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Caccioppoli-type inequalities for Dirac operators

Abstract

In this paper, we establish the Caccioppoli estimates for the nonlinear differential equation

$$ - \overline{D}\bigl( \vert Dv \vert ^{p-2}Dv\bigr) = \lambda \vert v \vert ^{p-2}v, \quad 1< p< \infty ,$$

where D is the Dirac operator. Moreover, we obtain general weighted versions of the Caccioppoli-type inequalities for the Dirac operators.

Introduction

In the Euclidean setting, we recall the Caccioppoli inequality

$$ \int _{\Omega } \phi ^{p} \vert \nabla v \vert ^{p}\,dx \leq p^{p} \int _{\Omega } v^{p} \vert \nabla \phi \vert ^{p}\,dx $$
(1.1)

for all nonnegative functions \(\phi \in C_{0}^{\infty }(\Omega )\), where a positive function v is a subsolution of the Dirichlet boundary value problem for p-Laplacian

$$ \textstyle\begin{cases} \nabla \cdot ( \vert \nabla v \vert ^{p-2}\nabla v) = \lambda \vert v \vert ^{p-2}v & \text{in } \Omega , \\ v=0 &\text{on } \partial \Omega . \end{cases} $$
(1.2)

See, for example, [4] for the problem in the Euclidean setting. We also refer to [1, 5, 6, 8, 10] and references therein for discussions on the Caccioppoli-type estimates in different settings.

The main aim of this paper is to obtain the Caccioppoli-type inequality for the nonlinear equation

$$ \textstyle\begin{cases} - \overline{D}( \vert Dv \vert ^{p-2}Dv) = \lambda \vert v \vert ^{p-2}v & \text{in } \Omega , \\ v=0 & \text{on } \partial \Omega , \end{cases} $$

where v is the subsolution, D is the usual Dirac operator, and is its conjugate. Also, we obtain weighted versions of the Caccioppoli-type inequality for the Dirac operator.

In what follows, we will work in \(\mathbb{H}\), the skew-field of the quaternion. This means that each element \(x'\in \mathbb{H}\) has the following representation:

$$ x'=x_{0}+\sum_{i=1}^{n}e_{i} x_{i}, $$

where \(1, e_{1},\ldots ,e_{n}\) are the basis elements of \(\mathbb{H}\). For these elements, we have the multiplication rules

  • \(e_{1}^{2} =\cdots =e_{n}^{2} = - 1\),

  • \(e_{i}e_{j}+e_{j}e_{i}=-2\delta +{ij}\) for all \(i,j=1,\ldots ,n\).

The conjugate element \(\overline{x'}\) is given by \(\overline{x'} = x_{0} -\sum_{i=1}^{n} e_{i} x_{i}\), and we have the properties

$$ \bigl\vert x' \bigr\vert ^{2}= x\overline{x'} = \overline{x'}x = x_{0}^{2}+\sum _{i=1}^{n} x_{i}^{2} $$

and

$$ \vert x \vert _{q}=\sqrt{\sum _{i=1}^{n} x_{i}^{2}} $$
(1.3)

for the norm on \(\mathbb{H}\).

We recall the usual Dirac operator, which factorizes the n-dimensional Laplace operator,

$$ Df=\sum_{i=1}^{n} e_{i} \frac{\partial f}{\partial x_{i}} $$

and its conjugate operator

$$ \overline{D}f=-\sum_{i=1}^{n} e_{i}\frac{\partial f}{\partial x_{i}}. $$

The products of these operators

$$ D\overline{D} = \overline{D} D = \Delta _{n}, $$

where \(\Delta _{n}\) is the Laplacian for functions defined over domains in \(\mathbb{R}^{n}\). For further discussions in this direction, we refer, for example, to [9] (see also [3] for theory of QDEs).

In Sect. 2, we discuss Picone’s identity for the Dirac operator. The main results of this paper are presented in Sect. 3.

