Entropy solutions of Dirichlet problem for a class of nonlinear elliptic fourth order equations with L¹-data
We consider the question on solvability and uniqueness of entropy solutions of Dirichlet problem for a class of nonlinear elliptic fourth order equations with L¹-right-hand sides. We restrict ourselves with equations of the fourth order, but it is not so signicant.
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Інститут прикладної математики і механіки НАН України
1999
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| Назва видання: | Нелинейные граничные задачи |
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| Цитувати: | Entropy solutions of Dirichlet problem for a class of nonlinear elliptic fourth order equations with L¹-data / A.A. Kovalevsky // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 46-54. — Бібліогр.: 12 назв. — англ. |
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irk-123456789-1692712020-06-10T01:26:22Z Entropy solutions of Dirichlet problem for a class of nonlinear elliptic fourth order equations with L¹-data Kovalevsky, A.A. We consider the question on solvability and uniqueness of entropy solutions of Dirichlet problem for a class of nonlinear elliptic fourth order equations with L¹-right-hand sides. We restrict ourselves with equations of the fourth order, but it is not so signicant. 1999 Article Entropy solutions of Dirichlet problem for a class of nonlinear elliptic fourth order equations with L¹-data / A.A. Kovalevsky // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 46-54. — Бібліогр.: 12 назв. — англ. 0236-0497 http://dspace.nbuv.gov.ua/handle/123456789/169271 en Нелинейные граничные задачи Інститут прикладної математики і механіки НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
We consider the question on solvability and uniqueness of entropy solutions of Dirichlet problem for a class of nonlinear elliptic fourth order equations with L¹-right-hand sides. We restrict ourselves with equations of the fourth order, but it is not so signicant. |
| format |
Article |
| author |
Kovalevsky, A.A. |
| spellingShingle |
Kovalevsky, A.A. Entropy solutions of Dirichlet problem for a class of nonlinear elliptic fourth order equations with L¹-data Нелинейные граничные задачи |
| author_facet |
Kovalevsky, A.A. |
| author_sort |
Kovalevsky, A.A. |
| title |
Entropy solutions of Dirichlet problem for a class of nonlinear elliptic fourth order equations with L¹-data |
| title_short |
Entropy solutions of Dirichlet problem for a class of nonlinear elliptic fourth order equations with L¹-data |
| title_full |
Entropy solutions of Dirichlet problem for a class of nonlinear elliptic fourth order equations with L¹-data |
| title_fullStr |
Entropy solutions of Dirichlet problem for a class of nonlinear elliptic fourth order equations with L¹-data |
| title_full_unstemmed |
Entropy solutions of Dirichlet problem for a class of nonlinear elliptic fourth order equations with L¹-data |
| title_sort |
entropy solutions of dirichlet problem for a class of nonlinear elliptic fourth order equations with l¹-data |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/169271 |
| citation_txt |
Entropy solutions of Dirichlet problem for a class of nonlinear elliptic fourth order equations with L¹-data / A.A. Kovalevsky // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 46-54. — Бібліогр.: 12 назв. — англ. |
| series |
Нелинейные граничные задачи |
| work_keys_str_mv |
AT kovalevskyaa entropysolutionsofdirichletproblemforaclassofnonlinearellipticfourthorderequationswithl1data |
| first_indexed |
2025-07-15T04:01:46Z |
| last_indexed |
2025-07-15T04:01:46Z |
| _version_ |
1837684078220935168 |
| fulltext |
ENTROPY SOLUTIONS OF DIRICHLET PROBLEM
FOR A CLASS OF NONLINEAR ELLIPTIC
FOURTH ORDER EQUATIONS WITH L1-DATA
c© Alexander Kovalevsky
1. Introduction
Last years some interesting works were published on solvability of nonlinear elliptic
second order equations with data which are not kept within usual scheme of the theory
of monotone operators [1]. First of all it concerns equations with L1-right-hand sides
[2]–[5], equations with measures [6]–[8] and problems leading in weakened statement to
so-called renormalized solutions [9]–[11]. Among the mentioned works the paper [4] is
especially distinguished. In that paper a theory of existence and uniqueness of entropy
solutions of Dirichlet problem for nonlinear elliptic second order equations with L1-data
was constructed. Note that the proof of solvability of the problem considered in [4] was
based on special estimates of solutions of Dirichlet problem for approximating equations
with bounded right-hand sides. The main tool to obtain those estimates consisted in
the use of test functions which are superpositions of solutions of the approximating
problems and of standard truncated functions Tk(s) = max{min{s, k},−k}, k > 0,
s ∈ R. Moreover superpositions of entropy solutions and of the standard truncated
functions played a significant part in the proof of uniqueness of an entropy solution. On
the whole the use of the functions Tk was in the base of the definition of new functional
classes which are wider than Sobolev spaces. Namely in these classes single-valued
entropy solvability of the problem under consideration was established.
