 G(Rn) to (6), i. e. there exists G" 2 EM (Rn) which satisfies (6').

Proof.
Denote by P ?" (x; D) = P
jffj<=m ~aff;"(x)Dff a representative of the adjoint operator to P"(x; D).

Let B" = maxfjr~aff;"(x)j; x 2 K1; jffj <= mg, where K1 = fx; d(x; ?) <= 1g and raff;" is the gradient of aff;". Since ~aff 2 G(Rn), there exist N1 > 0, C1 > 0 and ?1 > 0 such that jB"j <= C1"?N1 , " 2 (0; ?1).

Let " 2 (0; ?1) be given. Let ? be a cube fxj jxij <= b; i = 1; : : : ; ng, which contains ?. We divide it by hyperplanes

xk = bk=N"; i = 1; : : : ; n; k = 0; ?1; : : : ; ?(N" ? 1); N" 2 N;

into (2N")n cubes ?j , j = 1; : : : ; (2N")n. N" will be determinated later. Note, mes?j = (b=(2N"))n, where mes means measure. Denote by Xj the central point of the cube ?j , j = 1; : : : ; (2N")n.

6 M. NEDELJKOV, S. PILIPOVI?C

Let <=A denote the characteristic function of a set A. Put

~ j;" = <=?j ? OE"d and j;" = ~ j;"

(2N")n

j=1

~ j;"

?1
; j = 1; : : : ; (2N")n;

where d 2 R will be chosen later. Note that ~ j;" is equal to 1 on a set Kj ae ?j

which satisfies mes(supp( ~ j;" n Kj)) <= C0"d for some constant C0 independent on j.
We enumerate the family f?j ; j = 1; : : : ; (2N")ng in a way that the subfamily

f?j ; j = 1; : : : ; J"g covers ?. Then

J"
X

j=1
j;"(x) = 1; x 2 ? and mes(supp j;" n Kj ) = O("d):

Take N" 2 N such that B"(b=(2N"))n = "q, i. e.

N" = ((B"bn=2n)"?q)1=n;

where q will be chosen later. This implies that J" = O("?(q+N1)=n). Put ~
Aff;j;" = ~aff;"(Xj ). By Lagrange's mean value theorem

j~aff;"(x) ? ~aff;"(Xj)j <= B"

? b

2N"

?n
= "q ; x 2 ?j ; jffj <= m; j = 1; : : : ; J":

Now we determinate d by putting d = 2q.
Since H" 2 EM (?), there exist N2 2 N, C2 > 0 and ?2 > 0 such that

jH"(x) ~ j;"(x)j <= jH"(x)j <= C2"?N2 ; x 2 ?; " < ?2;

for every j <= J".
Let Gj;" 2 EM (Rn) be a representative of the solution to

Pj;"(D)Gj;" = (Am;j;"Dm1 +
m?1

k=0

jff0j<=k
Aff;j;"D0ff0
Dk1 )Gj;" = Hj;";

which exists by Theorem 1 in [10], where Pj;"(D) is the adjoint operator of

~Pj;"(D) = ~Am;j;"Dm1 +
X

jff0j<=k

~Aff;j;"D0ff0
Dk1 :

Put Gj;"(x) = Gj;"(x) ? <=j;"(x) ? O"(x), where <=j;"(x) = (<=Kj ? OE"d=4)(x) and (O")">0 is a family in EM (Rn) with the properties suppO" ae ? and
O"(x) = 1 for x 2 ??"d = fx 2 ?; d(x; {?) > "dg:

APPROXIMATED SOLUTIONS TO A DIRICHLET PROBLEM 7

We construct O" as follows. Let <=??"d=2 be the characteristic function of ??"d=2, " 2 (0; 1). Then O" = <=??"d=2 ? OE"d=2.
Note that

mesfx 2 Rn; x 2 supp ~ j;"; <=j;"(x) = 0g = O("d);

mes(? n ??"d) = O("d):

By repeating the part of the proof of Theorem 1 in [10], one can show that there exist M > 0 and N0 2 N0, which depend only on H and the growth order of the coefficients of P (x; D) on ? such that there exist C0 > 0 and ?0 > 0 such that for every j <= J"

jGj;"(x)j < C0"?M?N0 ; x 2 Rn; supp Gj;" \ supp Gk;" = ; for j 6= k " < ?0:

Let us note that C0 and ?0 are independent on j, j <= J". Also, since <=j;"O" 2

EM (Rn) and J" 2 C M , it follows P
j<=J" Gj;"<=j;"O" 2 EM (Rn).

