Abstract
We consider the interior regularity for weak solutions of second-order nonlinear elliptic systems with subquadratic growth under natural growth condition. We obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particularly the regularity we obtained is optimal.
Highlights
In this paper we consider optimal interior partial regularity for the weak solutions of nonlinear elliptic systems with subquadratic growth under natural growth condition of the following type: n− DαAαi x, u, Du Bi x, u, Du, i 1, . . . , N in Ω, α1 where Ω is a bounded domain in Rn, u and Bi taking values in RN, and Aαi ·, ·, · has value in RnN
E1 allows us to deduce the existence of a function ω t, s : 0, ∞ × 0, ∞ → 0, ∞ with ω t, 0 0 for all t such that t → ω t, s is monotone nondecreasing for fixed s, s → ω t, s is concave and monotone nondecreasing for fixed t, and such that for all x, u ∈ Ω × RN and p, q ∈ RnN, we have
In 11, we deal with the optimal partial regularity of the weak solution to 1.1 for the case m > 2 by the method of A-harmonic approximation technique, which is advantage to the result of
Summary
In this paper we consider optimal interior partial regularity for the weak solutions of nonlinear elliptic systems with subquadratic growth under natural growth condition of the following type: n. To define weak solution to 1.1 , one needs to impose certain structural and regularity conditions on Aαi and the inhomogeneity Bi, as well as to restrict u to a particular class of functions as follows, for 1 < m < 2, E1 Aαi x, u, p are differentiable functions in p and there exists L > 0 such that. In 11, 12 , we deal with the optimal partial regularity of the weak solution to 1.1 for the case m > 2 by the method of A-harmonic approximation technique, which is advantage to the result of 13. The purpose of this paper is to establish the optimal partial regularity of weak solution to 1.1 under natural growth condition with subquadratic growth, that is, the case of 1 < m < 2, directly.
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