Picone’s identity for the Dirac operator

Lemma 2.1

Let u, v be a differentiable functions defined a.e. in \(\Omega \subset \mathbb{H}\) such that \(v>0\) a.e. in Ω and \(u\geq 0\). Define

$$\begin{aligned} &R(u,v):= \vert Du \vert ^{p}- D \biggl(\frac{u^{p}}{v^{p-1}} \biggr) \vert Dv \vert ^{p-2}Dv, \\ &L(u,v):= \vert Du \vert ^{p}- p \biggl(\frac{u}{v} \biggr)^{p-1} \vert Dv \vert ^{p-2} DvDu \\ &\hphantom{L(u,v):=}{}+(p-1) \biggl(\frac{u}{v} \biggr)^{p} \vert Dv \vert ^{p}, \end{aligned}$$
(2.1)

where \(p>1\). Then

$$ L(u,v)=R(u,v)\geq 0. $$
(2.2)

Also, \(L(u,v)=0\) a.e. in Ω if and only if \(u=cv\) a.e. in Ω with positive constant c.

Proof of Lemma 2.1

A direct computation gives

$$\begin{aligned} R(u,v) &= \vert D u \vert ^{p} - D \biggl( \frac{u^{p}}{v^{p-1}} \biggr) \vert D v \vert ^{p-2}D v \\ &= \vert D u \vert ^{p} - \frac{pu^{p-1}Du v^{p-1}-u^{p} (p-1)v^{p-2}Dv}{(v^{p-1})^{2}} \vert Dv \vert ^{p-2}Dv \\ &= \vert D u \vert ^{p} - p\frac{u^{p-1}}{v^{p-1}} \vert Dv \vert ^{p-2}D v Du + (p-1)\frac{u^{p}}{v^{p}} \vert Dv \vert ^{p} \\ & = L(u,v). \end{aligned}$$

This proves the equality in (2.2). Now we rewrite \(L(u,v)\) to see that \(L(u,v)\geq 0\):

$$\begin{aligned} L(u,v) = {}& \vert D u \vert ^{p} - p \frac{u^{p-1}}{v^{p-1}} \vert D v \vert ^{p-1} \vert D u \vert + (p-1) \frac{u^{p}}{v^{p}} \vert D v \vert ^{p} \\ &{}+p \frac{u^{p-1}}{v^{p-1}} \vert D v \vert ^{p-2} \bigl( \vert D v \vert \vert D u \vert -D v Du \bigr) \\ ={}& S_{1} +S_{2}, \end{aligned}$$

where

$$\begin{aligned} S_{1}:={}& p \biggl[ \frac{1}{p} \vert D u \vert ^{p} + \frac{p-1}{p} \biggl( \biggl( \frac{u}{v} \vert D v \vert \biggr)^{p-1} \biggr)^{\frac{p}{p-1}} \biggr] \\ &{}- p \frac{u^{p-1}}{v^{p-1}} \vert D v \vert ^{p-1} \vert D u \vert \end{aligned}$$

and

$$ S_{2}:= p \frac{u^{p-1}}{v^{p-1}} \vert D v \vert ^{p-2} \bigl( \vert D v \vert \vert D u \vert -D v Du \bigr). $$

We can see that \(S_{2}\geq 0\) due to \(|Dv||D u| \geq D v Du\). To check that \(S_{1} \geq 0\), we need to use Young’s inequality

$$ ab \leq \frac{a^{p}}{p} + \frac{b^{q}}{q},\quad a\geq 0,b \geq 0, $$
(2.3)

where \(p>1\), \(q>1\), and \(\frac{1}{p}+\frac{1}{q}=1\). The equality holds if and only if \(a^{p}=b^{q}\), that is, if \(a = b^{\frac{1}{p-1}}\). Let us take \(a= |D u|\) and \(b = (\frac{u}{v}|Dv| )^{p-1}\) in (2.3) to get

$$ p \vert D u \vert \biggl(\frac{u}{v} \vert D v \vert \biggr)^{p-1} \leq p \biggl[ \frac{1}{p} \vert D u \vert ^{p} + \frac{p-1}{p} \biggl( \biggl(\frac{u}{v} \vert D v \vert \biggr)^{p-1} \biggr)^{\frac{p}{p-1}} \biggr]. $$
(2.4)

From this we see that \(S_{1} \geq 0\), which proves that \(L(u,v)=S_{1}+S_{2} \geq 0\). It is easy to see that \(u=cv\) implies \(R(u,v)=0\). Now let us prove that \(L(u,v)=0\) implies \(u=cv\). Due to \(u(x)\geq 0\) and \(L(u,v)(x_{0})=0\), \(x_{0}\in \Omega \), we consider the two cases \(u(x_{0})>0\) and \(u(x_{0}) =0\).