All above-stated concerns second order equations and as far as the author knows
any results were not before now obtained on solvability of higher order equations with
L1-data, although the development of corresponding theory has a great interests. This
circumstance was an inducement for the author to carry out some research. The main
its results are given in this paper.
We consider the question on solvability and uniqueness of entropy solutions of Dirich-
let problem for a class of nonlinear elliptic fourth order equations with L1-right-hand
sides. We restrict ourselves with equations of the fourth order, but it is not so signifi-
cant. As a matter of fact, the passage from second order equations to higher order ones
is more principal for us. We follow the approach of [4], however its realization meets a
series of difficulties. First of all it is connected with the fact that in the case of higher
order equations to obtain needful estimates for solutions of the approximating problems,
the functions Tk can not be used in the same way as it holds for second order equations.
In particular, in our case arises necessity to construct some functions substituting the
truncations Tk and to choose an energy space for the approximating problems. In so
doing the following requirements should be satisfied:
1) superpositions of new functions with solutions of the approximating problems are
twice differentiable and belong to chosen energy space;
2) the use of these superpositions as test functions in corresponding integral iden-
tities allows to estimate in a suitable way the terms connected with the second order
derivatives of the superpositions.
Some appropriate substitutions of the standard truncations and an energy space
for the approximating problems are established in the paper. By their means needful
estimates for solutions of the approximating problems are obtained and theorems on
existence and uniqueness of the solutions of the problem under consideration are proved.
Now we pass to the statement of initial assumptions of the paper.
Let n ∈ N, n > 2, and let Ω be a bounded open set of Rn.
We denote by Λ the set of all n-dimensional multiindices α such that |α| = 1 or
|α| = 2. We shall also use the following notations: Rn,2 is the space of all mappings
ξ : Λ −→ R; if u ∈ W 2,1(Ω), then ∇2u : Ω −→ Rn,2 and for every x ∈ Ω and α ∈ Λ,
(∇2u(x))α = Dαu(x).
Let p, q be real numbers such that
1 < p <
n
2
, (1.1)
2p < q < n . (1.2)
Let c1, c2 > 0, let g1, g2 be non-negative functions in Ω, g1, g2 ∈ L1(Ω), and let for
every α ∈ Λ, Aα : Ω×Rn,2 −→ R be a Carathéodory function. We shall assume that for
almost every x ∈ Ω and every ξ ∈ Rn,2,
∑
|α|=1
|Aα(x, ξ)|q/(q−1) +
∑
|α|=2
|Aα(x, ξ)|p/(p−1)
6 c1
{ ∑
|α|=1
|ξα|q +
∑
|α|=2
|ξα|p
}
+ g1(x) , (1.3)
∑
α∈Λ
Aα(x, ξ)ξα > c2
{ ∑
|α|=1
|ξα|q +
∑
|α|=2
|ξα|p
}
− g2(x) . (1.4)
Moreover we shall assume that for almost every x ∈ Ω and every ξ, ξ′ ∈ Rn,2, ξ 6= ξ′,
∑
α∈Λ
[
Aα(x, ξ)−Aα(x, ξ′)
]
(ξα − ξ′α) > 0 . (1.5)
Let F : Ω × R −→ R be a Carathéodory function. We shall study the following
problem:
∑
α∈Λ
(−1)|α|DαAα(x,∇2u) = F (x, u) in Ω , (1.6)
Dαu = 0 , |α| = 0, 1 , on ∂Ω . (1.7)
Precise definitions of solutions of this problem will be given in the main content of the
paper. Some additional conditions on the function F (such as in [4]) will be formulated
in theorems of existence and uniqueness of the solutions.