Let G" = P
j<=J" Gj;" and let 2 Hm(?). Then

(x)P"(x; D)

j<=J"
Gj;"(x)dx ?

(x)H"(x)dx = I1 + I2;

where

I1 =

(x)P"(x; D)

j<=J"
Gj;"(x)dx ?

(x)

j<=J"
Pj;"(D)Gj;"(x)dx

and

I2 =

(x)

j<=J"
Pj;"(D)Gj;"(x)dx ?

j<=J"

Z (x)Hj;"(x)dx:

The construction of Gj;" implies

jI2j <= J"C C"d=4 = O("q):

Let us estimate I1. Since supp G" ae ??"d , integration by parts gives

I1 =

G"(P ?" (x; D) ? ~Pj;"(D)) dx:

There holds

jI1j <= supfj~aff;"(x) ? ~
Aff;j;"j; j <= J"; x 2 Kj ; jffj <= mg

(7)

? sup

j<=J"

jGj;"(x)j ? jDff (x)jdx; jffj <= m

9 =

;

jffj<=m

j<=J"
sup

x2?

j~aff;"(x) ? ~Aff;j;"j

supp ~ j;"nKj
jGj;"(x)j ? jDff (x)jdx;

<= "q X
j<=J"
C C"?M?N0 mes?

+ ((m + 1)n=4)"d"?q=nC C3"?M?N0?N1=n mes ?

<= (C3 + C)C "q?M?N0?N1=n mes ?(1 + (m + 1)n);

8 M. NEDELJKOV, S. PILIPOVI?C

for " < ?, where C = k kHm(?). This implies

jI1j <= C"q?M?N0?N1=n; for " < ?:
Now, choose q such that

(8) q > M + N0 +N1=n + 1; jffj <= m:

This proves Theorem 2.

Remark. Since q in (8) can be chosen arbitrarily large, the above proof gives the

solution to the equation P (x; D)G
Hm(?)
ss s H for arbitrary large s, i. e.

jhP"(x; D)G"; i ? hH"; ij = o("s); as " ! 0

for every 2 Hm(?).

Proposition 1. Suppose that all the assumptions of Theorem 2 hold. Then Theo-
rem 2 holds true when the Hm(?)-association is replaced by the Gm;t(?)-s-association
in (6) for every t 2 N or t = 1 and every s >= 0.

Proof. It is enough to note that (7), with ? 2 Gm;t(?) instead of 2 Hm(?),
converges to zero if q satisfies the condition

q > M + N0 + N1=n + 1 + t + r + s (instead of (8)),
where r is determined by jaff;"j = O("?r), jffj <= m, M , N and N1 are given in the
proof of Theorem 2.

4. A class of generalized elliptic differential operators

We shall consider a family of equations of the form

(9)

P"(x; D)G" =

jffj<=2m
aff;"(x)DffG"(x) = H"(x); x 2 ?;

fDffG"j@?; ff 2 Nn0 g 2 NW (@?);

where:
1. ? is a bounded open set in Rn with smooth boundary @?.
2. aff;" 2 EM (Rn), jffj <= 2m, H" 2 EM (Rn).
3. For every " 2 (0; "0), P"(x; D) is uniformly strongly elliptic (cf. [12], Ch. 36,
(36.3)) and moreover, there exist C0 > 0 and p > 0 such that

C0"pkukHm <= Re(P"(x; D)u; u)L2 ; u 2 C1
0 ; " < "0:
Then, for every " < "0,

P"(x; D) : Hm0 (?) ! H?m(?) is a surjective isomorphism
and the solution to (9) satisfies(10)
kG"kHm <= C0"?pkH"kH?m :

(cf. Theorem 36.2 in [12] and Lemma 20.1 in [12].)

APPROXIMATED SOLUTIONS TO A DIRICHLET PROBLEM 9

Theorem 3. Let P"(x; D), H" and ? satisfy the above conditions. Then
1. For every s >= 0 there exists a solution Gs;" 2 EM (Rn) to (9) in the H2m
0 (?)-sassociated sense, i. e.