  1. (1)

    In the case \(u(x_{0})>0\), from \(L(u,v)(x_{0})=0\) it follows that \(S_{1}=0\) and \(S_{2}=0\). Then \(S_{1}=0\) implies

    $$ \vert D u \vert = \frac{u}{v} \vert D v \vert , $$
    (2.5)

    and \(S_{2}=0\) implies

    $$ \vert D v \vert \vert D u \vert - D v D u = 0, $$
    (2.6)

    Combination of (2.5) and (2.6) gives

    $$ \frac{D u}{D v} = \frac{u }{v}=c \quad \text{with } c\neq 0. $$
    (2.7)
  2. (2)

    Let us denote \(\Omega ^{*}:=\{x \in \Omega | u(x)=0 \}\). If \(\Omega ^{*} \neq \Omega \), then suppose that \(x_{0} \in \partial \Omega ^{*}\). Then there exists a sequence \(x_{k} \notin \Omega ^{*}\) such that \(x_{k} \rightarrow x_{0}\). In particular, \(u(x_{k}) \neq 0\), and hence by case (1) we have \(u(x_{k})=cv(x_{k})\). Passing to the limit, we get \(u(x_{0}) = c v(x_{0})\). Since \(u(x_{0})= 0\) and \(v(x_{0})\neq 0\), we get that \(c=0\). Then by case (1) again, since \(u =cv\) and \(u\neq 0\) in \(\Omega \backslash \Omega ^{*}\), it is impossible that \(c=0\). This contradiction implies that \(\Omega ^{*} = \Omega \).

This completes the proof of Lemma 2.1. □

Caccioppoli-type inequalities

Let us consider the Dirichlet boundary problem for the Dirac operator

$$ \textstyle\begin{cases} - \overline{D}( \vert Dv \vert ^{p-2}Dv) = \lambda \vert v \vert ^{p-2}v & \text{in } \Omega , \\ v=0 & \text{on } \partial \Omega . \end{cases} $$
(3.1)

We say that a weak solution of equation (3.1) means a function \(v \in W_{\mathrm{loc}}^{1,p}(\Omega )\) such that

$$ \int _{\Omega } \vert D v \vert ^{p-2} Dv D\phi \,dx' - \lambda \int _{\Omega } \vert v \vert ^{p-2}v \phi \,dx' =0 $$
(3.2)

for all functions \(\phi \in W_{0}^{1,p}(\Omega )\cap C(\Omega )\). The supsolution and subsolution of equation (3.1) mean a function \(v \in W_{\mathrm{loc}}^{1,p}(\Omega )\) such that

$$ \int _{\Omega } \vert D v \vert ^{p-2} Dv \overline{D} \phi \,dx' - \lambda \int _{ \Omega } \vert v \vert ^{p-2}v\phi \,dx' \geq 0 $$
(3.3)

and

$$ \int _{\Omega } \vert D v \vert ^{p-2} Dv \overline{D} \phi \,dx' - \lambda \int _{ \Omega } \vert v \vert ^{p-2}v\phi \,dx' \leq 0 $$
(3.4)

for the test functions \(\phi \in W_{0}^{1,p}(\Omega )\cap C(\Omega )\) with \(\phi \geq 0\), respectively.

If we take the test function as \(\phi =v\), then for the supsolution and subsolution, we have

$$ \int _{\Omega } \vert Dv \vert ^{p}\,dx' \geq \lambda \int _{\Omega } \vert v \vert ^{p}\,dx' $$
(3.5)

and

$$ \int _{\Omega } \vert Dv \vert ^{p}\,dx' \leq \lambda \int _{\Omega } \vert v \vert ^{p}\,dx'. $$
(3.6)

Now we are ready to establish a Caccioppoli-type inequality.

Theorem 3.1

Let \(\Omega \in \mathbb{H}\). Let v be a positive subsolution of equation (3.1) in Ω. For any fixed \(q>p-1\) and \(q< p<\infty \), we have

$$\begin{aligned} \int _{\Omega } v^{q-p} \phi ^{p} \vert D v \vert ^{p}\,dx' \leq \biggl( \frac{p}{q-p+1} \biggr)^{p} \int _{\Omega } v^{q} \vert D\phi \vert ^{p} \,dx' + \frac{\lambda p}{q-p+1} \int _{\Omega } v^{q} \phi ^{p} \,dx' \end{aligned}$$
(3.7)

for all nonnegative functions \(\phi \in C^{\infty }_{0}(\Omega )\).

Remark 3.2

Note that Theorem 3.1 for the Finsler norm was obtained in [2].