Let us give some remarks in connection with the conditions on the numbers p and
q. The condition (1.1) is one of possible for p. Introduction of the number q with
the restriction (1.2) is explained with the fact that under the condition (1.1) the space
◦
W 1,q
2,p(Ω) is a suitable energy space for problems approximating the problem (1.6), (1.7).
By this circumstance the conditions (1.3), (1.4) is also dictated.
In the case where p > n
2 solvability of the problem (1.6), (1.7) under corresponding
conditions on the coefficients Aα and under the same conditions on F as in given paper
can be established in the space
◦
W 2,p(Ω) on the base of known results of the theory of
monotone operators [1]. It follows from boundedness of the embedding of
◦
W 2,p(Ω) into
C(Ω), if n < 2p. The case where p = n
2 requires a separate consideration (a paper with
corresponding results is in preparation now).
Note that a class of nonlinear elliptic higher order equations under conditions on
coefficients of type (1.3), (1.4), but with right-hand sides which are kept within usual
framework of the theory of monotone operators, was introduced in the paper [12] devoted
to the study of regularity of solutions.
The author thanks the Editor for the opportunity to publish in the Proceedings this
paper. The main its results were obtained in the summer of 1998.
2. Functional class
◦
H1,q
2,p(Ω)
We denote by W 1,q
2,p (Ω) the set of all functions of W 1,q(Ω) having weak derivatives of
the second order from Lp(Ω). W 1,q
2,p (Ω) is a Banach space with the norm
‖u‖ = ‖u‖W 1,q(Ω) +
( ∑
|α|=2
∫
Ω
|Dαu|pdx
)1/p
.
We denote by
◦
W 1,q
2,p(Ω) the closure in W 1,q
2,p (Ω) of the set C∞0 (Ω).
We shall also use the following notations: if t ∈ [1, +∞], then | · |t is the norm in
Lt(Ω); if t ∈ [1, n), then t∗ = nt
n−t .
It is well known that
◦
W 1,q(Ω) ⊂ Lq∗(Ω) and there exists a positive constant c′
depending only on n, q and such that for every u ∈
◦
W 1,q(Ω)
|u|q∗ 6 c′
∑
|α|=1
|Dαu|q . (2.1)
Let for everyk ∈ N, ψk be the function on R such that
ψk(s) = s− sk+2 +
k + 1
k + 3
sk+3 , s ∈ R .
We define for every k ∈ N the function hk : R −→ R by
hk(s) =
s , if |s| 6 k ,
[
ψk
( |s| − k
k
)
+ 1
]
ksign s , if k < |s| < 2k ,
2k
k + 2
k + 3
sign s , if |s| > 2k .
For every k ∈ N we have hk ∈ C2(R), |hk| 6 2k, 0 6 h′k 6 1, |h′′k | 6 3 on R. Moreover,
if k, j ∈ N and j > 2k, then for every s ∈ R,
hk(hj(s)) = hk(s) . (2.2)
We denote by
◦
H1,q
2,p(Ω) the set of all functions u : Ω −→ R satisfying the condition:
∀k ∈ N, hk(u) ∈
◦
W 1,q
2,p(Ω).
The following properties hold:
◦
W 1,q
2,p(Ω) ⊂
◦
H1,q
2,p(Ω) ,
◦
H1,q
2,p(Ω) \ L1
loc(Ω) 6= ∅ .
Definition 2.1. If u ∈
◦
H1,q
2,p(Ω) and α ∈ Λ, then δαu is the function in Ω such that
δαu = Dαh1(u) in {|u| 6 1}
and ∀k ∈ N,
δαu = Dαh2k(u) in {2k−1 < |u| 6 2k}.
Lemma 2.2. Let u ∈
◦
H1,q
2,p(Ω). Then for every α ∈ Λ and k ∈ N,
δαu = Dαhk(u) a.e. in {|u| 6 k} .
The proof of the lemma is based on the use of the relation (2.2).
Due to Lemma 2.2 for every u ∈
◦
W 1,q
2,p(Ω) and α ∈ Λ, δαu = Dαu a.e. in Ω.
We set
r =
n(q − 1)
n− 1
,
c′′ = 2q∗/q(c′n)q∗ + 2nq .
Lemma 2.3. Let u ∈
◦
H1,q
2,p(Ω), M > 1 and let for every k ∈ N,
∫
{|u|<k}
{ ∑
|α|=1
|δαu|q +
∑
|α|=2
|δαu|p
}
dx 6 Mk .