(11) hP"(x; D)Gs;" ? H"; i = o("s); " ! 0; 2 H2m
0 (?)

and fDffGs;"j@?; ff 2 Nn0 g 2 NW (@?).
2. Let Ts;" be a solution to P"(x; D)Ts;" = R" in ? in the H2m
0 (?)-s-associated sense, s > p, and fDffTs;"j@?; ff 2 Nn0 g 2 NW (@?). Assume that

"pkR" ? H"kH?m ! 0 as " ! 0:

Then Ts;" and Gs;" are H?m(?)-associated.
3. Let P (x; D) have C1(Rn) coefficients and moreover satisfy

c0kukHm <= Re(P (x; D)u; u)L2 ; u 2 C1
0 (?);
where c0 does not depend on u. Let H 2 H?m(?) and U be the solution to

(12) P (x;D)U = H; U j@? = 0

(which exists by the Lax-Milgram Lemma), and let G0;" be the solution to

(13) P (x; D)G0;" = H" in ?; G"j@? = 0; where H" = H ? OE":

in H2m
0 -s-associated sense. Then U H?m(?)
ss G0.

4. Denote by ~G" to solution in the H2m-associated sense to (13) constructed in the proof of Theorem 2 and by G0;" the solution to (13) in H2m
0 (?)-associated sense.

Then ~G"

C10 (?)
ss G0;".

Proof.
1. Theorem 32.6 in [12] implies that for every fixed " < "0 there exists a solution g" 2 Hm0 (?) to the equation P"(x; D)g" = H"; in ?, that is

(14) hP"(x;D)g"; i = hH"; i; 2 Hm0 (?):

Let us adopt the notation for O" and ??" from the previous theorem. Then one can prove that Gs;" = g"O"s0=2 ? OE"s0=2 belongs to EM (Rn), i. e. it defines the generalized function Gs 2 G(Rn), where s0 will be chosen later.
Denote by P ?" the adjoint operator for P". Then

hP"(x; D)Gs;"; i = hGs;"; P ?" (x; D) i; 2 H2m
0 (?)

and the inequality which is to follow shows that for given s there exists s0 such that (11) holds.

jhg"; P ?" (x; D) i ? h(g"O"s0=2) ? OE"s0=2; P ?" (x; D) ij

<=mes(? n ??"s0 ) sup

x2?

jg"(x)j sup

x2?

jfij<=m

jaff;"(x)j ? k kH2m = o("s)

10 M. NEDELJKOV, S. PILIPOVI?C

Thus, (14) implies that (g"O"s0=2) ? OE"s0=2, is a solution to (11), i. e. Gs;" solves

P (x; D)Gs

H2m
0 (?)
ss s H in ?. Moreover, the construction of O"s0=2 implies that Gs satisfies the boundary condition Gsj@? = 0.
2. Let 2 H?m(?), " < "0, where "0 is chosen such that

Gs;"(x) = g"(x); Ts;"(x) := (T"O"s0=2 ? OE"s0=2)(x);

where Ts;"(x) = T (x) for x 2 supp and " < "0, g" satisfies (14) and T" solves (14) with R" instead of H". Then, by (10), there exists C0 > 0 such that

jh(Gs;"(x) ? Ts;"(x)); (x)ij
<=kg" ? T"kHmk kH?m <= C0"?pkR" ? F"kH?m ! 0; " ! 0:

3. There exists the unique solution U to (12) (in the weak sense) and

kUkHm <= c?1
0 kHkH?m :

Let Gs;" 2 EM (Rn) be the solution to

P (x; D)Gs;"
H2m
0 (?)
ss s H"( where H" = H ? OE")

constructed in the first part of Theorem 3. Then we have

jh(Gs;"(x) ? U(x)); (x)ij
<= jh(Gs;"(x) ? G0;"(x)); (x)ij + jh(G0;"(x) ? U(x)); (x)ij :

We already have proved that the first term in the right hand side converges to zero as " ! 0 and the second one is less or equal to c?1
0 kH" ? HkH?m . This proves the assertion.

4. Let 2 C1
0 (?) and ?(") be a family of open sets ?(") ae ? such that supp( ~G" ? G0;") ae ?("). There exists 1 2 C1(?) such that P ?(x; D) 1 = , where P ? is the adjoint operator for P . Thus for every 2 C1
0 (?) and " small enough

( ~G"(x) ? G0;"(x)) (x)dx =

?(")

( ~G"(x) ? G0;"(x)) (x)dx

?(")

( ~G"(x) ? G0;"(x))P ?(x; D) 1(x)dx

?(")

P (x; D)( ~G"(x) ? G0;"(x)) 1(x)dx ! 0 as " ! 0:

APPROXIMATED SOLUTIONS TO A DIRICHLET PROBLEM 11

5. Comparison of results for differential
operators with smooth coefficients

We shall examine the operators of the form

(15) Lu(x) =
n

i=1
Di

j=1
aij (x)Dju(x)

