The Weiss Monotonicity Formula and Its Consequences

Abstract

This chapter is dedicated to the monotonicity formula for the boundary adjusted energy introduced by Weiss in [52]. Precisely, for every Λ ≥ 0 and every u ∈ H ¹(B ₁).

This chapter is dedicated to the monotonicity formula for the boundary adjusted energy introduced by Weiss in [52]. Precisely, for every Λ ≥ 0 and every u ∈ H ¹(B ₁) we define

$$\displaystyle \begin{aligned} W_\Lambda(u):=\int_{B_1}|\nabla u|{}^2\,dx-\int_{\partial B_1}u^2\,d\mathcal{H}^{d-1}+\Lambda |\Omega_u\cap B_1|, \end{aligned}$$

where we recall that Ω_u := {u > 0}. In particular, we have

$$\displaystyle \begin{aligned} W_0(u)\!=\!\int_{B_1}|\nabla u|{}^2\,dx-\int_{\partial B_1}u^2\,d\mathcal{H}^{d-1}\qquad \text{and}\qquad W_\Lambda(u)=W_0(u)+\Lambda |\Omega_u\cap B_1|. \end{aligned}$$

This chapter is organized as follows:

In Sect. 9.1 we prove several preliminary results for the Weiss’ boundary adjusted energy, which hold for a general Sobolev function u defined on an open set $D\subset \mathbb {R}^d$. In particular, in Lemma 9.1 we prove that the function $(x_0,r)\mapsto W_\Lambda (u_{x_0,r})$ is continuous (where it is defined), where we recall that $u_{x_0,r}(x):=\frac 1ru(x_0+rx)$; in Lemma 9.2, we compute the derivative of $W_\Lambda (u_{x_0,r})$ with respect to r and we prove that

$$\displaystyle \begin{aligned} \frac{\partial }{\partial r}W_\Lambda(u_{x_0,r})=\frac{d}r\big(W_\Lambda(z_{x_0,r})-W_\Lambda(u_{x_0,r})\big)+\frac 1r\mathcal D(u_{x_0,r}), \end{aligned}$$

where $z_{x_0,r}$ is the one-homogeneous extension defined in Lemma 9.2 while the deviation $\mathcal D(u_{x_0,r})$ is defined as

$$\displaystyle \begin{aligned} \mathcal D(u_{x_0,r}):=\int_{\partial B_1}|x\cdot\nabla u_{x_0,r}-u_{x_0,r}|{}^2\,d\mathcal{H}^{d-1}, \end{aligned}$$

and measures at what extent the function is not one-homogeneous (see Lemma 9.3) and controls the oscillation of u from scale to scale, which is measured by the norm $\|u_{x_0,r}-u_{x_0,s}\|{ }_{L^2(\partial B_1)}$. Finally, in Proposition 9.4, as a direct consequence of the Weiss formula (Lemma 9.2), we obtain that, if u is a (local) minimizer of $\mathcal F_\Lambda$ in D, then the Weiss energy $W(u_{x_0,r})$ is monotone increasing in r.

In Sect. 9.2 we introduce the notion of stationary free boundary, that is, the free boundary ∂ Ω_u ∩ D of a function $u:D\to \mathbb {R}$, which is stationary for the functional $\mathcal F_\Lambda$ with respect to internal perturbations with vector fields compactly supported in D. In Lemma 9.5, we compute the variation of the energy $\mathcal F_\Lambda$ with respect to a compactly supported vector field $\xi \in C^\infty _c(D;\mathbb {R}^d)$, which is simply defined as

$$\displaystyle \begin{aligned} \delta\mathcal F_\Lambda(u,D)[\xi]:=\frac{\partial}{\partial t}\Big|{}_{t=0}\mathcal F_\Lambda(u_t,D), \end{aligned}$$

where $u_t:D\to \mathbb {R}$ is defined through the identity u _t(x + tξ(x)) = u(x). We say that a function is stationary (see Definition 9.7), if the first variation is zero with respect to any vector field, that is, if

$$\displaystyle \begin{aligned} \delta\mathcal F_\Lambda(u,D)[\xi]=0\qquad \text{for every}\qquad \xi\in C^\infty_c(D;\mathbb{R}^d). \end{aligned}$$

