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The one-phase bifurcation for the p-Laplacian

A bifurcation about the uniqueness of a solution of a singularly perturbed free boundary problem of phase transition associated with the p-Laplacian, subject to given boundary condition is proved in this paper. We show this phenomenon by proving the existence of a third solution through the Mountain Pass Lemma when the boundary data decreases below a threshold. In the second part, we prove the convergence of an evolution to stable solutions, and show the Mountain Pass solution is unstable in this sense.


Introduction
In this paper, one considers the phase transition problem of minimizing the p-functional which is a singular perturbation of the one-phase problem of minimizing the functional associated with the p-Laplacian where Γ ε (s) = Γ( s ε ) for ε > 0 and for a C ∞ function Γ defined by Γ(s) = 0 if s ≤ 0 1 if s ≥ 1, and 0 ≤ Γ(s) ≤ 1 for 0 < s < 1, and Q ∈ W 2,2 (Ω) is a positive continuous function on Ω such that inf Ω Q(x) > 0. Let β ε (s) = Γ ′ ε (s) = 1 ε β( s ε ) with β = Γ ′ . The domain Ω is always assumed to be smooth in this paper for convenience. As in the following we will fix the value of ε unless we specifically examine the influence of the value of ε on the critical boundary data and will not use the notation J p for a different purpose, we are going to abuse the notation by using J p for the functional J p,ε from now on.
In this paper, we take on the task of establishing in the general case when p = 2 the results proved in [CW] for the Laplacian when p = 2. The main difficulty in this generalization lies in the lack of sufficient regularity and the singular-degenerate nature of the p-Laplacian when p = 2. A well-known fact about p-harmonic functions is the optimal regularity generally possessed by them is C 1,α (e. g. [E] and [Le]). Thus we need to employ more techniques associated with the p-Laplacian, and in a case or two we have to make our conclusion slightly weaker. Nevertheless, we follow the overall scheme of approach used in [CW]. In the second section, we prove the bifurcation phenomenon through the Mountain Pass Theorem. In the third section, we establish a parabolic comparison principle. In the last section, we show the convergence of an evolution to a stable steady state in accordance with respective initial data.

A Third Solution
We first prove if the boundary data is small enough, then the minimizer is nontrivial. More precisely, let u 0 be the trivial solution of (1.3) and (1.4), being p-harmonic in the weak sense, and u 2 be a minimizer of the p-functional (1.1), and set σ M = max ∂Ω σ(x) and σ m = min ∂Ω σ(x).
If σ M is small enough, then u 0 = u 2 .
In fact, we pick u ∈ W 1,p (Ω) so that where Ω δ = {x ∈ Ω : dist(x, ∂Ω) > δ} and δ > 0 is a small constant independent of ε and σ so that Ω δ Q(x) dx has a positive lower bound which is also independent of ε and σ. Using an approximating domain if necessary, we may assume Ω δ possesses a smooth boundary. Clearly, It is well-known that So, for all small ε > 0, Let B denote the Banach space W 1,p 0 (Ω) we will work with. For every v ∈ B, we write u = v + u 0 and adopt the norm v B = Ω |∇v| p 1 p = Ω |∇u − ∇u 0 | p 1 p . We define the functional Set v 2 = u 2 − u 0 . Clearly, I[0] = 0 and I[v 2 ] ≤ 0 on account of the definition of u 2 as a minimizer of J p . If I[v 2 ] < 0 which is the case if σ M is small, we will apply the Mountain Pass Lemma to prove there exists a critical point of the functional I which is a weak solution of the problem (1.3) and (1.4). The Fréchet derivative of I at v ∈ B is given by which is obviously in the dual space B * of B in light of the Hölder's inequality. Equivalently We see that I ′ is Lipschitz continuous on any bounded subset of B with Lipschitz constant depending on ε, p, and sup Q. In fact, for any v, w, and ϕ ∈ B, In addition, Therefore I ′ is Lipschitz continuous on bounded subsets of B.
We note that f ∈ B * if and only if there exist f 0 , f 1 , f 2 , ..., f n ∈ L p ′ (Ω), where Next we justify the Palais-Smale condition on the functional I. Suppose {v k } ⊂ B is a Palais-Smale sequence in the sense that (Ω) to B * is compact due to the fact W 1,p 0 (Ω) ⊂⊂ L p (Ω) ⊂ B * following from the Rellich-Kondrachov Compactness Theorem. Then there exists f ∈ L p (Ω) ⊂ B * such that for a subsequence, still denoted by {v k }, of {v k }, it holds that Recall that As a consequence, Obviously, that {I[v k ]} is bounded implies that a subsequence of {v k }, still denoted by {v k } by abusing the notation without confusion, converges weakly in B = W 1,p 0 (Ω). In particular,

