Caccioppoli-type estimates for a class of nonlinear differential operators
We establish Caccioppoli-type estimates for a class of nonlinear differential equations with the aid of a differential identity that generalizes the well-known multidimensional Picone’s formula. In special cases, these estimates give the Finsler $p$-Laplacian, the $p$-Laplacian and the pseudo-$p$-La...
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| author | Tiryaki, A. Тирякі, А. |
| author_facet | Tiryaki, A. Тирякі, А. |
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| description | We establish Caccioppoli-type estimates for a class of nonlinear differential equations with the aid of a differential identity
that generalizes the well-known multidimensional Picone’s formula. In special cases, these estimates give the Finsler $p$-Laplacian, the $p$-Laplacian and the pseudo-$p$-Laplacian. |
| first_indexed | 2026-03-24T02:09:49Z |
| format | Article |
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UDC 517.9
A. Tiryaki (Izmir Univ., Turkey)
CACCIOPPOLI-TYPE ESTIMATES FOR A CLASS
OF NONLINEAR DIFFERENTIAL OPERATORS
ОЦIНКИ ТИПУ КАЧЧIОППОЛI ДЛЯ ОДНОГО КЛАСУ
НЕЛIНIЙНИХ ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
We establish Caccioppoli-type estimates for a class of nonlinear differential equations with the aid of a differential identity
that generalizes the well-known multidimensional Picone’s formula. In special cases, these estimates give the Finsler
p-Laplacian, the p-Laplacian and the pseudo-p-Laplacian.
Встановлено оцiнки типу Каччiопполi для одного класу нелiнiйних диференцiальних рiвнянь за допомогою дифе-
ренцiальної тотожностi, що узагальнює вiдому багатовимiрну формулу Пiконе. В частинних випадках цi оцiнки
дають p-лапласiан Фiнслера, p-лапласiан та p-псевдолапласiан.
1. Introduction. The Caccioppoli inequality is an important tool for the regularity estimate of elliptic
partial differential equations (see [1, 2] and references therein). In the basic Caccioppoli inequality,
L2-norm of the gradient of a harmonic (or subharmonic) function v is estimated in terms of the
L2-norm of v itself. Such an estimate may not hold for arbitrary functions v, but we can typically
use them for solutions of elliptic equations or systems in which integrals of higher derivatives can
be bounded in terms of integrals of lower derivatives, usually over a slightly larger set. One of the
alternative ways to establish the basic Caccioppoli inequality is to derive it from Picone’s identity
[3, 4]:
\bigm\| \bigm\| \nabla u
\bigm\| \bigm\| 2
2
-
\biggl\langle
\nabla
\biggl(
u2
v
\biggr)
,\nabla v
\biggr\rangle
=
\bigm\| \bigm\| \bigm\| \nabla u - u
v
\nabla v
\bigm\| \bigm\| \bigm\| 2
2
, (1.1)
where u and v are differentiable functions in a domain \Omega \subset Rn, v(x) \not = 0 in \Omega , \| \cdot \| 2, \nabla and \langle , \rangle
denote the Euclidean norm, the usual gradient and the inner product in Rn, respectively.
Let v > 0 be a weak (continuous) solution of
\bigtriangleup v = 0 in \Omega
and \eta \in C\infty
0 (\Omega ) be a nonnegative test function.
If we integrate (1.1) by substituting u = \eta v, and use Young’s inequality and Cauchy – Schwartz
inequality, respectively, then we can obtain the desired estimate\int
\Omega
\bigm\| \bigm\| \eta \nabla v
\bigm\| \bigm\| 2
2
dx \leq 4
\int
\Omega
\bigm\| \bigm\| v\nabla \eta
\bigm\| \bigm\| 2
2
dx.