  • For the case \(q=p\) and \(\lambda =0\) in Theorem 3.1, we have

    $$ \int _{\Omega } \phi ^{p} \vert Dv \vert ^{p}\,dx' \leq p^{p} \int _{\Omega } v^{p} \vert D \phi \vert ^{p}\,dx'. $$
    (3.8)
  • For the case \(q=0\) in Theorem 3.1, we have

    $$ \int _{\Omega } \phi ^{p} \vert D\log v \vert ^{p}\,dx' \leq \biggl( \frac{p}{1-p} \biggr)^{p} \int _{\Omega } \vert D\phi \vert ^{p} \,dx' + \frac{\lambda p}{1-p} \int _{\Omega } \phi ^{p}\,dx'. $$
    (3.9)

Proof of Theorem 3.1

Let us begin the proof by replacing \(u = v^{\frac{q}{p}}\phi \) in \(L(u,v)\), which gives

$$\begin{aligned} \int _{\Omega } L\bigl(v^{q/p}\phi ,v\bigr) \,dx' = {}& \int _{\Omega } \bigl\vert D\bigl(v^{q/p} \phi \bigr) \bigr\vert ^{p}\,dx' + (p-1) \int _{\Omega } v^{q-p}\phi ^{p} \vert Dv \vert ^{p}\,dx' \\ &{} - p \int _{\Omega } \bigl(v^{\frac{q-p}{p}}\phi \bigr)^{p-1} \vert Dv \vert ^{p-2}D\bigl(v^{q/p} \phi \bigr) \,dx' \\ ={}& \int _{\Omega } \bigl\vert D\bigl(v^{q/p}\phi \bigr) \bigr\vert ^{p}\,dx' + (p-1) \int _{\Omega } v^{q-p} \phi ^{p} \vert Dv \vert ^{p}\,dx' \\ &{} -p \int _{\Omega } \bigl(v^{\frac{q-p}{p}}\phi \bigr)^{p-1} \vert Dv \vert ^{p-2} \biggl( \frac{q}{p} v^{\frac{q-p}{p}}\phi Dv+ v^{q/p} Dv D\phi \biggr)\,dx' \\ ={}& \int _{\Omega } \bigl\vert D\bigl(v^{q/p}\phi \bigr) \bigr\vert ^{p}\,dx' - (q-p+1) \int _{\Omega } v^{q-p} \phi ^{p} \vert Dv \vert ^{p}\,dx' \\ &{} +p \int _{\Omega } \bigl(v^{\frac{q-p}{p}}\phi \bigr)^{p-1} \vert Dv \vert ^{p-1} v^{q/p} \vert D \phi \vert \,dx'. \end{aligned}$$

In the last line, we have used the Schwarz inequality. Now we apply the Young inequality of the form

$$ a b^{p-1} \leq \frac{a^{p}}{p\tau ^{p-1}} + \frac{p-1}{p} \tau b^{p},\quad a,b\geq 0, \tau >0. $$

By choosing \(a= v^{q/p}|D\phi |\) and \(b=v^{\frac{q-p}{p}}\phi |Dv|\) and using inequality (3.6) we arrive at

$$\begin{aligned} 0\leq{ }& \int _{\Omega } \bigl\vert D\bigl(v^{q/p}\phi \bigr) \bigr\vert ^{p}\,dx' - (q-p+1) \int _{ \Omega } v^{q-p}\phi ^{p} \vert Dv \vert ^{p}\,dx' \\ &{}+ \tau ^{1-p} \int _{\Omega } v^{q} \vert D\phi \vert ^{p} \,dx' + (p-1)\tau \int _{ \Omega } v^{q-p} \phi ^{p} \vert Dv \vert ^{p}\,dx' \\ \leq {}& \lambda \int _{\Omega } v^{q} \phi ^{p} \,dx'- \bigl(q-p+1- \tau (p-1)\bigr) \int _{\Omega } v^{q-p}\phi ^{p} \vert Dv \vert ^{p}\,dx' \\ &{}+\tau ^{1-p} \int _{\Omega } v^{q} \vert D\phi \vert ^{p} \,dx'. \end{aligned}$$

