A Caccioppoli-type estimate for very weak solutions to obstacle problems with weight
© Hongya and Jinjing; licensee Springer. 2011
Received: 3 March 2011
Accepted: 17 September 2011
Published: 17 September 2011
This paper gives a Caccioppoli-type estimate for very weak solutions to obstacle problems of the -harmonic equation with , where 1 < p < ∞ and w(x) be a Muckenhoupt A1 weight.
Mathematics Subject Classification (2000) 35J50, 35J60
KeywordsCaccioppoli-type estimate very weak solution obstacle problem Mucken-houpt weight -harmonic equation
for all balls B in R n .
The weighted Sobolev class W1,p(E, w) consists of all functions f, and its first generalized derivatives belong to L p (E, w). The symbols and are self-explanatory.
The function ψ is an obstacle, and θ determines the boundary values.
whenever and H comes from the Hodge decomposition (1.3).
The local and global higher integrability of the derivatives in obstacle problems with w(x) ≡ 1 was first considered by Li and Martio  in 1994, using the so-called reverse Hölder inequality. Gao and Tian  gave a local regularity result for weak solutions to obstacle problem in 2004. Recently, regularity theory for very weak solutions of the -harmonic equations with w(x) ≡ 1 have been considered , and the regularity theory for very solutions of obstacle problems with w(x) ≡ 1 have been explored in . This paper gives a Caccioppoli-type estimate for solutions to obstacle problems with weight, which is closely related to the local regularity theory for very weak solutions of the -harmonic equation (1.2).
where 0 < ρ < R < +∞ and C is a constant depends only on n, p and β/α.
2 Preliminary Lemmas
The following lemma comes from  which is a Hodge decomposition in weighted spaces.
where γ depends only on p.
We also need the following lemma in the proof of the main theorem.
3 Proof of the main theorem
Since u - T k (u) is a very weak solution to the -obstacle problem, we derive, by
where c is the constant given by Lemma 2. This ends the proof of the main theorem.
The authors would like to thank the referee of this paper for helpful suggestions.
Research supported by NSFC (10971224) and NSF of Hebei Province (A2011201011).
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