Then for every k ∈ N,
meas{|u| > k} 6 c′′Mn/(n−q)k−r∗ ,
meas
{ ∑
|α|=1
|δαu| > k
}
6 c′′Mn/(n−1)k−r ,
meas
{ ∑
|α|=2
|δαu| > k
}
6 c′′Mn/(n−1)k−pr/q .
This result is proved by means of (2.1).
3. Entropy solutions
Introduce the notation: if u ∈
◦
H1,q
2,p(Ω), then δ2u : Ω −→ Rn,2 and for every x ∈ Ω
and α ∈ Λ, (δ2u(x))α = δαu(x).
Definition 3.1. An entropy solution of the problem (1.6), (1.7) is a function u ∈
◦
H1,q
2,p(Ω) satisfying the following conditions:
1) F (x, u) ∈ L1(Ω);
2) there exist c > 0, b ∈ (1, r) and γ > 0 such that for every ϕ ∈ C∞0 (Ω) and k ∈ N,
∫
{|u−ϕ|<2k}
{∑
α∈Λ
Aα(x, δ2u)(δαu− δαϕ)
}
h′k(u− ϕ)dx
6
∫
Ω
F (x, u)hk(u− ϕ)dx + c
[
1 + ‖ϕ‖W 1,b(Ω)
]b
k−γ . (3.1)
In the further considerations we shall denote by ci, i = 3, 4, . . . , positive constants
which depend only on n, p, q, c1, c2, |g1|1, |g2|1 and on meas Ω.
Lemma 3.2. Let u be an entropy solution of the problem (1.6), (1.7). Then there
exists c > 0 such that for every k ∈ N,
∫
{|u|6k}
{ ∑
|α|=1
|δαu|q +
∑
|α|=2
|δαu|p
}
dx 6 c3[ |F (x, u)|1 + c + 1 ]k .
This result is proved with the use of (1.4) and of the properties of the functions hk.
From Lemmas 3.2, 2.3 and (1.3) we deduce the following result.
Lemma 3.3. Let u be an entropy solution of the problem (1.6), (1.7). Then
1) for every λ ∈ (0, r∗), u ∈ Lλ(Ω);
2) for every α, |α| = 1, and λ ∈ (0, r), δαu ∈ Lλ(Ω);
3) for every α, |α| = 2, and λ ∈ (
0, pr
q
)
, δαu ∈ Lλ(Ω);
4) for every α, |α| = 1, and λ ∈ (
0, r
q−1
)
, Aα(x, δ2u) ∈ Lλ(Ω);
5) for every α, |α| = 2, and λ ∈ (
0, pr
q(p−1)
)
, Aα(x, δ2u) ∈ Lλ(Ω);
6) for every α ∈ Λ, Aα(x, δ2u) ∈ L1(Ω).
Proposition 3.4. Let u be an entropy solution of the problem (1.6), (1.7). Then
for every function ϕ ∈ C∞0 (Ω),
lim sup
k→∞
[∫
{|u−ϕ|<k}
{∑
α∈Λ
Aα(x, δ2u)(δαu− δαϕ)
}
dx−
∫
Ω
F (x, u)hk(u− ϕ)dx
]
6 0 .
This result is proved by means of (1.4), Lemmas 3.2, 2.3, and of the assertion 6) of
Lemma 3.3.
Lemma 3.5. Let u be an entropy solution of the problem (1.6), (1.7). Then there
exist c > 0, b ∈ (1, r) and γ > 0 such that for every ϕ ∈
◦
W 1,q
2,p(Ω) ∩ L∞(Ω) and k ∈ N
the inequality (3.1) holds.
The proof of this fact is based on the use of an approximation of a function ϕ ∈
◦
W 1,q
2,p(Ω) ∩ L∞(Ω) by an uniformly bounded sequence of functions of the class C∞0 (Ω).
Lemma 3.6. Let u be an entropy solution of the problem (1.6), (1.7). Then there
exist c > 0, b ∈ (1, r) and γ > 0 such that for every k ∈ N, k > 2, and m ∈ N the
following inequality holds:
c4
∫
{k6|u|6k+m}
{ ∑
|α|=1
|δαu|q +
∑
|α|=2
|δαu|p
}
dx
6 m
∫
{|u|>k/4}
{|F (x, u)|+ g1 + g2
}
dx +
[|F (x, u)|1 + c + 1
]n/(n−q)
k−1
+c
[
1 + |u|b +
∑
|α|=1
|δαu|b
]b
m−γ .