More generally, if p > 1 is fixed and v > 0 is a weak continuous solution (or subsolution) of the
p-harmonic equation
\mathrm{d}\mathrm{i}\mathrm{v}
\bigl(
\| \nabla v\| p - 2
2 \nabla v
\bigr)
= 0 in \Omega ,
c\bigcirc A. TIRYAKI, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 10 1429
1430 A. TIRYAKI
then the Lp-version of the Caccioppoli estimate\int
\Omega
\bigm\| \bigm\| \eta \nabla v
\bigm\| \bigm\| p
2
dx \leq pp
\int
\Omega
\bigm\| \bigm\| v\nabla \eta
\bigm\| \bigm\| p
2
dx
([5], see also inequality (5.27) in [6] and Corollary A6 in [7]) can be easily obtained from the
p-Laplacian generalization of Picone’s identity
\bigm\| \bigm\| \nabla u
\bigm\| \bigm\| p
2
-
\biggl\langle
\nabla
\biggl(
| u| p
vp - 1
\biggr)
,
\bigm\| \bigm\| \nabla v
\bigm\| \bigm\| p - 2
2
\nabla v
\biggr\rangle
= \Phi p(u, v),
where
\Phi p(u, v) :=
\bigm\| \bigm\| \nabla u
\bigm\| \bigm\| p
2
+ (p - 1)
\| u\| p
vp
\| \nabla v\| p2 - p
| u| p - 1
vp - 1
\Bigl\langle
\nabla u, \| \nabla v\| p - 2
2 \nabla v
\Bigr\rangle
\geq 0
by setting u = \eta v and making use of Young’s and Hölder inequality [8, 9].
Recently, Jaroš [10] extended the Caccioppoli inequality to a class of differential operators of the
form
\Delta H,pv := \mathrm{d}\mathrm{i}\mathrm{v}
\bigl(
H(\nabla v)p - 1\nabla \xi H(\nabla v)
\bigr)
, (1.2)
where p > 1, H : Rn \rightarrow [0,\infty ), n \geq 2 is a convex function (see, for example, [11]) of the class
C1(Rn\setminus \{ 0\} ) which is positively homogeneous of degree 1 and \nabla and \nabla \xi stand for usual gradient
operators with respect to the variables x and \xi , respectively.
We refer to the operator \Delta H,p as the Finsler p-Laplacian (or the anisotropic p-Laplacian). A
prototype of H satisfying the above conditions is the lr -norm
H(\xi ) = \| \xi \| r =
\Biggl(
n\sum
i=1
| \xi i| r
\Biggr) 1/r
, r > 1,
for which the operator defined by (1.2) has the form
\Delta r,pv := \mathrm{d}\mathrm{i}\mathrm{v}
\bigl(
\| \nabla v\| p - r
r \nabla rv
\bigr)
, (1.3)
where
\nabla rv :=
\Biggl( \bigm| \bigm| \bigm| \bigm| \partial v\partial x1
\bigm| \bigm| \bigm| \bigm| r - 2 \partial v
\partial x1
, . . . ,
\bigm| \bigm| \bigm| \bigm| \partial v\partial xn
\bigm| \bigm| \bigm| \bigm| r - 2 \partial v
\partial xn
\Biggr)
. (1.4)
The class of operators of the form (1.3) includes the usual p-Laplacian and the so-called pseudo-
p-Laplace operator as the special cases corresponding to r = 2 and p \in (1,\infty ) and r = p > 1,
respectively. Clearly, if p = r = 2, then (1.3) reduces to the standard Laplacian \Delta .
Anisotropic elliptic problems involving this kind of operator have recently been studied in several
papers including [6, 12 – 22].
Recently, Jaroš obtained the following interesting result related to Caccioppoli inequality:
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 10
CACCIOPPOLI-TYPE ESTIMATES FOR A CLASS OF NONLINEAR DIFFERENTIAL OPERATORS 1431
Theorem 1.1 (Caccioppoli-type inequality) [10]. Let v > 0 be a weak subsolution of
- \Delta H,pv = g(x)| v| p - 2v in \Omega .
Then, for any fixed q > p - 1 and w = vq/p, the inequality\int
\Omega
H(\eta \nabla w)pdx \leq
\biggl(
q
q - p+ 1
\biggr) p \int
\Omega
H(w\nabla \eta )pdx+
qpp1 - p
q - p+ 1
\int
\Omega
g(x)wp\eta pdx (1.5)
holds for all 0 \leq \eta \in C\infty
0 (\Omega ), where 0 \leq g \in L\infty
Loc(\Omega ).
The purpose of this paper is to extend the Caccioppoli inequality to a class of A-harmonic
operators of the form
\Delta Av := \mathrm{d}\mathrm{i}\mathrm{v}
\bigl(
\nabla \xi A(x,\nabla v)
\bigr)
, (1.6)
where A : \Omega \times Rn \rightarrow R+ is measurable function such that \xi \rightarrow A(x, \xi ) is continuously differentiable
function, convex and homogeneous of degree p > 1 with A(x, \xi ) \leq A1(x)A2(\xi ). A typical example
of functions A satisfying the above hypothesis is
A(x, \xi ) =
1
p
\| \xi \| pr , p > 1, r > 1, \xi \in Rn,
or more generally
A(x, \xi ) =
1
p
H(\xi )p, p > 1.