Then by setting
]. In particular, if p = 2, {v k } is a Cauchy sequence in W 1,2 0 (Ω) and hence converges. We will apply the following elementary inequalities associated with the p-Laplacian, [L], to the general case p = 2: (2.9) and < |b| p−2 b − |a| p−2 a, b − a >≥ 2 2−p |b − a| p , p ≥ 2. (2.10) We assume first 1 < p < 2. Let K = 2pM + 2J p [u 0 ]. Then the first elementary inequality (2.9) implies Therefore, {v k } is a Cauchy sequence in B and hence converges. Suppose p > 2. The second elementary inequality (2.10) implies which in turn implies {v k } is a Cauchy sequence in B and hence converges, on account of (2.8). The Palais-Smale condition is verified for 1 < p < ∞ for the functional I on the Banach space W 1,p 0 (Ω). Before we continue the main proof, let us state an elementary result closely related to the p-Laplacian, which follows readily from the Fundamental Theorem of Calculus.
Lemma 2.1 For any a and b ∈ R n , it holds We are now in a position to show there is a closed mountain ridge around the origin of the Banach space B that separates v 2 from the origin with the energy I as the elevation function, which is the content of the following lemma.
Lemma 2.2 For all small ε > 0 such that Cε ≤ 1 2 σ m for a large universal constant C, there exist positive constants δ and a independent of ε, such that, for every Proof. It suffices to prove If not, one may pick z ∈ Λ. Let AC([a, b], S) be the set of absolutely continuous As shown in [CW], there is a minimizing path γ x 0 for the distance d(x 0 , z).
Suppose the domain Ω is convex or star-like about z. For any x 0 ∈ ∂Ω, let γ = γ x 0 be a minimizing path of d(x 0 , z). Then it is clear that γ is a straight line segment and γ(t) = z for t ∈ [0, 1). Furthermore, for any two distinct points x 1 and x 2 ∈ ∂Ω, the corresponding minimizing paths do not intersect in Ω\{z}. For this reason, we can carry out the following computation. Clearly v(x 0 ) = 0 and v(γ (1) ( 2.13) For each x 0 ∈ ∂Ω, let e(x 0 ) be the unit vector in the direction of x 0 − z and ν(x 0 ) the outer normal to ∂Ω at x 0 . Then ν(x 0 ) · e(x 0 ) > 0 everywhere on ∂Ω. Hence the above inequality (2.13) implies where the second and third inequalities are due to the application of the Hölder's inequality, and the constant C depends on n and p. The second equality follows from the two representation formulas If we take δ sufficiently small and independent of ε in the preceding inequality the measure |{u > 0}| of the positive domain would be greater than that of Ω, which is impossible, provided that for a constant C which depends on n, p and |Ω|, but not on z or v. Hence Λ must be empty. So we need to justify the inequality (2.14). To fulfil that condition, for e = e(x 0 ), where B is the unit ball about z and dσ(e) is the surface area element on the unit sphere ∂B which is invariant under rotation and reflection. Clearly, Consequently, in order to prove (2.14), one needs only to prove ∂B l 2 (e) dσ(e) ≥ C(n, p, |Ω|).
(2.15) Next, we show the integral on the left-hand-side of (2.15) is minimal if Ω is a ball while its measure is kept unchanged. In fact, this is almost obvious if one notices the following fact. Let π be any hyperplane passing through z, and x 1 and x 2 be the points on ∂Ω which lie on a line perpendicular to π. Let x * 1 and x * 2 be the points on the boundary ∂Ω π , where Ω π is the symmetrized image of Ω about the hyperplane π, which lie on the line x 1 x 2 . Let 2a = |x 1 x 2 | = |x * 1 x * 2 | and d be the distance from z to the line x 1 x 2 . Then for some t in −a ≤ t ≤ a, it holds that As a consequence, if Ω * is the symmetrized ball with measure equal to that of Ω, then ∂B l 2 (e) dσ(e) ≥ ∂B (l * (e)) 2 dσ(e) = C(n, |Ω|), where l * is the length from z to a point on the boundary ∂Ω * which is constant. This finishes the proof of the fact that Λ = ∅.
In case the domain Ω is not convex, the minimizing paths of d(x 1 , z) and d(x 2 , z) for distinct x 1 , x 2 ∈ ∂Ω may partially coincide. We form the set DA(∂Ω) of the points x 0 on ∂Ω so that a minimizing path γ of d(x 0 , z) satisfies γ(t) ∈ Ω\{z} for t ∈ (0, 1). We call a point in DA(∂Ω) a directly accessible boundary point. Let Ω 1 be the union of these minimizing paths for the directly accessible boundary points. It is not difficult to see that |Ω 1 | > 0 and hence H n−1 (DA(∂Ω)) > 0. Then we may apply the above computation to the star-like domain Ω 1 with minimal modification. We have (2.16) For small enough δ, this raises a contradiction |Ω| > |Ω|. So Λ = ∅.