Thus we have the following inequality:

$$\begin{aligned} \int _{\Omega } v^{q-p}\phi ^{p} \vert Dv \vert ^{p}\,dx'\leq{}& \frac{\tau ^{1-p}}{(q-p+1- \tau (p-1))} \int _{\Omega } v^{q} \vert D\phi \vert ^{p} \,dx' \\ &{}+ \frac{\lambda }{(q-p+1- \tau (p-1))} \int _{\Omega } v^{q} \phi ^{p} \,dx'. \end{aligned}$$
(3.10)

Taking a suitable constant \(\tau = \frac{q-p+1}{p}\) leads to

$$\begin{aligned} \int _{\Omega } v^{q-p} \phi ^{p} \vert D v \vert ^{p}\,dx' \leq \biggl( \frac{p}{q-p+1} \biggr)^{p} \int _{\Omega } v^{q} \vert D\phi \vert ^{p} \,dx' + \frac{\lambda p}{q-p+1} \int _{\Omega } v^{q} \phi ^{p} \,dx'. \end{aligned}$$

This proves the theorem. □

Weighted versions

Let us consider the following weighted operator:

$$ \Delta _{p,w}f = \overline{D}\bigl(w(x) \vert Df \vert ^{p-2}Df\bigr),\quad 1< p< \infty , $$
(4.1)

where \(0\leq w \in C^{1}(\mathbb{H})\).

Theorem 4.1

Let \(2\leq p< \infty \). Let \(0 \leq F \in C^{\infty }(\mathbb{H})\) and \(0\leq \eta \in L_{\mathrm{loc}}^{1}(\mathbb{H})\) be such that

$$ \eta F^{p-1}\leq - \Delta _{p,w}F\quad \textit{a.e. in } \mathbb{H}. $$
(4.2)

Then we have

$$\begin{aligned} \int _{\mathbb{H}} \eta (x) \bigl\vert f(x) \bigr\vert ^{p}\,dx' + C_{p} \int _{\mathbb{H}} w(x) \bigl\vert F(x) \bigr\vert ^{p} \bigl\vert D(f/F) \bigr\vert ^{p}\,dx' \leq \int _{\mathbb{H}} w(x) \bigl\vert Df(x) \bigr\vert ^{p} \,dx' \end{aligned}$$
(4.3)

for all real-valued functions \(f \in C_{0}^{\infty }(\mathbb{H})\). Here \(C_{p}\) is a positive constant.

Note that the Carnot group version of Theorem 4.1 was obtained in [7].

Proof of Theorem 4.1

For all \(a,b \in \mathbb{R}^{n}\), there exists a positive number \(C_{p}\) such that

$$ \vert a \vert ^{p} + C_{p} \vert b \vert ^{p} + p \vert a \vert ^{p-2} a\cdot b \leq \vert a+b \vert ^{p},\quad 2 \leq p < \infty . $$
(4.4)

Using this by taking \(a=g(x)DF(x)\) and \(b=F(x)Dg(x)\), we get

$$\begin{aligned} &\bigl\vert g(x) \bigr\vert ^{p} \bigl\vert DF(x) \bigr\vert ^{p} + C_{p} \bigl\vert F(x) \bigr\vert ^{p} \bigl\vert Dg(x) \bigr\vert ^{p} +F(x) \bigl\vert DF(x) \bigr\vert ^{p-2}DF(x) \cdot D \bigl\vert g(x) \bigr\vert ^{p} \\ &\quad \leq \bigl\vert g(x) DF(x) +F(x) Dg(x) \bigr\vert ^{p} = \bigl\vert Df(x) \bigr\vert ^{p}, \end{aligned}$$
(4.5)

where \(g = f/F\). It follows that

$$\begin{aligned} \int _{\mathbb{H}} w(x) \bigl\vert Df(x) \bigr\vert ^{p} \,dx' \geq {}& \int _{\mathbb{H}} w(x) \bigl\vert DF(x) \bigr\vert ^{p} \bigl\vert g(x) \bigr\vert ^{p}\,dx' \\ & {}+ C_{p} \int _{\mathbb{H}} w(x) \bigl\vert Dg(x) \bigr\vert ^{p} \bigl\vert F(x) \bigr\vert ^{p}\,dx' \\ & {}- \int _{\mathbb{H}} \overline{D} \bigl(w(x)F(x) \bigl\vert DF(x) \bigr\vert ^{p-2}DF(x)\bigr) \bigl\vert g(x) \bigr\vert ^{p} \,dx' \\ \geq {}& C_{p} \int _{\mathbb{H}} w(x) \bigl\vert Dg(x) \bigr\vert ^{p} \bigl\vert F(x) \bigr\vert ^{p}\,dx' \\ & {}+ \int _{\mathbb{H}} - \overline{D}\bigl(w(x) \bigl\vert DF(x) \bigr\vert ^{p-2}DF(x)\bigr) F(x) \bigl\vert g(x) \bigr\vert ^{p}\,dx'. \end{aligned}$$