In these two important particular cases, (1.6) reduces to (1.3) and (1.2), respectively.
2. Preliminaries. In this section we give some properties of the A-harmonic operator and
general norms in Rn which will be used in the sequel. The proofs can be obtained similarly as in
[15] or [16].
Let \langle , \rangle denote the usual inner product in Rn and A : \Omega \times Rn \rightarrow R+ be a measurable function
such that \xi \rightarrow A(x, \xi ) is convex and homogeneous of degree p > 1 so that
A(x, t\xi ) = | t| pA(x, \xi ) (2.1)
for all t \in R and (x, \xi ) \in \Omega \times Rn. If we assume that A \in C1(\Omega \times Rn\setminus \{ 0\} ), then from (2.1) it
follows that
\langle \xi ,\nabla \xi A(x, t\xi )\rangle = pA(x, \xi )
for all (x, \xi ) \in \Omega \times Rn [23].
Let A2 be an arbitrary norm in Rn. If we define the dual norm A0 of A2 by
A0(u) = p
- 1
p \mathrm{s}\mathrm{u}\mathrm{p}
\langle u, \xi \rangle
A2(\xi )
1
p
for u \in Rn
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 10
1432 A. TIRYAKI
and if we assume A2 \in C1
\bigl(
Rn\setminus \{ 0\}
\bigr)
, then from (2.1) we obtain
\nabla \xi A2(t\xi ) = \mathrm{s}\mathrm{g}\mathrm{n}\mathrm{t} | t| p\nabla \xi A2(\xi ) for all \xi \not = 0 and t \not = 0 (2.2)
and
\langle \xi ,\nabla \xi A2(\xi )\rangle = pA2(\xi ) for all \xi \in Rn,
where the left-hand side is defined to be 0 if \xi = 0.
Moreover,
A0
\Bigl(
\nabla \xi A2(\xi )
1
p
\Bigr)
= p
- 1
p for all \xi \in Rn\setminus \{ 0\} .
Similarly, if A0 is of class C1 for u \not = 0, then
A2
\Bigl(
\nabla A0(u)
1
p
\Bigr)
= p
- 1
p for all u \in Rn\setminus \{ 0\} .
From (2.2), we obtain the Hölder-type inequality
\langle u, \xi \rangle \leq p
1
pA0(u)A2(\xi )
1
p for all u, \xi \in Rn.
We will also need the following lemmas.
Lemma 2.1 [23]. Let the function \xi \rightarrow A(x, \xi ) be continuously differentiable convex and ho-
mogeneous of degree p > 1, that is, A(x, t\xi ) = | t| pA(x, \xi ) for all t \in R and (x, \xi ) \in \Omega \times Rn.
Then
A(x, \eta ) + (p - 1)A(x, \xi ) \geq \langle \nabla \xi A(x, \xi ), \eta \rangle (2.3)
for all \xi , \eta \in Rn. If in addition, \xi \rightarrow A(x, \xi ) is a strictly convex function, then the equality in (2.3)
holds if and only if \xi = \eta .
Lemma 2.2 (A generalization of Picone’s identity) [23]. Let the function \xi \rightarrow A(x, \xi ) be con-
tinuously differentiable convex and homogeneous of degree p > 1. Assume that u and v are differen-
tiable in a given domain \Omega \subset Rn with v(x) \not = 0 in \Omega and denote
\Phi A(u, v) := A(x,\nabla u) + (p - 1)A
\Bigl(
x,
u
v
\nabla v
\Bigr)
-
\Bigl\langle
\nabla \xi A
\Bigl(
x,
u
v
\nabla v
\Bigr)
,\nabla u
\Bigr\rangle
.
Then
A(x,\nabla u) - 1
p
\biggl\langle
\nabla \xi A(x,\nabla v),\nabla
\biggl(
| u| p
| v| p - 2v
\biggr) \biggr\rangle
= \Phi A(u, v) (2.4)
and \Phi A(u, v) \geq 0 a.e. in \Omega . If , in addition, the function \xi \rightarrow A(x, \xi ) is strictly convex in Rn, then
\Phi A(u, v) = 0 a.e. in \Omega if and only if u\nabla v = v\nabla u in \Omega .