The proof of this lemma takes into account the special behaviour of the functions hk
in (−∞, k) ∪ (k, +∞) and the equality
p− 1
p
+
2
q
+
q − 2p
qp
= 1
which allows to use in a suitable way Hölder and Young inequalities in estimates.
4. H-solutions
Definition 4.1. An H-solution of the problem (1.6), (1.7) is a function u ∈
◦
H1,q
2,p(Ω)
satisfying the following conditions:
1) F (x, u) ∈ L1(Ω);
2) for every α ∈ Λ, Aα(x, δ2u) ∈ L1(Ω);
3) for every function ϕ ∈ C∞0 (Ω),
∫
Ω
{∑
α∈Λ
Aα(x, δ2u)δαϕ
}
dx =
∫
Ω
F (x, u)ϕdx .
Theorem 4.2. Let u be an entropy solution of the problem (1.6), (1.7). Then u is
an H-solution of the same problem.
As far as the proof of this theorem is concerned we emphasize the importance of the
same things which are mentioned after Lemma 3.6.
5. Uniqueness and existence
Theorem 5.1. Let for almost every x ∈ Ω the function F (x, ·) be nonincreasing on
R. Let u, v be entropy solutions of the problem (1.6), (1.7). Then u = v a.e. in Ω.
The proof of this result is based on the use of the inequalities (1.5), (2.1), of the
properties of the functions hk and of Lemmas 3.5, 3.6.
Theorem 5.2. Let the following conditions be satisfied:
1) for almost every x ∈ Ω the function F (x, ·) is nonincreasing on R;
2) for every s ∈ R the function F (·, s) belongs to L1(Ω).
Then there exists an H-solution of the problem (1.6), (1.7).
We set
p1 =
3n− 2
n + p− 1
p .
In virtue of (1.1) we have p1 ∈ (2p, n).
Theorem 5.3. Let the conditions 1), 2) of Theorem 5.2 be satisfied and let q > p1.
Then there exists an entropy solution of the problem (1.6), (1.7).
A common part of the proof of these theorems is connected with obtaining of some
estimates for solutions of problems approximating the problem (1.6), (1.7). We state
below only first two of them. We shall assume that the conditions 1), 2) of Theorem
5.2 are satisfied.
We set f = F (·, 0) and define for every l ∈ N the function Fl : Ω× R −→ R by
Fl(x, s) = hl(f(x)− F (x, s)) , (x, s) ∈ Ω× R .
By the condition 1) of Theorem 5.2 we have: if l ∈ N, then
for almost every x ∈ Ω the function Fl(x, ·) is nondecreasing on R . (5.1)
In virtue of the condition 2) of the same theorem f ∈ L1(Ω). Therefore there exists
{fl} ⊂ C∞0 (Ω) such that |fl − f |1 → 0 and ∀l ∈ N, |fl|1 6 |f |1 + 1.
From (1.3)–(1.5), (5.1) and from results of the theory of monotone operators we
obtain: if l ∈ N, then there exists a function ul ∈
◦
W 1,q
2,p(Ω) such that ∀v ∈
◦
W 1,q
2,p(Ω),
∫
Ω
{∑
α∈Λ
Aα(x,∇2ul)Dαv + Fl(x, ul)v
}
dx =
∫
Ω
flvdx . (5.2)
Lemma 5.4. For every k, l ∈ N the following inequalities hold:
∫
{|ul|6k}
{ ∑
|α|=1
|Dαul|q +
∑
|α|=2
|Dαul|p
}
dx 6 c5[|f |1 + 1]k , (5.3)
c6
∫
{|ul|>2k}
|Fl(x, ul)|dx 6
∫
{|ul|>k}
|f |dx + [|f |1 + 1]k−1 + |fl − f |1 . (5.4)
To prove (5.3) we first define a sequence of functions χk ∈ C2(R) which have the
properties: χk(s) = s if |s| 6 k; |χk| 6 3k, 0 < χ′k 6 1 and |χ′′k | 6 (8/k)χ′k on R. Then
we put the function χk(ul) in (5.2) instead of v, and using (1.3), (1.4), (3.2), Young
inequality, (5.1) and the properties of the functions χk, we obtain the required estimate.
