Abstract
This chapter is dedicated to the regularity of the flat free boundaries. In particular, we will show how the improvement of flatness (proved in previous section) implies the C 1, α regularity of the free boundary (see Fig. 8.1). The results of this section are based on classical arguments and are well-known to the specialists in the field. The main result of the chapter is the following.
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This chapter is dedicated to the regularity of the flat free boundaries. In particular, we will show how the improvement of flatness (proved in previous section) implies the C 1, α regularity of the free boundary (see Fig. 8.1). The results of this section are based on classical arguments and are well-known to the specialists in the field. The main result of the chapter is the following.
There are dimensional constants ε > 0 and δ > 0 such that the following holds:
Suppose that \(u:B_1\to \mathbb {R}\) satisfies the following conditions:
-
(a)
u is a non-negative continuous function and a viscosity solution of (7.1) in B 1;
-
(b)
u is ε-flat in B 1 , that is,
$$\displaystyle \begin{aligned} (x_d-{\varepsilon})_+\le u(x)\le (x_d+{\varepsilon})_+\qquad \mathit{\text{for every}}\qquad x\in B_1. \end{aligned}$$
Then, there is α > 0 such that the free boundary ∂ Ω u is C 1, α regular in the cylinder \(B_{\delta }^{\prime }\times (-\delta ,\delta )\) . Precisely, there is a function \(g:B_\delta ^{\prime }\to (-\delta ,\delta )\) such that:
-
(i)
g is C 1, α regular in the (d − 1)-dimensional ball \(B_\delta ^{\prime }\subset \mathbb {R}^{d-1}\);
-
(ii)
the set \(\Omega _u\cap \big (B_\delta ^{\prime }\times (-\delta ,\delta )\big )\) is the supergraph of g, that is,
$$\displaystyle \begin{aligned} \displaystyle\Omega_u\cap \big(B_\delta^{\prime}\times (-\delta,\delta)\big)=\big\{x=(x',x_d)\in B_\delta^{\prime}\times (-\delta,\delta)\ :\ x_d>g(x')\big\}. \end{aligned}$$
Moreover, g (and so, ∂ Ω u ) is C 1, α regular, for any α ∈ (0, 1∕2).
FormalPara ProofThe existence of a function \(g: B_\delta ^{\prime }\subset \mathbb {R}^{d-1}\), which is C 1, α regular, for some α > 0, for which (ii) holds, is a consequence of:
-
Theorem 7.4, in which we show that the improvement of flatness (Condition 8.3) holds for viscosity solutions (with constants σ = C d κ);
-
Lemma 8.4, in which we show that the improvement of flatness implies the uniqueness of the blow-up limit and the decay of the blow-up sequence:
$$\displaystyle \begin{aligned} \|u_{r,x_0}-u_{x_0}\|{}_{L^\infty(B_1)}\le C_dr^\gamma\quad \text{for every}\quad r<{1}/{2}\quad \text{and every}\quad x_0\in B_{{1}/{2}}, \end{aligned} $$(8.1)where γ is such that κ γ = σ;
-
Proposition 8.6, in which we show that if (8.1) holds, then ∂ Ωu is C 1, α regular in B 1∕2, where \(\alpha =\frac {\gamma }{1+\gamma }\).
In particular, we notice that by choosing κ small enough, we can take γ as close to 1 (and so, α as close to 1∕2) as we want. □
As a consequence, we obtain the regularity of the free boundary for minimizers of \(\mathcal F_\Lambda \).
Let D be a bounded open set in \(\mathbb {R}^d\) and let \(u:D\to \mathbb {R}\) be a (non-negative) minimizer of \(\mathcal F_\Lambda \) in D. Then, every regular point x 0 ∈ Reg(∂ Ω u) ⊂ D has a neighborhood \(\mathcal U\) such that \(\partial \Omega _u\cap \mathcal U\) is a C 1, α regular manifold, for every α ∈ (0, 1∕2).
FormalPara ProofNotice that, up to replacing u(x) by v(x) = Λ−1∕2 u(x), we may assume that Λ = 1. By the definition of Reg(∂ Ωu) (see Sect. 6.4), there is a sequence r n → 0 such that the blow-up sequence \(u_{r_n,x_0}\) converges uniformly (in B 1) to a function \(u_0:\mathbb {R}^d\to \mathbb {R}\) of the form
for some unit vector \(\nu \in \mathbb {R}^d\). Then, by Proposition 6.2, for n large enough, we have
This means that \(u_{r_n,x_0}\) is 2ε-flat in B 1, that is,
Now, taking ε small enough and applying Theorem 7.4, Proposition 7.1 and Theorem 8.1, we get the claim. □
This chapter is organized as follows.