Remark 2.1. In the special case, where A(x, \xi ) =
1
p
H(\xi )p, p > 1, Lemmas 2.1 and 2.2 reduce
to Lemmas 2.1 and 2.2 in [10], respectively. In [10], this special formula was used to establish
the Caccioppoli-type inequality (1.5) for the subsolution (resp. the supersolution) of the nonlinear
equation involving a Finsler p-Laplace operator \bigtriangleup H,p. Similarly in the special case, where A(x, \xi ) =
=
1
p
\| \xi \| pr , where p, r > 1, \xi \in Rn, the identity (2.4) reduces to
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 10
CACCIOPPOLI-TYPE ESTIMATES FOR A CLASS OF NONLINEAR DIFFERENTIAL OPERATORS 1433
\| \nabla u\| pr -
\biggl\langle
\nabla
\biggl(
| u| p
| v| p - 2v
\biggr)
, \| \nabla v\| p - r
r \nabla rv
\biggr\rangle
= \Phi p,r(u, v),
where \nabla rv is defined in (1.4) and
\Phi p,r(u, v) := \| \nabla u\| pr + (p - 1)
| u| p
| v| p
\| \nabla v\| pr - p
| u| p - 2u
| v| p - 2v
\bigl\langle
\nabla u, \| \nabla v\| p - 2
r \nabla rv
\bigr\rangle
\geq 0
which is the special case of (2.4).
3. Caccioppoli-type estimates. In this section we establish Caccioppoli-type estimates for
positive sub- and supersolutions of nonlinear equations involving anisotropic elliptic operators \bigtriangleup A
where A(x, \xi ) \leq A1(x)A2(\xi ).
Let \Omega be a domain in Rn and A : \Omega \times Rn \rightarrow [0,\infty ) be a special Caratheodory function, i.e.,
A1 \in C(\Omega ) with A1(x) > 0, A2 is measurable nonnegative function for all \xi \in Rn. Moreover we
assume that A2(\xi ) is differentiable and satisfies the properties given in Section 2.
Consider the equation
- 1
p
\bigtriangleup Av = g(x)| v| p - 2v + f(x), (3.1)
where p > 1, 0 \leq g \in L\infty (\Omega ), 0 \leq f \in Lp\prime (\Omega ) and p\prime =
p - 1
p
.
As usual we will say that a continuous function v \in W 1,p(\Omega ) is a (weak) solution of Eq. (3.1) in
a domain \Omega \subset Rn if it satisfies
1
p
\int
\Omega
\bigl\langle
\nabla \xi A(x,\nabla u),\nabla \eta
\bigr\rangle
dx -
\int
\Omega
g(x)| v| p - 2v\eta dx =
\int
\Omega
f(x)\eta dx (3.2)
for all \eta \in W 1,p
0 (\Omega ).
(Weak) subsolution and supersolution (3.1) are defined analogously using the nonnegative test
functions \eta \in W 1,p
0 (\Omega ) by replacing `` = "" in (3.2) with `` \leq "" and `` \geq "", respectively.
We define the functional JA as
JA(u; \Omega ) :=
\int
\Omega
A(x,\nabla u)dx -
\int
\Omega
g(x)| u| pdx, u \in W 1,p
0 (\Omega )
associated with (3.1).
Let v > 0 be a (continuous) weak subsolution of (3.1) in \Omega and u \in W 1,p
0 (\Omega ). Then we can
choose
\eta =
| u| p
vp - 1
as a test function in (3.1) and conclude by (2.4), that
JA(u; \Omega ) \leq
\int
\Omega
\biggl\{
A(x,\nabla u) - 1
p
\biggl\langle
\nabla \xi A(x,\nabla u),\nabla
\biggl(
| u| p
vp - 1
\biggr) \biggr\rangle \biggr\}
dx+
\int
\Omega
f(x)
| u| p
vp - 1
dx =
=
\int
\Omega
\Phi A(u, v)dx+
\int
\Omega
f(x)
| u| p
vp - 1
dx. (3.3)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 10
1434 A. TIRYAKI
Clearly, for positive supersolution v of (3.1) and any u \in W 1,p
0 (\Omega ), the reversed inequality
JA(u; \Omega ) \geq
\int
\Omega
\Phi A(u, v)dx+
\int
\Omega
f(x)
| u| p
vp - 1
dx.
holds true. If, in particular, v \in W 1,p
0 (\Omega ) is a positive solution of (3.1), then (3.3) becomes an
equality.