In Lemma 9.6 we show that if u is a minimizer of $\mathcal F_\Lambda$ in D, then it is stationary in D. Then, in Lemma 9.8, we prove that every stationary function satisfies an equipartition-of-the-energy identity; in Lemma 9.9, we prove that the equipartition of the energy is sufficient for the monotonicity of the Weiss energy. In particular, the monotonicity formula holds for stationary free boundaries. The result of Sect. 9.2 are fundamental for the proof of Theorem 1.9, but we do not need them in the proof of Theorem 1.4, where we can use directly Proposition 9.4.

In Sect. 9.3 we give the sufficient conditions for the homogeneity of the blow-up limits of a function $u:D\to \mathbb {R}$ (Lemma 9.10). We then apply this result to minimizers of $\mathcal F_\Lambda$ (Proposition 9.12), but we will also use it in the context of Theorem 1.9. This is why the exposition contains the intermediate Lemma 9.11.

In Sect. 9.4 we prove that the only one-homogeneous global solutions in dimension two are the half-plane solutions (see Proposition 9.13). In particular, this means that d ^∗≥ 3.

In Sect. 9.5 we give another proof of the fact that the minimizers of $\mathcal F_\Lambda$ are viscosity solutions (Proposition 7.1). Our main result is Proposition 9.18, which applies to minimizers of $\mathcal F_\Lambda$, but also in the context of Theorem 1.9.

Finally, in Sect. 9.6, we use the Weiss monotonicity formula to relate the energy density

$$\displaystyle \begin{aligned} \lim_{r\to 0}W(u_{x_0,r}), \end{aligned}$$

of a minimizer u of $\mathcal F_\Lambda$, to the Lebesgue density

$$\displaystyle \begin{aligned} \lim_{r\to0}\frac{|\Omega_u\cap B_r(x_0)|}{|B_r|}, \end{aligned}$$

of the set Ω_u, at every point of the free boundary x ₀ ∈ ∂ Ω_u (Lemma 9.20). Moreover, we characterize the regular part of the free boundary Reg(∂ Ω_u) in terms of the energy and the Lebesgue densities (Lemma 9.22). We will not use the results from Sect. 9.6 in the proofs of Theorems 1.2, 1.4, 1.9 and 1.10, but they remain an interesting application of the monotonicity formula and the homogeneity of the blow-up limits and were used, for instance, in the analysis of the vectorial free boundaries (see [41]).

9.1 The Weiss Boundary Adjusted Energy

Let u ∈ H ¹(B _r(x ₀)) be a given function on the ball $B_r(x_0)\subset \mathbb {R}^d$ and consider the rescaling

$$\displaystyle \begin{aligned} u_{r,x_0}\in H^1(B_1)\qquad \text{where}\qquad u_{r,x_0}(x)=\frac 1r u(x_0+rx). \end{aligned}$$

We notice that the different terms of the energy W _Λ have the following scaling properties:

$$\displaystyle \begin{aligned} \int_{B_1}|\nabla u_{r,x_0}|{}^2\,dx & =\frac 1{r^d}\int_{B_r(x_0)}|\nabla u|{}^2\,dx\ ,\\ \int_{\partial B_1}u_{r,x_0}^2\,d\mathcal{H}^{d-1} & =\frac 1{r^{d+1}}\int_{\partial B_r(x_0)} u^2\,d\mathcal{H}^{d-1} \end{aligned} $$

$$\displaystyle \begin{aligned} \text{and}\qquad \big|\{u_{x_0,r}>0\}\cap B_1\big|=\frac 1{r^d}\big|\{u>0\}\cap B_r(x_0)\big|\ . \end{aligned}$$

Thus, we have

$$\displaystyle \begin{aligned} W_\Lambda(u_{x_0,r})=\frac 1{r^d}\int_{B_r(x_0)}|\nabla u|{}^2\,dx-\frac 1{r^{d+1}}\int_{\partial B_r(x_0)} u^2\,d\mathcal{H}^{d-1}+\frac{\Lambda}{r^d}\big|\{u>0\}\cap B_r(x_0)\big|. \end{aligned}$$

In particular, since u is a Sobolev function, the function $(x_0,r)\mapsto W_\Lambda (u_{x_0,r})$ is continuous, where it is defined. We give the precise statement in the following lemma.

