Variational Integral and Some Inequalities of a Class of Quasilinear Elliptic System
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Advances in Applied Clifford Algebras
Variational Integral and Some Inequalities of a Class of Quasilinear Elliptic System Yueming Lu∗ and Pan Lian Communicated by Paula Cerejeiras Abstract. This paper is concerned with properties for a class of degenerate elliptic equations in Clifford analysis. Here we obtain a direct proof of the existence and uniqueness for the Dirac equations by the method of variational integral. Also, we get the Poincar´e inequalities for the case q < 1. Mathematics Subject Classification. Primary 32W50, Secondary 35B05. Keywords. Poincar´e inequality, Dirac operator, Variational integral.
1. Introduction As well known, many researchers have done deep research on the nonlinear systems − divA(x, ∇u(x)) = 0,
(1.1)
and got rich results in recent years. It was also successfully applied in the field of elastic mechanics and engineering, etc. The weighted Poincar´e and Sobolev inequalities are crucial to develop this second order degenerate quasilinear elliptic equations. Here A satisfies the growth condition A(x, ξ) · ξ ≈ ω(x)|ξ|p , these corresponding weights are called p-admissible weights, and it includes the Ap -weights of Muckenhoupt and certain powers of the Jacobian of a quasiconformal mapping (see [17] and the references therein). As it developed, many researchers generalized the Eq. (1.1) to high dimension case which has This paper was supported by Natural Science Foundation of Heilongjiang Province (CN) (No. A2018009), University Nursing Program for Young Scholars with Creative Talents in Heilongjiang Province (CN) (UNPYSCT-2018206). ∗ Corresponding
author.
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Adv. Appl. Clifford Algebras
attracted much interest, such as A-harmonic equations in the differential form setting d∗ A(x, du) = 0.
(1.2)
These equations are intimately connected to the fields such as potential theory, quasi-conformal mappings and the theory of elasticity and so on, see [2,3,11,12]. After that, a class of Dirac equations DA(x, Du) = 0
(1.3)
was introduced to generalized the system (1.1) in the Clifford-valued function spaces, where the operator D is the Euclidean Dirac operator, see [5,15,17– 19,21]. These equations were considered as nonlinear Dirac equations, as well as a generalization of Eq. (1.2). Up to now, the focus in this line of the research was on the existence and some properties of the weak solutions to Eq. (1.3) with the theory of the monotone theory, and the Poincar´e inequality for q < 1 has not been proved before. This paper is organized as follows. Section 2 is the preliminaries part. In Sect. 3, a direct proof of the existence and uniqueness for a kind of degenerate A-Dirac equations is given by the methods of variational integral, Based on [9], both variational integral and obstacle problem are redefined and proved in Clifford-valued function space to show the main results, this is also the luminescent spot of this article. In Sect. 4, we discuss the Lp (Ω, μ)-norm of the solutions of degenerate A-Dirac equations. Furthermore, we get the Poincar´e type inequal
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