Then we have the following theorem.
Theorem 3.1 (Caccioppoli-type inequality). Let v > 0 be weak subsolution of (3.1) in \Omega . Then
for any fixed q > p - 1 and w = v
q
p , the inequality
qp
q - p+ 1
\int
\Omega
A(x, \eta \nabla w)dx - (p - 1)(q + 1)
q - p+ 1
\int
\Omega
A1(x)A2(\eta \nabla w)dx \leq
\leq
\biggl(
q
q - p+ 1
\biggr) p \int
\Omega
A1(x)A2(w\nabla \eta )dx+
qpp1 - p
q - p+ 1
\int
\Omega
g(x)wp\eta pdx+
qpp1 - p
q - p+ 1
\int
\Omega
f(x)\eta dx
(3.4)
holds for all 0 \leq \eta \in C\infty
0 (\Omega ).
Proof. Let v be a positive subsolution of (3.1) in \Omega . Fix a nonnegative function \eta \in C\infty
0 (\Omega ).
Then u := v
q
p \eta belongs to W 1,p
0 (\Omega ) and we can use it as a test function in (3.3) to get
JA(v
q
p \eta ; \Omega ) \leq
\int
\Omega
A(x,\nabla v
q
p \eta )dx+ (p - 1)
\int
\Omega
A
\Bigl(
x, v
q - p
p \eta \nabla v
\Bigr)
dx -
-
\int
\Omega
\Bigl\langle
\nabla \xi A
\bigl(
x, v
q - p
p \eta \nabla v
\bigr)
,\nabla
\bigl(
v
q
p \eta
\bigr) \Bigr\rangle
dx+
\int
\Omega
f(x)\eta dx. (3.5)
By using
\nabla
\bigl(
v
q
p \eta
\bigr)
=
q
p
v
q
p
- 1
\eta \nabla v + v
q
p\nabla \eta
and
A(x, \xi ) \leq A1(x)A2(\xi )
in the above inequality (3.5), we have
JA
\bigl(
v
q
p \eta ; \Omega
\bigr)
\leq
\int
\Omega
A
\bigl(
x,\nabla v
q
p \eta
\bigr)
dx+ (p - 1)
\int
\Omega
A1(x)A2(v
q - p
p \eta \nabla v)dx -
-
\int
\Omega
\biggl\langle
\nabla \xi A
\bigl(
x, v
q - p
p \eta \nabla v
\bigr)
,
q
p
v
q - p
p \eta \nabla v
\biggr\rangle
dx+
+
\int
\Omega
\Bigl\langle
\nabla \xi A1(x)A2
\bigl(
v
q - p
p \eta \nabla v
\bigr)
, v
q
p\nabla \eta
\Bigr\rangle
dx+
\int
\Omega
f(x)\eta dx. (3.6)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 10
CACCIOPPOLI-TYPE ESTIMATES FOR A CLASS OF NONLINEAR DIFFERENTIAL OPERATORS 1435
On the other hand by using properties of A0 and A2, we can easily obtain\Bigl\langle
\nabla \xi A1(x)A2(v
q - p
p \eta \nabla v), v
q
p\nabla \eta
\Bigr\rangle
= A1(x)
\Bigl\langle
\nabla \xi
\Bigl[
A2(v
q - p
p \eta \nabla v)
1
p
\Bigr] p
, v
q
p\nabla \eta
\Bigr\rangle
\leq
\leq pA1(x)A2
\bigl(
v
q - p
p \eta \nabla v
\bigr) p - 1
p A2
\bigl(
v
q
p\nabla \eta
\bigr) 1
p .
By using this inequality in (3.6), we get
JA(v
q
p \eta ; \Omega ) \leq
\int
\Omega
A
\bigl(
x,\nabla v
q
p \eta
\bigr)
dx -
- (1 - p)
\int
\Omega
A1(x)A2
\bigl(
v
q - p
p \eta \nabla v
\bigr)
dx - q
\int
\Omega
A
\bigl(
x, v
q - p
p \eta \nabla v
\bigr)
dx+
+p
\int
\Omega
A1(x)A2
\bigl(
v
q - p
p \eta \nabla v
\bigr) p - 1
p A2(v
q
p\nabla \eta )
1
pdx+
\int
\Omega
f(x)\eta dx.