In Sect. 8.1, we prove that the improvement of flatness (Condition 8.3) implies the uniqueness of the blow-up limit and gives a (polynomial) rate of convergence of the blow-ups in L ∞(B 1).
In Sect. 8.2, we prove that the uniqueness of the blow-up limit and the polynomial rate of convergence of the blow-up sequence imply the regularity of the free boundary. We notice that the uniqueness of the blow-up limit and the rate of convergence of the blow-up sequence can be obtained also by different arguments, for instance, via an epiperimetric inequality. In fact, the result of this section can be used also in combination with Theorem 12.1, which is an alternative way to the regularity of the free boundary.
8.1 Improvement of Flatness, Uniqueness of the Blow-Up Limit and Rate of Convergence of the Blow-Up Sequence
Condition 8.3 (Improvement of Flatness)
Let \(u:B_1\to \mathbb {R}\) be a non-negative function. There are constants κ ∈ (0, 1), σ ∈ (0, 1), C 0 > 0 and ε 0 > 0 such that:
Lemma 8.4 (Uniqueness of the Blow-Up Limit)
Suppose that \(u:B_1\to \mathbb {R}\) is a continuous non-negative function satisfying Condition 8.3. Then, there are constant ε 1 > 0, γ > 0 and C 1 > 0 (depending on ε 0 , κ, σ and C 0 ) such that if
for some ν ∈ ∂B 1 , then for every x 0 ∈ ∂ Ω u ∩ B 1∕2 there is a unique unit vector
such that
where the function \(u_{x_0}\) is defined as
Precisely, we can take γ, ε 1 and C 1 as follows:
Proof
Let \({\varepsilon }_1=\frac {{\varepsilon }_0}4.\) Notice that if u is ε 1-flat in B 1, then
for every x 0 ∈ ∂ Ωu ∩ B 1∕2.
Let x 0 ∈ ∂ Ωu ∩ B 1∕2 be fixed,
By the improvement of flatness, there is a sequence of unit vectors ν n ∈ ∂B 1 such that
and
In particular, for every 1 ≤ n < m, we have
This implies that there is a vector ν ∞∈ ∂B 1 such that
Thus,
which implies that
Now, we set
Let r ≤ 1∕2 be arbitrary and let \(n\in \mathbb {N}\) be such that
Then, there is ρ ∈ (κ, 1] such that r = ρr n. Since \(u_{r_n,x_0}\) satisfies
we get that \(u_{r,x_0}=u_{\rho r_n,x_0}\) satisfies
which implies that
and finally gives that
Since κ γ = σ, we get that
from which, we deduce
which concludes the proof. □
8.2 Regularity of the One-Phase Free Boundaries
Condition 8.5
(Uniqueness of the Blow-Up Limit and Rate of Convergence of the Blow-Up Sequence) The function \(u:B_1\to \mathbb {R}\) satisfies this condition if it is non-negative and if there are constants C 1 > 0 and γ > 0 such that, for every x 0 ∈ ∂ Ω u ∩ B 1∕2 there is a unique function \(u_{x_0}:B_1\to \mathbb {R}\) such that:
-
(i)
there is \(\nu _{x_0}\in \partial B_1\) such that \(u_{x_0}(x)=(\nu _{x_0}\cdot x)_+\) for every x ∈ B 1;
-
(ii)
\(\|u_{r,x_0}-u_{x_0}\|{ }_{L^\infty (B_1)}\le C_1r^\gamma \) for every r ≤ 1∕2.
Proposition 8.6 (The Condition 8.5 Implies the Regularity of ∂ Ωu)
Let \(u:B_1\to \mathbb {R}\) be a non-negative function such that:
-
(a)
u is Lipschitz continuous on B 1 and \(L=\|\nabla u\|{ }_{L^\infty (B_1)}\);
-
(b)
u is non-degenerate in the sense that there is a constant η > 0 such that
$$\displaystyle \begin{aligned} \mathit{\text{if}}\quad y_0\in\overline\Omega_u\cap\partial B_{{1}/{2}}\,,\quad \mathit{\text{then}}\quad \|u\|{}_{L^\infty(B_r(y_0))}\ge \eta r\,,\quad \mathit{\text{for every}}\quad r\in(0,{1}/{2}). \end{aligned}$$ -
(c)
u satisfies Condition 8.5 for some γ > 0 and C 1 > 0.