To prove (5.4) we utilize some other test function in (5.2).
Remark 5.5. Some modifications in the definition of entropy solution allow to prove
the result of existence of (modified) entropy solution without the condition q > p1.
6. W -solutions
Definition 6.1. A W -solution of the problem (1.6), (1.7) is a function u ∈
◦
W 2,1(Ω)
satisfying the following conditions:
1) F (x, u) ∈ L1(Ω);
2) for every α ∈ Λ, Aα(x,∇2u) ∈ L1(Ω);
3) for every ϕ ∈ C∞0 (Ω),
∫
Ω
{∑
α∈Λ
Aα(x,∇2u)Dαϕ
}
dx =
∫
Ω
F (x, u)ϕdx .
We set
p2 =
np
n(p− 1) + 1
.
In virtue of (1.1) we have p2 ∈ (1, n).
Theorem 6.2. Let the conditions 1), 2) of Theorem 5.2 be satisfied and let q > p2.
Then there exists a W -solution of the problem (1.6), (1.7).
This result is established simultaneously with the proof of Theorem 5.2. We only note
that under the condition q > p2 a limit function of the solutions ul of the approximating
problems belongs to
◦
W 2,1(Ω).
Remark 6.3. We have p2 > 2p if and only if p < 3
2 − 1
n . Therefore if p > 3
2 − 1
n ,
due to (1.2) we obtain q > p2.
From Theorem 6.2 and Remark 6.3 we get the following result.
Corollary 6.4. Let the conditions 1), 2) of Theorem 5.2 be satisfied and let p > 3
2− 1
n .
Then there exists a W -solution of the problem (1.6), (1.7).
References
1. Lions J.L., Quelques méthodes de résolution des problèmes aux limites non linéaires, Paris: Dunod,
1969.
2. Bénilan Ph., On the p-Laplacian in L1, Conference at the Analysis Seminar. Dept. of Math., Univ.
Wisconsin at Madison (1987).
3. Boccardo L., Gallouët Th., Vazquez J.L., Nonlinear elliptic equations in RN without growth re-
strictions on the data, J. Diff. Eqns 105 (1993), 334–363.
4. Bénilan Ph., Boccardo L., Gallouët Th., Gariepy R., Pierre M., Vazquez J.L., An L1-theory of
existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa
Cl. Sci. 22 (1995), 241–273.
5. Rakotoson J.M., Resolution of the critical cases for problems with L1-data, Asymptotic Analysis
6 (1993), 229–246.
6. Boccardo L., Gallouët Th., Non-linear elliptic and parabolic equations involving measure data, J.
Funct. Anal. 87 (1989), 149–169.
7. Boccardo L., Gallouët Th., Nonlinear elliptic equations with right-hand side measures, Comm.
Partial Diff. Eqns 17 (1992), 641–655..
8. Rakotoson J.M., Generalized solutions in a new type of sets for problems with measures as data,
Diff. Integral Eqns 6 (1993), 27–36.
9. Boccardo L., Diaz J.I., Giachetti D., Murat F., Existence of a solution for a weaker form of a non
linear elliptic equation, Recent advances in nonlinear elliptic and parabolic problems (Proceedings,
Nancy, 1988). Pitman Res. Notes Math. Ser. 208 (1989), 229–246.
10. Boccardo L., Diaz J.I., Giachetti D., Murat F., Existence and regularity of renormalized solutions
for some elliptic problems involving derivatives of nonlinear terms, J. Diff. Eqns 106 (1993), 215–
237.
11. Murat F., Soluciones renormalizadas de EDP elipticas no lineales, Publ. Laboratoire d’Analyse
Numérique, R 93023. Univ. Paris 6 (1993).
12. Skrypnik I.V., On quasilinear elliptic equations of higher order with continuous generalized solu-
tions, Diff. Eqns 14 (1978), 1104–1118.
Institute of Applied Mathematics and Mechanics
National Academy of Sciences of Ukraine
Rosa Luxemburg St. 74
340114 Donetsk
Ukraine
E-mail: alexkvl@iamm.ac.donetsk.ua
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