A +

n
X

i=1
bi(x)Diu + c(x)u;

where aij ; bi; c 2 L1(?), aij = aji, i; j = 1; : : : ; n, and ? is a bounded open set. Assume that there exist >= > 0 and ? > 0 such that

(16) >=j?j2 <=
n
X

i;j=1
aij (x)?i?j <= ?; c(x) < 0 for all x 2 ? and ? 2 Rn:

By [13], there holds:

For every h 2 H?1(?); there exists a unique weak solution
g 2 H10(?) to the Dirichlet problem(17)
Lg = h in ?; g = 0 on @?:

The boundary condition in (17) can be replaced by gj@? = zj@?, z 2 H1(?). In this case, we just have to put h ? Lz instead of h in the equation, because Lz 2 H?1(?). Note that g = z on @? means g ? z 2 H10(?).
Also we need the estimates

(18) kgkH1 <= C1kLgkH?1; kgkH1 <= C?1
1 kL?gkH?1 ;

where C1 is the coercive constant for L?.

Recall, ??" = fx 2 ?; d(x; {?) >= "g and <=??" is its characteristic function. Note, H" = h<=??" ? OE", where OE 2 C1
0 ,

R
OE(x)dx = 1, defines the generalized function H 2 G(Rn).
Consider the Dirichlet's problems:

(19) Lg = h in ?; gj@? = 0;

(20) LZ" = H" in ?; Z"j@? = 0; " 2 (0; 1) is fixed,

LG H2
ss H in ?; Gj@? = 0; that is

(21)

lim

"!0

(LG" ? H") dx = 0; for every 2 H2(?); Gj@? 2 NW (@?):

12 M. NEDELJKOV, S. PILIPOVI?C

Theorem 4. Let L satisfy (15), (16) and have coefficients which are smooth in a neighbourhood of ? and h 2 H?1. The generalized solution G to (21) constructed in Theorem 2 is C0-associated with the classical solution g to (19).
Proof. Note that H" converges to h in H?1(?) as " ! 0.
From (17) and (18) it follows

(22) kg ? Z"kH1 ! 0; as " ! 0;

where g satisfies (19) and Z" satisfies (20). Then (22) implies
Z (g(x) ? Z"(x))?(x)dx ! 0; as " ! 0; ? 2 C0(?) \ H10(?):

We have to prove that
Z

? (G"(x) ? Z"(x))?(x)dx ! 0; as " ! 0:

The boundary value problem

(23) L? = ? in ?; j@? = 0

has a solution in C2(?) \ H10(?). This follows from Theorem 6.8 in [5], since L? satisfies the assumptions of this theorem.

We have

G"(x)?(x)dx ==

G"(x)L? (x)dx =

LG"(x) (x)dx;(24)

Z"(x)?(x)dx =

? Z"(x)L? (x)dx = L(Z"; )

:= ?
n

j;k=1

ajk(x)DjZ"(x)Dk (x)dx +
n

j=1

bj(x)DjZ"(x) (x)dx

? c(x)Z"(x) (x); 2 H10(?):dx

Since L(Z"; ) = hH"; i, for 2 H10(?) (and thus for 2 C2(?) \ H10(?)), Theorem 3 implies

(G"(x) ? Z"(x))?(x)dx =

? (LG"(x) ? H"(x)) (x)dx ! 0; as " ! 0:

This proves the theorem.

APPROXIMATED SOLUTIONS TO A DIRICHLET PROBLEM 13

6. Comparison of results for strictly elliptic
differential operators with bounded coefficients

Up to the end we shall suppose that ? is a bounded open set with a smooth boundary.

Assume that L is an operator of the form (15) which satisfies condition (16). Then L? is the operator of the same form and satisfies the same conditions.

We regularize the coefficients of L considering them as distributions. We use a non-negative function OE 2 C1
0 , 0 2 supp OE,
R
OE(x)dx = 1. Put

OE"(x) = (? log ")nOE(?x= log "); " 2 (0; 1); x 2 Rn:

Then we define L?" to be the operator whose coefficients are given by

(25) ~ai;j ? OE"(x) = ai;j;"(x); ~bi ? OE"(x) = bi;"(x) and ~c ? OE"(x) = c"(x); x 2 R;

where ~y is a function equal to y in ??" and zero outside ?.
Note,

>=j?j2 <=
n
X

i;j=1
ai;j;"(x)?i?j <= ?; c"(x) < 0;

for all x 2 ?, ? 2 Rn and " < "0. Also all of ai;j;", bi;" and c" tend in L1-norm to ai;j , bi, c, respectively, as " ! 0.