Then, there is ρ > 0 such that ∂ Ω u is a C 1, α manifold in B ρ , where \(\alpha :=\frac {\gamma }{1+\gamma }\).
Precisely, there are ρ > 0 and a C 1, α -regular function \(g:B^{\prime }_\rho \to (-\rho ,\rho )\) such that, up to a rotation of the coordinate system of \(\mathbb {R}^d\) , we have
Lemma 8.7 (Flatness of the Free Boundary ∂ Ωu)
Let \(u:B_1\to \mathbb {R}\) be a non-negative function such that
-
(a)
u satisfies the Condition 8.5 with constants C 1 and γ.
-
(b)
u is non-degenerate, that is, there is a constant η > 0 such that
$$\displaystyle \begin{aligned} \mathit{\text{if}}\quad y_0\in\overline\Omega_u\cap\partial B_{{1}/{2}}\,,\quad \mathit{\text{then}}\quad \|u\|{}_{L^\infty(B_r(y_0))}\ge \eta r\,,\quad \mathit{\text{for every}}\quad r\in(0,{1}/{2}). \end{aligned}$$
Then, there are constants C > 0 and r 0 > 0 such that, for every x 0 ∈ ∂ Ω u ∩ B 1∕2 , we have
for every r ∈ (0, r 0), where \(\Omega _{x_0,r}:=\{u_{x_0,r}>0\}\).
Proof
In order to prove the first part of (8.2), we notice that
implies that
This gives the first inclusion of (8.2) for any constant C ≥ C 1. In order to prove the second inclusion in (8.2), we suppose that there is a point y ∈ B 1 such that
This implies that \(\tilde y:=\frac {y}2\in B_{{1}/{2}}\) is such that
The non-degeneracy of u now implies that
Notice that \(u_{x_0}=0\) on \(B_\rho (\tilde y)\). On the other hand, choosing r 0 such that
we get that ρ ≤ 1∕2 and so \(B_\rho (\tilde y)\subset B_1\). Thus, we have that
which is a contradiction, if we choose
which concludes the proof by taking
□
Lemma 8.8 (Oscillation of ν)
Let \(u:B_1\to \mathbb {R}\) be a Lipschitz continuous function and let \(L=\|\nabla u\|{ }_{L^\infty (B_1)}\) . Suppose that u satisfies the Condition 8.5 with the constants C 1 and γ. Then, there are constants R ∈ (0, 1), α and C such that
Precisely, one can take
Proof
Let \(\alpha :=\frac {\gamma }{1+\gamma }\). Let x 0, y 0 ∈ B R ∩ ∂ Ωu and r := |x 0 − y 0|1−α. Then, for every x ∈ B 1, we have
which gives that
On the other hand, Condition 8.5 gives that
We notice that in order to apply Condition 8.5 we need that r ≤ 1∕2 and R ≤ 1∕2. We choose R such that (2R)1−α ≤ 1∕2. Thus, by the triangular inequality and the fact that r γ = |x 0 − y 0|α, we obtain
The conclusion now follows by a general argument. Indeed, for any \(v_1,v_2 \in \mathbb {R}^d\), we have
which implies that
Applying the above estimate to \(v_1=\nu _{x_0}\) and \(v_2=\nu _{y_0}\), we get (8.3). □
Proof of Proposition 8.6
We first notice that, for every ε > 0, there exists R > 0 such that, for x 0 ∈ ∂ Ωu ∩ B R we have
where for a vector ν ∈ ∂B 1, we denote by \(\mathcal C_\varepsilon ^+(x_0,\nu )\) and \(\mathcal C_\varepsilon ^-(x_0,\nu )\) the cones
(see Fig. 8.2).
Indeed, the flatness estimate (8.2) implies (8.4) by taking R such that CR γ ≤ ε, where C and γ are the constants from Lemma 8.7.