Consequently,
where the last inequality is a consequence of the elementary inequality The verified Palais-Smale condition and the preceding lemma allow us to apply the Mountain Pass Theorem as stated, for example, in [J] to conclude that there is a v 1 ∈ B such that I[v 1 ] = c, and for any ϕ ∈ B = W 1,p 0 (Ω), where u 1 = v 1 + u 0 . So u 1 is a weak solution of the problem (1.3) and (1.4). In essence, the Mountain Pass Theorem is a way to produce a saddle point solution. Therefore, in general, u 1 tends to be an unstable solution in contrast to the stable solutions u 0 and u 2 .
In this section, we have proved the following theorem.

A Comparison Principle for Evolution
In this section, we prove a comparison theorem for the following evolution problem.
The comparison principle for weak sub-and super-solutions is stated as follows.
Uniqueness of a weak solution of (3.1) follows from the comparison principle, Theorem 3.1, immediately.
Define, for j = 1, 2, v j (x, t) = e −λt w j (x, t), where the constant λ > 2K. Then w j (x, t) = e λt v j (x, t), and it is clear that w 1 ≤ w 2 in Ω × R T is equivalent to v 1 ≤ v 2 in Ω × R T . In addition, for any ϕ ∈ T + , the following inequalities hold: Consequently, it holds for any ϕ ∈ T + We take ϕ = (v 1 − v 2 ) + = max{v 1 − v 2 , 0} as the test function, since it vanishes on the boundary of Ω × R T . Then On the other hand, As a consequence, it holds that We call into play two elementary inequalities ( [L]) associated with the p-Laplacian: (1 ≤ p ≤ 2), and < |b| p−2 b − |a| p−2 a, b − a >≥ 2 2−p |b − a| p (p ≥ 2) for any a, b ∈ R n .
By applying them with b = ∇v 1 and a = ∇v 2 in the preceding inequalities, we obtain One can easily see in either case the respective inequality is true only if the measure of the set {v 1 > v 2 } is zero. The proof is complete.
In the next lemma, we show the strict inequality on the boundary data can be relaxed to a non-strict one.
Proof. For any δ > 0, takeδ > 0 such thatδ ≤ δ 4K and definẽ Thenw 1 < w 1 ≤ w 2 on ∂ p (Ω × R n ), and for any ϕ ∈ T + , the following holds: The preceding lemma impliesw 1 ≤ w 2 in Ω × R T for small T and for any small δ > 0, and whence the conclusion of this lemma. Now the parabolic comparison theorem (3.1) follows from the preceding lemma quite easily as shown by the following argument: Let T 0 > 0 be any small value of T in the preceding lemma so that the conclusion of the preceding lemma holds. Then w 1 ≤ w 2 on Ω × (0, T 0 ). In particular, w 1 ≤ w 2 on ∂ p (Ω × (T 0 , 2T 0 )). The preceding lemma may be applied again to conclude that w 1 ≤ w 2 on Ω × (T 0 , 2T 0 ). And so on. This recursion allows us to conclude that w 1 ≤ w 2 on Ω × R T .