Lemma 9.1 (Continuity of the Function $(x_0,r)\mapsto W_\Lambda (u_{x_0,r})$)

Let D be a bounded open set in $\mathbb {R}^d$ and let u ∈ H ¹(D). Let δ > 0 and let D _δ be the set

$$\displaystyle \begin{aligned} D_\delta:=\big\{x\in D\ :\ \mathrm{dist}\,(x,\partial D)<\delta\big\}. \end{aligned}$$

Then, the function

$$\displaystyle \begin{aligned} \Phi_u: D_\delta\times (0,\delta)\to \mathbb{R}\ ,\qquad \Phi_u(x_0,r):=W_\Lambda(u_{x_0,r}), \end{aligned}$$

is continuous.

Proof

The continuity of the terms

$$\displaystyle \begin{aligned} (x_0,r)\mapsto \frac 1{r^d}\int_{B_r(x_0)}|\nabla u|{}^2\,dx\qquad \text{and}\qquad (x_0,r)\mapsto \frac 1{r^d}\big|\{u>0\}\cap B_r(x_0)\big|, \end{aligned}$$

follows by the fact that if $f:D\to \mathbb {R}$ is a function in L ¹(D), then the map

$$\displaystyle \begin{aligned} (x_0,r)\mapsto \int_{B_r(x_0)}f(x)\,dx, \end{aligned}$$

is continuous, which in turn follows by the dominated convergence theorem. In order to prove the continuity of the function

$$\displaystyle \begin{aligned} (x_0,r)\mapsto \frac 1{r^{d+1}}\int_{\partial B_r(x_0)} u^2\,d\mathcal{H}^{d-1}, \end{aligned}$$

we consider the sequence to (x _n, r _n) ∈ D _δ × (0, δ) converging to a point (x ₀, r ₀) ∈ D _δ × (0, δ). We first notice that reasoning as above, we have

$$\displaystyle \begin{aligned} \lim_{n\to\infty}\|\nabla u_{x_n,r_n}\|{}_{L^2(B_1)} & =\|\nabla u_{x_0,r_0}\|{}_{L^2(B_1)}\quad \text{ and } \\ \lim_{n\to\infty}\|u_{x_n,r_n}\|{}_{L^2(B_1)} & =\|u_{x_0,r_0}\|{}_{L^2(B_1)}. \end{aligned} $$

Next, we notice that $u_{x_n,r_n}$ converges weakly in H ¹(B ₁) to $u_{x_0,r_0}$. In fact, for any $\phi \in C^\infty _c(B_1)$ we have

$$\displaystyle \begin{aligned} \lim_{n\to\infty}\int_{B_1}\nabla\phi\cdot\nabla u_{x_n,r_n}\,dx & =\lim_{n\to\infty}\int_{B_1}\nabla\phi (x)\cdot\nabla u (x_n+r_n x)\,dx\\ & =\lim_{n\to\infty}\int_{B_1}\nabla\phi \Big(\frac{y-x_n}{r_n}\Big)\cdot\nabla u (y)\,dy\\ & = \int_{B_1}\nabla\phi \Big(\frac{y-x_0}{r_0}\Big)\cdot\nabla u (y)\,dy\\ & =\int_{B_1}\nabla\phi\cdot\nabla u_{x_0,r_0}\,dx. \end{aligned} $$

Now, since the norm of $u_{x_n,r_n}$ converges to the norm of $u_{x_0,r_0}$, we get that

$$\displaystyle \begin{aligned} u_{x_n,r_n}\to u_{x_0,r_0}\qquad \text{strongly in}\qquad H^1(B_1). \end{aligned}$$