Using (4.2), this implies that

$$\begin{aligned}& \int _{\mathbb{H}} \eta (x) \bigl\vert g(x) \bigr\vert ^{p} \bigl\vert F(x) \bigr\vert ^{p}\,dx' + C_{p} \int _{ \mathbb{H}} w(x) \bigl\vert Dg(x) \bigr\vert ^{p} \bigl\vert F(x) \bigr\vert ^{p}\,dx' \\& \quad \leq \int _{\mathbb{H}} w(x) \bigl\vert Df(x) \bigr\vert ^{p} \,dx'. \end{aligned}$$
(4.6)

Since \(g =f/F\), we arrive at

$$ \int _{\mathbb{H}} \eta (x) \bigl\vert f(x) \bigr\vert ^{p}\,dx' + C_{p} \int _{\mathbb{H}} w(x) \bigl\vert F(x) \bigr\vert ^{p} \bigl\vert D(f/F) \bigr\vert ^{p}\,dx' \leq \int _{\Omega } w(x) \bigl\vert Df(x) \bigr\vert ^{p} \,dx', $$
(4.7)

which proves (4.3). □

Remark 4.2

For \(p=2\), we have equality in inequality (4.4) with \(C_{2}=1\), that is, the above proof gives the following remainder formula:

$$\begin{aligned} \int _{\mathbb{H}} w(x) \bigl\vert F(x) \bigr\vert ^{2} \bigl\vert D(f/F) \bigr\vert ^{2}\,dx' = \int _{ \mathbb{H}} w(x) \bigl\vert Df(x) \bigr\vert ^{2} \,dx' - \int _{\mathbb{H}} \eta (x) \bigl\vert f(x) \bigr\vert ^{2}\,dx'. \end{aligned}$$
(4.8)

Remark 4.3

For \(1< p<2\), inequality (4.4) can be stated as for all \(a,b \in \mathbb{R}^{n}\), there exists a positive number \(C_{p}\) such that

$$ \vert a \vert ^{p} +C_{p} \frac{ \vert b \vert ^{p}}{( \vert a \vert + \vert b \vert )^{2-p}} + p \vert a \vert ^{p-2}a\cdot b \leq \vert a+b \vert ^{p},\quad 1< p< 2. $$
(4.9)

In turn, from the proof it follows that

$$\begin{aligned} & \int _{\mathbb{H}} w(x) \bigl\vert Df(x) \bigr\vert ^{p} \,dx' \\ &\quad \geq \int _{\mathbb{H}} \eta (x) \bigl\vert f(x) \bigr\vert ^{p}\,dx' \\ &\qquad {}+ C_{p} \int _{\mathbb{H}} w(x) \biggl( \biggl\vert \frac{f(x)}{F(x)}DF(x) \biggr\vert + F \biggl\vert D\frac{f(x)}{F(x)} \biggr\vert \biggr)^{p-2} \bigl\vert F(x) \bigr\vert ^{2} |D (f(x)/F(x)|^{2}\,dx' \end{aligned}$$
(4.10)

for all real-valued functions \(f \in C_{0}^{\infty }(\mathbb{H})\).

Proposition 4.4

For \(f \in C_{0}^{\infty }(\mathbb{H})\), we have

$$ \int _{\Omega } \frac{ \vert f(x) \vert ^{p}}{ \vert x \vert _{q}^{p}}\,dx'\leq \biggl( \frac{p}{\gamma -p-2} \biggr)^{p} \int _{\Omega } \bigl\vert Df(x) \bigr\vert ^{p} \,dx', $$
(4.11)

where \(1< p<\gamma -2\) and \(\gamma \leq 2+n\) with \(\gamma \in \mathbb{R}\).