Applying Young’s inequality in the form
a
1
p bp - 1 \leq 1
p
1
\tau p - 1
a+
p - 1
p
\tau bp, a, b \geq 0, \tau > 0,
we have
JA(v
q
p \eta ; \Omega ) \leq
\int
\Omega
A
\bigl(
x,\nabla v
q
p \eta
\bigr)
dx -
-
\bigl[
1 - p+ (1 - p)\tau
\bigr] \int
\Omega
A1(x)A2
\bigl(
v
q - p
p \eta \nabla v
\bigr)
dx - q
\int
\Omega
A
\bigl(
x, v
q - p
p \eta \nabla v
\bigr)
dx+
+
1
\tau p - 1
\int
\Omega
A1(x)A2
\bigl(
v
q
p\nabla \eta
\bigr)
dx+
\int
\Omega
f(x)\eta dx.
Now making use of the definition of JA we obtain
q
\int
\Omega
A
\bigl(
x, v
q - p
p \eta \nabla v
\bigr)
dx \leq (p - 1)(1 + \tau )
\int
\Omega
A1(x)A2
\bigl(
v
q - p
p \eta \nabla v
\bigr)
dx+
+
1
\tau p - 1
\int
\Omega
A1(x)A2
\bigl(
v
q
p\nabla \eta
\bigr)
dx
\int
\Omega
g(x)vq\eta pdx+
\int
\Omega
f(x)\eta dx,
which after choosing the constant \tau :=
q - p+ 1
p
as in [10], leads to
qp
q - p+ 1
\int
\Omega
A
\bigl(
x, v
q - p
p \eta \nabla v
\bigr)
dx - (p - 1)(q + 1)
q - p+ 1
\int
\Omega
A1(x)A2
\bigl(
v
q - p
p \eta \nabla v
\bigr)
dx \leq
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 10
1436 A. TIRYAKI
\leq
\biggl(
p
q - p+ 1
\biggr) p \int
\Omega
A1(x)A2
\bigl(
v
q
p\nabla \eta
\bigr)
dx+
+
p
q - p+ 1
\int
\Omega
g(x)vq\eta pdx+
p
q - p+ 1
\int
\Omega
f(x)\eta dx.
Finally, the substitution of w = v
q
p yields (3.4) as claimed.
Theorem 3.1 is proved.
An analogous result as Theorem 3.1 above holds for positive supersolution of (3.1) and a range
of q smaller than p - 1. The proof is similar to that of Theorem 3.1, hence omitted.
Theorem 3.2. Let v > 0 be a weak supersolution of (3.1) in \Omega . Then, for any fixed q < p - 1,
the inequality
(p - 1)(q + 1)
p - q - 1
\int
\Omega
A1(x)A2
\bigl(
v
q - p
p \eta \nabla v
\bigr)
dx - qp
p - q - 1
\int
\Omega
A(x, v
q - p
p \eta \nabla v)dx \leq
\leq
\biggl(
p
p - q - 1
\biggr) p \int
\Omega
A1(x)A2(v
q
p\nabla \eta )dx -
- p
p - q - 1
\int
\Omega
g(x)vq\eta pdx - p
p - q - 1
\int
\Omega
f(x)\eta dx (3.7)
holds for all \eta \in C\infty
0 (\Omega ) with \eta \geq 0.
Now we consider the equality case, that is, A(x, \xi ) = A1(x)A2(\xi ). From Theorems 3.1 and 3.2
we have the following results.
Theorem 3.3 (Caccioppoli-type inequality). Let v > 0 be weak subsolution of (3.1) in \Omega . Then,
for any fixed q > p - 1 and w = v
q
p , the inequality\int
\Omega
A1(x)A2(\eta \nabla w)dx \leq
\biggl(
q
q - p+ 1
\biggr) p \int
\Omega
A1(x)A2(w\nabla \eta )dx+
+
qpp1 - p
q - p+ 1
\int
\Omega
g(x)wp\eta pdx+
qpp1 - p
q - p+ 1
\int
\Omega
f(x)\eta dx
holds for all 0 \leq \eta \in C\infty
0 (\Omega ).
Corollary 3.1. Let q = p, g(x) \equiv 0, and f(x) \equiv 0 in \Omega . If v > 0 is a weak subsolution of (3.1)
in \Omega , then \int
\Omega
A1(x)A2(\eta \nabla v)dx \leq pp
\int
\Omega
A1(x)A2(v\nabla \eta )dx
for any nonnegative \eta \in C\infty
0 (\Omega ).