By the trace inequality, we have that

$$\displaystyle \begin{aligned} u_{x_n,r_n}\to u_{x_0,r_0}\qquad \text{strongly in}\qquad L^2(\partial B_1), \end{aligned}$$

which concludes the proof. □

Lemma 9.2 (Derivative of the Weiss’ Energy)

Let D be a bounded open set in $\mathbb {R}^d$ and let u ∈ H ¹(D). Let x ₀ ∈ D and δ = dist(x ₀, ∂D). Then, the function Φ _u(x ₀, ⋅) is differentiable almost everywhere on (0, δ) and for (almost) every r ∈ (0, δ), we have

$$\displaystyle \begin{aligned} \frac{\partial }{\partial r}W_\Lambda(u_{x_0,r}) & =\frac{d}r\big(W_\Lambda(z_{x_0,r})-W_\Lambda(u_{x_0,r})\big)\\ & \quad +\frac 1r\int_{\partial B_1}|x\cdot \nabla u_{x_0,r}-u_{x_0,r}|{}^2\,d\mathcal{H}^{d-1},\end{aligned} $$

(9.1)

where $z_{x_0,r}:B_1\to \mathbb {R}$ is the one-homogeneous extension of $u_{x_0,r}$ in B ₁:

$$\displaystyle \begin{aligned} z_{x_0,r}(x):=|x|\,u_{x_0,r}\left({x}/{|x|}\right). \end{aligned}$$

Proof

Without loss of generality we can assume x ₀ = 0. We recall that u _r := u _0,r.

We first notice that the function r↦| Ω_u ∩ B _r| is differentiable almost everywhere and that for almost every r ∈ (0, δ) we have

$$\displaystyle \begin{aligned} \frac{\partial}{\partial r}\left(\frac 1{r^d}|\Omega_u\cap B_r|\right) & =-\frac{d}{r^{d+1}}|\Omega_u\cap B_r|+\frac 1{r^d}\mathcal{H}^{d-1}(\Omega_u\cap \partial B_r), \end{aligned} $$

which can be written as

$$\displaystyle \begin{aligned} \frac{\partial}{\partial r}\left(\frac 1{r^d}|\Omega_u\cap B_r|\right)=-\frac{d}{r}|\Omega_{u_r}\cap B_1|+\frac{d}{r}|\Omega_{z_r}\cap B_1|.\end{aligned} $$

(9.2)

In fact, we have

$$\displaystyle \begin{aligned} |\Omega_{z_r}\cap B_1|=\int_0^1 \mathcal{H}^{d-1}(\Omega_{u_r}\cap \partial B_1)s^{d-1}\,ds=\frac{1}{d} \mathcal{H}^{d-1}(\Omega_{u_r}\cap \partial B_1)=\frac{r^{d-1}}{d} \mathcal{H}^{d-1}(\Omega_{u}\cap \partial B_r). \end{aligned}$$

Thus, (9.2) implies that it is sufficient to prove (9.1) in the case Λ = 0.

As above, we notice that the function $\displaystyle r\mapsto \int _{B_r}|\nabla u|{ }^2\,dx$ is differentiable almost-everywhere and that we have

$$\displaystyle \begin{aligned} \frac{\partial}{\partial r}\left(\frac 1{r^d}\int_{B_r}|\nabla u|{}^2\,dx\right) & =-\frac{d}{r^{d+1}}\int_{B_r}|\nabla u|{}^2\,dx+\frac 1{r^d}\int_{\partial B_r}|\nabla u|{}^2\,d\mathcal{H}^{d-1}\notag\\ & =-\frac{d}{r^{d+1}}\int_{B_r}|\nabla u|{}^2\,dx+\frac 1r\int_{\partial B_1}|\nabla u_r|{}^2\,d\mathcal{H}^{d-1}.{} \end{aligned} $$

(9.3)

In order to deal with the boundary term, we first compute

$$\displaystyle \begin{aligned} \frac{\partial}{\partial r}\left(\frac 1{r^{d-1}}\int_{\partial B_r}u^2(x)\,d\mathcal{H}^{d-1}(x)\right) & =\frac{\partial}{\partial r}\int_{\partial B_1}u(ry)^2\,d\mathcal{H}^{d-1}(y)\\ & =2\int_{\partial B_1}u(ry)\,y\cdot \nabla u(ry)\,d\mathcal{H}^{d-1}(y)\\ & =2r\int_{\partial B_1}u_r\,(x\cdot \nabla u_r)\,d\mathcal{H}^{d-1}(x) \end{aligned} $$