Proof of Proposition 4.4

In Theorem 4.1, we take \(w=1\) and

$$ F_{\varepsilon } = \vert x_{\varepsilon } \vert _{q}^{-\frac{\gamma -p-2}{p}} = \bigl( (x_{1}+\varepsilon )^{2} + \cdots + (x_{n}+\varepsilon )^{2} \bigr)^{-\frac{\gamma -p-2}{2p}} $$

for a given \(\varepsilon >0\). A direct computation gives

$$\begin{aligned}& D \vert x_{\varepsilon } \vert _{q}^{\alpha } = \alpha \vert x_{\varepsilon } \vert _{q}^{ \alpha -2}\sum _{i=1}^{n} e_{i}x_{i}, \\& \vert D \vert x_{\varepsilon }|_{q}^{\alpha } |_{q}^{p-2} = \vert \alpha \vert ^{p-2} \vert x_{ \varepsilon } \vert _{q}^{(\alpha -2)(p-2)+(p-2)}, \\& \overline{D} \bigl( \vert D \vert x_{\varepsilon }|_{q}^{\alpha } |_{q}^{p-2}D \vert x_{ \varepsilon }|_{q}^{\alpha } \bigr) =\overline{D} \Biggl(\alpha \vert \alpha \vert ^{p-2} \vert x_{ \varepsilon } \vert _{q}^{\alpha p - \alpha -p}\sum _{i=1}^{n}e_{i}x_{i}\Biggr) \\& \hphantom{\overline{D} \bigl( \vert D \vert x_{\varepsilon }|_{q}^{\alpha } |_{q}^{p-2}D \vert x_{ \varepsilon }|_{q}^{\alpha } \bigr) } = \alpha \vert \alpha \vert ^{p-2}(\alpha p- \alpha -p+n) \vert x_{\varepsilon } \vert _{q}^{ \alpha p -\alpha -p}. \end{aligned}$$

By taking \(\alpha =-\frac{\gamma -p-2}{2p}\) we compute

$$\begin{aligned} - \Delta _{p,1} F_{\varepsilon } &=- \overline{D} \bigl( \vert DF_{ \varepsilon } \vert ^{p-2}DF_{\varepsilon } \bigr) \\ & =- \overline{D} \bigl( \vert D \vert x_{\varepsilon }|_{q}^{- \frac{\gamma -p-2}{p}}|_{q}^{p-2}D \vert x_{\varepsilon }|_{q}^{- \frac{\gamma -p-2}{p}} \bigr) \\ & = \frac{\gamma -p-2}{p} \biggl\vert \frac{\gamma -p-2}{p} \biggr\vert ^{p-2} \biggl( \frac{\gamma -p-2}{p} - \gamma +2 +n \biggr) \vert x_{\varepsilon } \vert _{q}^{- \frac{(\gamma -p-2)(p-1)}{p}-p} \\ &= \biggl( \biggl\vert \frac{\gamma -p-2}{p} \biggr\vert ^{p} + \frac{\gamma -p-2}{p} \biggl\vert \frac{\gamma -p-2}{p} \biggr\vert ^{p-2}(- \gamma +2+n) \biggr) \vert x_{\varepsilon } \vert _{q}^{- \frac{(\gamma -p-2)(p-1)}{p}-p}. \end{aligned}$$

If \(1< p<\gamma -2\) and \(\gamma \leq 2+n\), then the above expression gives

$$ - \Delta _{p,1} F_{\varepsilon } \geq \biggl\vert \frac{\gamma -p-2}{p} \biggr\vert ^{p} \frac{1}{ \vert x_{\varepsilon } \vert _{q}^{p}}F_{\varepsilon }^{p-1}, $$
(4.12)

that is, according to the assumption in Theorem 4.1, we can put

$$ \eta (x) = \biggl\vert \frac{\gamma -p-2}{p} \biggr\vert ^{p} \frac{1}{ \vert x_{\varepsilon } \vert _{q}^{p}}, $$
(4.13)

which shows that (4.3) implies (4.11). □

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References

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Acknowledgements

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Funding

This research was funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP08855571). The authors were also supported by the Nazarbayev University grant SST2018047.

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GO wrote the initial draft after calculation of results, and AK originated the idea of this research. Both authors read and approved the final manuscript.

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Correspondence to Gulaiym Oralsyn.

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Kashkynbayev, A., Oralsyn, G. Caccioppoli-type inequalities for Dirac operators. J Inequal Appl 2022, 31 (2022). https://doi.org/10.1186/s13660-022-02766-4

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  • DOI: https://doi.org/10.1186/s13660-022-02766-4

MSC

  • 22E30
  • 43A80

Keywords

  • Caccioppoli inequality
  • Quaternions
  • Dirac operator
  • Picone identity