Let ν 0 be the normal vector at the origin 0 ∈ ∂ Ωu. Without loss of generality we can suppose that ν 0 = e d. In particular, if u 0(x) = (x ⋅ ν 0)+ is the blow-up limit in zero, then
Let ε ∈ (0, 1) and R > 0 be as in (8.4) and set
Let \(x'\in B_\rho ^{\prime }\). Then, by (8.4), we have:
-
the vertical section
$$\displaystyle \begin{aligned} \mathcal S_+^{x'}:=\{(x',t)\in B_R \ :\ u(x',t)>0\} \end{aligned}$$contains the segment
$$\displaystyle \begin{aligned} \{(x',t)\in B_R \ :\ t>{\varepsilon} R\}; \end{aligned}$$ -
the closed set
$$\displaystyle \begin{aligned} \mathcal S_0^{x'}:=\{(x',t)\in B_R \ :\ u(x',t)=0\} \end{aligned}$$contains the segment
$$\displaystyle \begin{aligned} \{(x',t)\in B_R \ :\ t<-{\varepsilon} R\}. \end{aligned}$$
This implies that the function
is well defined for \(x'\in B_\rho ^{\prime }\) (see Fig. 8.3).
Let δ ≤ ρ. Let \(x_0^{\prime }\in B_\delta ^{\prime }\) and let \(t_0:=g(x_0^{\prime })\). By definition, we have
Moreover, by construction, we have
Thus,
We next claim that, for δ small enough, we have that
Indeed, applying (8.4) for the point x 0, we have
so, it is sufficient to prove that
Let \(x\in \mathcal C_{2\varepsilon }^\pm (x_0,e_d)\). Then,
where:
-
for the first inequality we used the definition of \(\mathcal C_{2\varepsilon }^\pm (x_0,e_d)\) and the following estimate, which is a consequence of Lemma 8.8:
$$\displaystyle \begin{aligned} |\nu_{x_0}-e_d|\le C|x_0|{}^\alpha \le C \big(\sqrt{ 2}\,\delta\big)^\alpha\,; \end{aligned}$$ -
for the second in equality, we choose δ such that \(C \big (\sqrt { 2}\,\delta \big )^\alpha \le \varepsilon \).
This proves (8.5). As a consequence, we obtain that the sections \(\mathcal S_+^{x'}\) and \(\mathcal S_0^{x'}\) are segments:
and so, the free boundary is precisely the graph of g, that is,
We next prove that the function \(g:B_\delta ^{\prime }\to \mathbb {R}\) is Lipschitz continuous on \(B_\delta ^{\prime }\). Also this follows by the uniform cone condition (8.5). Indeed, let
Since \(x_1\notin \mathcal C^+_{2\varepsilon }(x_2,e_d)\), we have that
Analogously, \(x_2\notin \mathcal C^+_{2\varepsilon }(x_1,e_d)\) implies that
and the two estimates give
and finally, choosing ε ≤ 1∕4, we get
which concludes the proof of the Lipschitz continuity of g.
We will next show that g is differentiable. Indeed, let \(x_0^{\prime }\in B_\delta ^{\prime }\). Now, the improvement of flatness at \(x_0=(x_0^{\prime },g(x_0^{\prime }))\) implies that
for any x = (x′, g(x′)) with \(x'\in B_\delta ^{\prime }\). For the sake of simplicity, we set \(\nu :=\nu _{x_0}\) and \(\nu =(\nu ',\nu _d)\in \mathbb {R}^{d-1}\times \mathbb {R}\). Since
we get that
This implies that g is differentiable at \(x_0^{\prime }\) and that \(\nabla g(x_0^{\prime })=\frac {\nu '}{\nu _d}\). Finally, the α-Hölder continuity of \(\nabla g:B_\delta ^{\prime }\to \mathbb {R}^{d-1}\) follows by the γ-Hölder continuity of the map x↦ν x. Indeed, for any \(x',y'\in B_\delta ^{\prime }\), x = (x′, g(x′)) and y = (y′, g(y′)) we have that
which implies the Hölder continuity of all the components of the map \(B_\delta ^{\prime }\ni x\mapsto \nu _x\in \mathbb {R}^d\) and thus, of the gradient ∇g. This concludes the proof of Proposition 8.6. □
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Velichkov, B. (2023). Regularity of the Flat Free Boundaries. In: Regularity of the One-phase Free Boundaries. Lecture Notes of the Unione Matematica Italiana, vol 28. Springer, Cham. https://doi.org/10.1007/978-3-031-13238-4_8
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