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 10
CACCIOPPOLI-TYPE ESTIMATES FOR A CLASS OF NONLINEAR DIFFERENTIAL OPERATORS 1437
Theorem 3.4. Let r > 0 be a weak supersolution of (3.1) in \Omega . Then, for any fixed q < p - 1,
the inequality \int
\Omega
A1(x)A2
\bigl(
v
q
p
- 1
\eta \nabla v
\bigr)
dx \leq
\biggl(
p
p - q - 1
\biggr) p \int
\Omega
A1(x)A2
\bigl(
v
q
p\nabla \eta
\bigr)
dx -
- p
p - q - 1
\int
\Omega
g(x)vq\eta pdx - p
p - q - 1
\int
\Omega
f(x)\eta dx
holds for all \eta \in C\infty
0 (\Omega ) with \eta \geq 0.
The particular case of the above theorem when q = 0 is interesting in the sense that the right-hand
side of (3.7) does not contain v. The result specializes as follows.
Corollary 3.2 (Logarithmic Caccioppoli-type inequality). Let v > 0 be weak subsolution of (3.1)
in \Omega . Then \int
\Omega
A1(x)A2(\eta \nabla logv)dx \leq
\biggl(
p
p - 1
\biggr) p \int
\Omega
A1(x)A2(\nabla \eta )dx -
- p
p - 1
\int
\Omega
g(x)\eta pdx - p
p - 1
\int
\Omega
f(x)\eta dx (3.8)
whenever 0 \leq \eta \in C\infty
0 (\Omega ).
If A(\xi ) =
1
p
\| \xi \| p2, p > 1, \xi \in Rn, and g(x) \equiv 0 and f(x) \equiv 0 in \Omega , then (3.8) reduces to the
well-known logarithmic Caccioppoli inequality for the positive p-superharmonic functions [24].
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Received 16.03.16
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 10
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| id | umjimathkievua-article-1646 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:09:49Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/73/325b0271c12910effe6c6d4f63a4f873.pdf |
| spelling | umjimathkievua-article-16462019-12-05T09:21:55Z Caccioppoli-type estimates for a class of nonlinear differential operators Оцiнки типу Каччiопполi для одного класу нелiнiйних диференцiальних рiвнянь Tiryaki, A. Тирякі, А. We establish Caccioppoli-type estimates for a class of nonlinear differential equations with the aid of a differential identity that generalizes the well-known multidimensional Picone’s formula. In special cases, these estimates give the Finsler $p$-Laplacian, the $p$-Laplacian and the pseudo-$p$-Laplacian. Встановлено оцiнки типу Каччiопполi для одного класу нелiнiйних диференцiальних рiвнянь за допомогою диференцiальної тотожностi, що узагальнює вiдому багатовимiрну формулу Пiконе. В частинних випадках цi оцiнки дають $p$-лапласiан Фiнслера, $p$-лапласiан та $p$-псевдолапласiан. Institute of Mathematics, NAS of Ukraine 2018-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1646 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 10 (2018); 1429-1438 Український математичний журнал; Том 70 № 10 (2018); 1429-1438 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1646/628 Copyright (c) 2018 Tiryaki A. |
| spellingShingle | Tiryaki, A. Тирякі, А. Caccioppoli-type estimates for a class of nonlinear differential operators |
| title | Caccioppoli-type estimates for a class of nonlinear differential operators |
| title_alt | Оцiнки типу Каччiопполi для одного класу нелiнiйних диференцiальних рiвнянь |
| title_full | Caccioppoli-type estimates for a class of nonlinear differential operators |
| title_fullStr | Caccioppoli-type estimates for a class of nonlinear differential operators |
| title_full_unstemmed | Caccioppoli-type estimates for a class of nonlinear differential operators |
| title_short | Caccioppoli-type estimates for a class of nonlinear differential operators |
| title_sort | caccioppoli-type estimates for a class of nonlinear differential operators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1646 |
| work_keys_str_mv | AT tiryakia caccioppolitypeestimatesforaclassofnonlineardifferentialoperators AT tirâkía caccioppolitypeestimatesforaclassofnonlineardifferentialoperators AT tiryakia ocinkitipukaččioppolidlâodnogoklasunelinijnihdiferencialʹnihrivnânʹ AT tirâkía ocinkitipukaččioppolidlâodnogoklasunelinijnihdiferencialʹnihrivnânʹ |