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Perturbation formula for the two-phase membrane problem
- Farid Bozorgnia^{1}Email author
https://doi.org/10.1186/1687-1847-2011-19
© Bozorgnia; licensee Springer. 2011
- Received: 11 January 2011
- Accepted: 29 June 2011
- Published: 29 June 2011
Abstract
A perturbation formula for the two-phase membrane problem is considered. We perturb the data in the right-hand side of the two-phase equation. The stability of the solution and the free boundary with respect to perturbation in the coefficients and boundary value is shown. Furthermore, continuity and differentiability of the solution with respect to the coefficients are proved.
Keywords
- Free boundary problems
- Two-phase membrane
- Perturbation
Introduction
By Ω^{+}(u) and Ω ^{ - } (u) we denote the sets {x ∈ Ω: u(x) > 0} and {x ∈ Ω: u(x) < 0}, respectively. Also, Λ(u) denotes the set {x ∈ Ω: u(x) = 0}.
The regularity of the solution, the Hausdorff dimension and the regularity of the free boundary are discussed in [2–5]. In [5], on the basis of the monotonicity formula due to Alt, Caffarelli, and Friedman, the boundedness of the second-order derivatives D ^{2} u of solutions to the two-phase membrane problem is proved. Moreover, in [3], a complete characterization of the global two-phase solution satisfying a quadratic growth at a two-phase free boundary point and at infinity is given. In [4] it has been shown that if λ ^{+} and λ ^{ - } are Lipschitz, then, in two dimensions, the free boundary in a neighborhood of each branch point is the union of two C ^{1}-graphs. Also, in higher dimensions, the free boundary has finite (n - 1)-dimensional Hausdorff measure. Numerical approximation for the two-phase problem is discussed in [6].
In this article, by perturbation we mean the perturbation of the coefficients λ ^{+} and λ ^{ - } and the perturbation of the boundary values g. The case of the one phase obstacle problem has been studied in [7].
- 1< p < ∞ (see [8]). Define the map
- 1.
The stability of solution with respect to boundary value and coefficients is shown.
- 2.Let . By , we mean the solution of problem (1.2) with coefficients (λ ^{+} + εh _{1}) and (λ ^{-} + εh _{2}). If we Consider the map T : (λ ^{+}, λ ^{ - } ) ↦ u, for given parameters λ ^{+} and λ ^{+} and a fixed Dirichlet condition, then the Gateaux derivative of this map is characterized in . More precisely, it is shown in Theorem 3.4 that
- 3.(Theorem 3.5) Assuming that all free boundary points are one-phase points (points such that ∇u = 0), a stability result for the free boundary in the flavor of [7] is proved which says that
Were Γ^{±} = ∂ {±u(x) > 0} ∩ Ω. The function δ is constructed as a solution of certain Dirichlet problem in . The vector v _{1} stands for the exterior unit normal vector to .
The structure of article is organized as follows. In the next section, stability of solution with respect to boundary value and coefficients is studied. In Section 3, we prove that the map T is Lipschitz continuous (Theorem 3.1) and differentiable (Theorem 3.4).
Preliminary analysis and stability results
In this section, we state some lemmas which have been proved in the case of one-phase obstacle problem (see [9]). The following proposition shows the stability in L ^{∞}-norm. In what follows, we will denote by B _{ r } (x _{0}) the ball of radius r centered at x _{0} and, for simplicity, we use the notation B _{ r } = B _{ r } (0).
which is impossible. Therefore, .
□
Proof. Let ε > 0,; we will show that u ^{ ε } ≤ u. Set D = {x ∈ B _{1} : u ^{ ε } (x) > u(x)}. If u ^{ ε } ≤ 0, on D, then u < 0 on D and Δu = - λ ^{ - } ≤ - (λ ^{ - } - ε) ≤ Δu ^{ ε } : On the other hand, if u ^{ ε } > 0; then Δu ^{ ε } = λ ^{+} + ε ≥ Δu. Therefore, Δu ^{ ε } ≥ Δu and, by maximum principle, D = ∅.
□
Remark 1. An analysis similar to Lemma 2.2 shows that if the coefficients λ ^{±} be perturbed by ±ε, then |u ^{ ε } - u| ≤ Cε.
where u is the minimizer of (1.1) with data g and potential λ ^{±}.
which implies that c* ≥ c _{0}. Finally, from the equality of the minima c _{0} = c = c*, we also deduce the strong convergence of u _{ k } in H ^{1}(Ω). □
Perturbation formula for the free boundary
In this section, we prove the continuity and differentiability of the map T. The case of one-phase obstacle problem was studied by Stojanovic [7].
We first prove the following lemma:
In the last equation, we have used (1.16).
□
Thus we completed the proof of Theorem 3.1.
□
The proof of Theorem 3.4 uses the following theorem, proved by I. Blank in [9].
- (1)
When all the points are regular one-phase points (cf. Theorem 3.3).
- (2)
When all the points are two-phase points with |∇u| = 0 (branching points).
- (3)
When |∇u| is uniformly bounded from below (cf. Estimate 1.19).
Although we could not prove this theorem for the two-phase case in general, there are grounds, however, to suggest that it holds true in this case as well.
Now we shall prove that the map is differentiable in the following sense:
In Equation 1.20, denotes the (n - 1)-dimensional Hausdorff measure.
Now let x _{0} be a one-phase regular point for and where x _{ ε } has minimal distance to x _{0}.
which shows that at one-phase regular points.
We deduce that, satisfies (1.20).
□
Straightforward calculations show that if , then the set {x ∈ Ω: u(x) = 0} has a positive measure. In this setting, an interesting question is which conditions in higher dimensions will imply that the zero set has positive measure in B _{1}.
By Weiss [1], we know that the Hausdorff dimension of Γ = ∂{u > 0} ∪ ∂{u < 0} is less than or equal to n - 1 and by Edquist et al. [2] the regularity of the free boundary is C ^{1}. Let d Γ denote the measure ; the restriction of the (n - 1)-dimensional Hausdorff measure on the set Γ. Moreover, let v _{1} be the unit normal exterior to ∂{u > 0} and v _{2} be the unit normal to ∂{u < 0} exterior to {u < 0}.
Declarations
Acknowledgements
The author thanks Henrik Shahgholian for initiating this work and for useful suggestions. Moreover, the author would like to express his great sense of gratitude to the referees for carefully reading the article and coming with many helpful suggestions.
Authors’ Affiliations
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