Thus, we have

$$\displaystyle \begin{aligned} \frac{\partial}{\partial r}\left(\frac 1{r^{d+1}}\int_{\partial B_r}u^2\,d\mathcal{H}^{d-1}\right) & =-\frac{2}{r^{d+2}}\int_{\partial B_r}u^2\,d\mathcal{H}^{d-1}+\frac 2{r}\int_{\partial B_1}u_r\,(x\cdot \nabla u_r)\,d\mathcal{H}^{d-1}.{} \end{aligned} $$

(9.4)

Now, we notice that for every r such that u _r = z _r ∈ H ¹(∂B ₁), we can write the function $z_r:B_1\to \mathbb {R}$ in polar coordinates ρ ∈ (0, 1], $\theta \in \mathbb S^{d-1}$ as z _r(ρ, θ) = ρ z _r(1, θ) and we have

$$\displaystyle \begin{aligned} W_0(z_r) & =\int_{B_1}|\nabla z_r|{}^2\,dx-\int_{\partial B_1}z_r^2\,d\mathcal{H}^{d-1}\notag\\ & =\int_0^1r^{d-1}\,dr\int_{\mathbb S^{d-1}}\left(z_r^2(1,\theta)+|\nabla_\theta z_r|{}^2\right)\,d\theta-\int_{\mathbb S^{d-1}}z_r^2(1,\theta)\,d\theta\notag\\ & =\frac 1d\int_{\mathbb S^{d-1}}|\nabla_\theta z_r|{}^2\,d\theta-\frac{d-1}{d}\int_{\mathbb S^{d-1}}z_r^2(1,\theta)\,d\theta\notag\\ & =\frac 1d\int_{\partial B_1}\left(|\nabla u_r|{}^2-(x\cdot\nabla u_r)^2\right)\,d\mathcal{H}^{d-1}-\frac{d-1}{d}\int_{\partial B_1}u_r^2\,d\mathcal{H}^{d-1}.{} \end{aligned} $$

(9.5)

Now, putting together (9.3), (9.4) and (9.5), we get that

$$\displaystyle \begin{aligned} \frac{\partial }{\partial r}W_0(u_{x_0,r})=\frac{d}r\big(W_0(z_{x_0,r})-W_0(u_{x_0,r})\big)+\frac 1r\int_{\partial B_1}|x\cdot \nabla u_{x_0,r}-u_{x_0,r}|{}^2\,d\mathcal{H}^{d-1}, \end{aligned}$$

which concludes the proof. □

We now define the deviation $\mathcal D$ as

$$\displaystyle \begin{aligned} \mathcal D(\phi):=\int_{\partial B_1}|x\cdot\nabla \phi-\phi|{}^2\,d\mathcal{H}^{d-1}. \end{aligned}$$

Thus, (9.1) can be written as

$$\displaystyle \begin{aligned} {} \frac{\partial }{\partial r}W_\Lambda(u_{x_0,r})=\frac{d}r\big(W_\Lambda(z_{x_0,r})-W_\Lambda(u_{x_0,r})\big)+\frac 1r\mathcal D(u_{x_0,r}). \end{aligned}$$

In the next lemma we show that the deviation $\mathcal D(u_{x_0,r})$ controls the oscillation of u.