Convergence of Evolution
Define S to be the set of weak solutions of the stationary problem (1.3) and (1.4). The p-harmonic function u 0 is the maximum element in S, while u 2 denotes the least solution which may be constructed as the infimum of super-solutions. We also use the term nonminimal solution with the same definition in [CW]. That is, u a non-minimal solution of the problem (1.3) and (1.4) if it is a viscosity solution but not a local minimizer in the sense that for any δ > 0, there exists v in the admissible set of the functional J p with v = σ on ∂Ω such that v − u L ∞ < δ, and J p (v) < J p (u).
In this section, we consider the evolutionary problem and will apply the parabolic comparison principle (3.1) proved in Section 3 to prove the following convergence of evolution theorem. One just notes that the parabolic problem (3.1) includes the above problem (4.1) as a special case so that the comparison principle (3.1) applies in this case.
Theorem 4.1 If the initial data v 0 falls into any of the categories specified below, the corresponding conclusion of convergence holds.
Proof. We first take care of case 4. We may take new initial data a smooth functionṽ 0 so that D 2ṽ 0 < −KI and |∇ṽ 0 | ≥ δ > 0 onΩ. According to the parabolic comparison principle (3.1), it suffices to prove the solutionw generated by the initial dataṽ 0 converges locally uniformly to u 0 if we also takeṽ 0 large than v 0 , which can easily be done. So we use v 0 and w for the new functionsṽ 0 andw without any confusion.
For any V ⊂⊂ Ω and any nonnegative function ϕ which is independent of the time variable t and supported in V , it holds that Mϕ for some M = M(n, p, K, δ) > 0.
The Hölder continuity of ∇w up to t = 0 as stated in [DiB], then implies V |∇w| p−2 ∇w · ∇ϕ ≥ M 2 V ϕ for any small t in (0, t 0 ), and any nonnegative function ϕ which is independent of t, supported in V and subject to the condition for a fixed constant A > 0 and some t 0 > 0 dependent on A. Then the sub-solution condition on w for any small t 2 > t 1 in (0, t 0 ), and any nonnegative function ϕ which is independent of t, supported in V and subject to (4.2). In particular, V wϕ t 2 t 1 ≤ 0 for any nonnegative function ϕ independent of t, supported in V and subject to (4.2). So w(x, t 2 ) ≤ w(x, t 1 ) for any x ∈ Ω and 0 ≤ t 1 ≤ t 2 . Then the parabolic comparison principle readily implies w is decreasing in t for t in [0, ∞). Therefore w(x, t) → u ∞ (x) locally uniformly as t → ∞ and hence u ∞ is a solution of (1.3) and (1.4). Furthermore, the parabolic comparison principle also implies w(x, t) ≥ u 0 (x) at any time t > 0. Consequently, u ∞ = u 0 as u 0 is the greatest solution of (1.3) and (1.4).
Next, we briefly explain the proof for case 1. We may take a new smooth initial datã v 0 such thatṽ 0 is very large negative, D 2ṽ 0 ≥ KI and |∇ṽ 0 | ≥ δ onΩ for large constant K > 0 and constant δ > 0. It suffices to prove the solutionw generated by the initial datã v 0 converges to u 2 locally uniformly onΩ as t → ∞. Following a computation exactly parallel to that in case 4, we can prove w is increasing in t in [0, ∞). So w converges locally uniformly to a solution u ∞ of (1.3) and (1.4). As u ∞ ≤ u 2 and u 2 is the least solution of (1.3) and (1.4), we conclude u ∞ = u 2 .
In case 2, we may replace v 0 by a strict super-solution of △ p v − Qβ ε (v) = 0 in Ω between u 2 andū 2 , by employing the fact that u 2 is the infimum of super-solutions of (1.3) and (1.4). Using v 0 as the initial data, we obtain a solution w(x, t) of (4.1). Then one argues as in case 4 that for any V ⊂⊂ Ω, there exist constants A > 0 and t 0 > 0 such that for t 1 < t 2 with t 1 , t 2 ∈ [0, t 0 ), V wϕ | t 2 t 1 ≤ 0 for any nonnegative function ϕ independent of t, supported in V and subject to the condition V |∇ϕ| V ϕ ≤ A. As a consequence, w(x, t 1 ) ≥ w(x, t 2 ) (x ∈ Ω). Then the parabolic comparison principle implies w is decreasing in t over [0, +∞). Therefore w(x, t) converges locally uniformly to some function u ∞ as t → ∞ which solves (1.3) and (1.4). Clearly u 2 (x) ≤ w(x, t) ≤ū 2 (x) Therefore J ε,p (w ε (·, t) ≤ J ε,p (v 0 ), which in turn implies J p (w(·, t) ≤ J p (v 0 ) < J p (u 1 ).
In conclusion, w does not converge to u 1 as t → ∞.