Caccioppoli-type inequalities for Dirac operators

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The main aim of this paper is to obtain the Caccioppoli-type inequality for the nonlinear equation where v is the subsolution, D is the usual Dirac operator, and D is its conjugate. Also, we obtain weighted versions of the Caccioppoli-type inequality for the Dirac operator.
In what follows, we will work in H, the skew-field of the quaternion. This means that each element x ∈ H has the following representation: where 1, e 1 , . . . , e n are the basis elements of H. For these elements, we have the multiplication rules • e 2 1 = · · · = e 2 n = -1, • e i e j + e j e i = -2δ + ij for all i, j = 1, . . . , n. The conjugate element x is given by x = x 0 -n i=1 e i x i , and we have the properties for the norm on H.
We recall the usual Dirac operator, which factorizes the n-dimensional Laplace operator, and its conjugate operator The products of these operators where n is the Laplacian for functions defined over domains in 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
where p > 1. Then This proves the equality in (2.2). Now we rewrite L(u, v) to see that L(u, v) ≥ 0: We can see that S 2 ≥ 0 due to |Dv||Du| ≥ DvDu. To check that S 1 ≥ 0, we need to use Young's inequality (2.4) From this we see that S 1 ≥ 0, which proves that L(u, v) = S 1 + S 2 ≥ 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) ≥ 0 and L(u, v)(x 0 ) = 0, x 0 ∈ , we consider the two cases u(x 0 ) > 0 and u(x 0 ) = 0.

Caccioppoli-type inequalities
Let us consider the Dirichlet boundary problem for the Dirac operator for the test functions φ ∈ W 1,p 0 ( ) ∩ C( ) with φ ≥ 0, respectively. If we take the test function as φ = v, then for the supsolution and subsolution, we have |Dv| p dx ≥ λ |v| p dx (3.5) and Now we are ready to establish a Caccioppoli-type inequality.
Remark 3.2 Note that Theorem 3.1 for the Finsler norm was obtained in [2].
• For the case q = p and λ = 0 in Theorem 3.1, we have • For the case q = 0 in Theorem 3.1, we have Proof of Theorem 3.1 Let us begin the proof by replacing u = v q p φ in L(u, v), which gives In the last line, we have used the Schwarz inequality. Now we apply the Young inequality of the form By choosing a = v q/p |Dφ| and b = v q-p p φ|Dv| and using inequality (3.6) we arrive at Thus we have the following inequality: Taking a suitable constant τ = q-p+1 This proves the theorem.

Weighted versions
Let us consider the following weighted operator: where 0 ≤ w ∈ C 1 (H).