A minimum problem with two-phase free boundary in Orlicz spaces
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A minimum problem with two-phase free boundary in Orlicz spaces Jun Zheng · Zhihua Zhang · Peihao Zhao
Received: 1 August 2012 / Accepted: 9 August 2013 / Published online: 11 September 2013 © Springer-Verlag Wien 2013
Abstract We consider the optimization problem of minimizing Jγ (u) = (G(|∇u|) + λ+ (u + )γ + λ− (u − )γ + f u) dx in the class of functions W 1,G () with u − ϕ ∈ function ϕ, where W 1,G () is the class of weakly differentiable W01,G () for a given functions with G(|∇u|) dx < ∞. The conditions on the function G allow for a different behavior at 0 and at ∞. We give a rather complete description of regularity theory for a family of two-phase variational free boundary problems. For 0 < γ ≤ 1, 1,α continuous. For γ = 0, we obtain we prove that every minimizer u γ of Jγ (u) is Cloc local Lipschitz continuity for any minimizer u 0 of J0 (u). Here we also consider an asymptotic problem as γ → 0. At last, we establish sharp geometric estimates for the free boundary corresponding to the minimizer u 0 of J0 (u). Keywords Free boundary problem · Orlicz spaces · Minimizer · Regularity theorem · Hausdorff measure
Communicated by J. Escher. The work is partially supported by NSFC 10971088. J. Zheng (B) · P. Zhao Basic Course Department, Emei Campus, Southwest Jiaotong University, Leshan, Sichuan 614202, China e-mail: [email protected] P. Zhao e-mail: [email protected] Z. Zhang School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 610054, Sichuan, China
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Mathematics Subject Classification (2010) 49K20
35B65 · 35R35 · 35J20 · 35J65 ·
1 Introduction Given a smooth bounded domain in Rn (n ≥ 2), ϕ ∈ L ∞ () with ϕ + = 0 and G(|∇ϕ|) dx < ∞, and some integrable function f in , we consider the problem of minimizing the functional Jγ (u) =
(G(|∇u|) + Fγ (u) + f u) dx → min,
in the class of functions K = {u ∈ L 1 () : G(|∇u|) dx < ∞, u = ϕ on ∂}, where Fγ (u) = λ+ (u + )γ +λ− (u − )γ for γ ∈ (0, 1] and F0 (u) = λ+ χ{u>0} +λ− χ{u≤0} for γ = 0, 0 ≤ λ− < λ+ < ∞, u ± = max{±u, 0}. In fact, the minimizer, denoted by u γ , satisfies (in a certain weak sense) the following Euler–Lagrange equation G u := div
g(|∇u|) ∇u = γ (λ+ (u + )γ −1 χ{u>0} + λ− (u − )γ −1 χ{u 0} ∪ ∂{u γ < 0}) ∩ , between the positive and negative phases of a minimizer. In this work we intend to extend several regularity properties of the minimizer of the optimal problem when γ ∈ [0, 1] and of the free boundary of the limiting function u 0 , obtained as γ → 0, which is proven to be a minimum of J0 , to a large class of elliptic operators under the natural condition which generalizes the Ladyzhenskaya–Uraltseva operators, for some positive constants δ, g0 0 0. g(t)
(1.2)
The operator G not only includes the case of the p-Laplace p (δ = g0 = p−1 > 0), but also the interesting case of a variable exponent p = p(t) > 0 p u := div (|∇u| p(|∇u|)−2 ∇u), corresponding to set g(t) = t p(t)−1 , for which (1.2) holds if δ ≤ t (ln t) p (t) + p(t) − 1 ≤ g0 for all
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