The representatives of a generalized function which correspond to an L1-function and which are obtained by such regularizations are called slowly varying moderate functions, for short SVM functions. Note that if F" is an SVM function, then there exists s 2 R such that for every ff 2 Nn0 there exists Cff such that

F" <= Cffj log "js; x 2 ?:

Let ? 2 C1(?)\H10(?) be given. Then Theorem 6.14 and 6.17 in [5] imply that for every " 2 (0; 1) a solution ?" to

(26) L?"?" = ? in ?; ?"j@? = 0:

belongs to C1(?) \ C2(?).
By (6.42) in [5],

sup

jfij<=2

x2?

jDfi?"(x)j <= C sup
x2?
j?(x)j; " 2 (0; 1):

where the constant C depends only on >=, ? and the diameter of the set ? (let us remind that these constants are the same for regularized equation and the original one).
Applying the partial derivative Dk to both sides of (26) we obtain

L?"(Dk?") = Dk? ?
n

i=1
Di

j=1
(Dkai;j;")Dj?"

A+

n
X

i=1
(Dkbi;")Di?" + (Dkc)?":

14 M. NEDELJKOV, S. PILIPOVI?C

The right hand side consists of members which are products of moderate functions. Continuing this process, we obtain that all the derivatives of ?" have the same property. Thus ?" 2 EM(?). Moreover (27) proves that the solution ? to (26), given by the representative ?" belongs to G2;0(?) if ? 2 C1(?) \ H10(?).
By Proposition 1, the solution in H2-associated sense to the Dirichlet problem

given in Theorem 2 is also the solution to the same problem in the sense of G2;0
ss . This implies

(28)

? (G"(x) ? Z"(x)?(x)dx ! 0; as " ! 0:

This can be proved like (23), because there exists ?" 2 G2;1(?) such that

L??" = ?;

and then (28) follows as in (24).
Note that for every g 2 H1(?) and 2 H10(?)

jh(L" ? L)g; ij <=
n
X

i=1

m
X

j=1
kai;j;" ? ai;jkL1kDjgkL2kDi kL2

i=1
kbi;" ? bikL1kDigkL2 + kc" ? ckL1kgkL2

!
k kL2 ! 0;

as " ! 0.
This proves

Theorem 5. Let L be a differential operator of the form (15) with the coefficients in L1(?). Then the generalized solution G to (21) is C1(?) \ H10(?)-associated to the classical solution g 2 H10(?) to (19), where the coefficients of the regularized operator L" are given by (25).

References

1. H. A. Biagioni, J. F. Colombeau, Whitney's extension theorem for generalized functions, J. Math. Anal. Appl. 10, 2 (1986).
2. H. A. Biagioni, A Nonlinear Theory of Generalized Functions, Springer-Verlag, Berlin Heidelberg New York London Paris Tokyo Hong Kong, 1990.
3. J. F. Colombeau, Elementary Introduction in New Generalized Functions, North Holland, 1985.
4. Yu. V. Egorov, A contribution to the theory of generalized functions, Russian Math. Surveys 45:5 (1990), 1-49.
5. D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg New York Tokyo, 1983.
6. M. Nedeljkov, S. Pilipovi?c, Convolution in new generalized function spaces, Part I, II, Publ. Inst. Math. de Belgrad 52(66) (1992), 95-105 and 105-9.
7. M. Nedeljkov, S. Pilipovi?c, D. Scarpal?ezos, Division problem and partial differential equations with constant coefficients in Colombeau's space of new generalized functions, Monatsch. Math. 122, 2 (1996), 157-170.
8. M. Nedeljkov, S. Pilipovi?c, Hypoelliptic differential operators with generalized constant coefficients, to appear in Proc. Edinb. Math. Soc..

APPROXIMATED SOLUTIONS TO A DIRICHLET PROBLEM 15

9. M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Res. Not. Math. 259, Longman Sci. Techn., Essex, 1992.
10. S. Pilipovi?c, D. Scarpal?ezos, Differential operators with generalized constant coefficients in Colombeau algebra, Port. Math. 53, 3 (1996), 305-324.
11. E. E. Rosinger, Nonlinear Partial Differential Equations - An Algebraic View of Generalized Solutions, North Holland, Amsterdam, 1990.
12. F. Treves, Basic Linear Partial Differential Equations, Academic Press, New York San Francisco London, 1975.