Lemma 9.3 (The Deviation Controls the Oscillation of the Blow-Up Sequence)

Let D be a bounded open set in $\mathbb {R}^d$ and let u ∈ H ¹(D). Let x ₀ ∈ D and δ = dist(x ₀, ∂D). Then, for almost every 0 < r < R < δ, we have

$$\displaystyle \begin{aligned} \|u_{x_0,R}-u_{x_0,r}\|{}_{L^2(\partial B_1)}^2\le \frac 1r\int^R_r \mathcal D(u_{x_0,s})\, ds. \end{aligned}$$

In particular, if $\mathcal D(u_{x_0,s})=0$ for every s ∈ (0, δ), then the function $u_{x_0,\delta }:B_1\to \mathbb {R}$ is one-homogeneous, that is

$$\displaystyle \begin{aligned} u(x_0+rx)=ru(x_0+x)\quad \mathit{\text{for every}}\quad |x|\le \delta\quad \mathit{\text{and every}}\quad r\le 1. \end{aligned}$$

Proof

We set for simplicity, x ₀ = 0 and $u_r:=u_{x_0,r}$. For any x ∈ ∂B ₁, we have

$$\displaystyle \begin{aligned} \frac{u(Rx)}R-\frac{u(rx)}r=\int_r^R\left(\frac{x\cdot (\nabla u)(sx)}{s}-\frac{u(sx)}{s^2}\right)\,ds=\int_r^R\frac 1s\big(x\cdot \nabla u_s(x)-u_s(x)\big)\,ds. \end{aligned}$$

Integrating over the sphere ∂B ₁ and using the Cauchy-Schwarz inequality, we obtain

$$\displaystyle \begin{aligned} \int_{\partial B_1} |u_R - u_r|{}^2 \,d\mathcal{H}^{d-1} & \leq \int_{\partial B_1} \left(\int^R_r \frac{1}{s} |x \cdot \nabla u_s - u_s |\,ds \right)^2 \,d\mathcal{H}^{d-1} \\ & \leq \int_{\partial B_1} \left(\int^R_r s^{-2}ds\right) \left( \int^R_r |x \cdot \nabla u_s - u_s |{}^2 ds \right) \,d\mathcal{H}^{d-1} \\ & = \left(\frac 1r-\frac 1R\right )\int^R_r \mathcal D(u_s)\, ds. \end{aligned} $$

which concludes the proof. □

We conclude this subsection with the following proposition.

Proposition 9.4 (Weiss Monotonicity Formula)

Let D be a bounded open set in $\mathbb {R}^d$ and let u ∈ H ¹(D) be a minimizer of $\mathcal F_\Lambda$ in D. Let x ₀ ∈ D and $\delta _{x_0}=\mathit{\text{dist}}(x_0,\partial D)$ . Then the function $r\mapsto W_\Lambda (u_{x_0,r})$ is non-decreasing on the interval $(0,\delta _{x_0})$.

Proof

By Lemma 9.2 we have that

$$\displaystyle \begin{aligned} \frac{\partial}{\partial r}W_\Lambda(u_{x_0,r})\ge \frac{d}r\big(W_\Lambda(z_{x_0,r})-W_\Lambda(u_{x_0,r})\big). \end{aligned}$$

Now, since $u_{x_0,r}$ is a minimizer of $\mathcal F_\Lambda$ in B ₁ and since by definition $z_{x_0,r}=u_{x_0,r}$ on ∂B ₁, we get that $\displaystyle \frac {\partial }{\partial r}W_\Lambda (u_{x_0,r})\ge 0$, which concludes the proof. □

9.2 Stationary Free Boundaries

In this section we introduce the notion of a stationary free boundary (Definition 9.7) and we prove a monotonicity formula for the Weiss energy (Proposition 9.9).

Lemma 9.5 (First Variation of the Energy)

Suppose that $D\subset \mathbb {R}^d$ is a bounded open set and that u ∈ H ¹(D). Let $\xi \in C^\infty _c(D ;\mathbb {R}^d)$ be a given vector field with compact support in D and let Ψ _t be the diffeomorphism

$$\displaystyle \begin{aligned} \Psi_t(x)=x+t\xi (x)\quad \mathit{\text{for every}}\quad x\in D. \end{aligned}$$

Then,

• (i) for t small enough, Ψ _t : D → D is a diffeomorphism and setting $\Phi _t:=\Psi _t^{-1}$ , the function u _t := u ∘ Φ _t is well-defined and belongs to H ¹(D);

• (ii) the function t↦∫_D|∇u _t|² dx is differentiable at t = 0 and

  $$\displaystyle \begin{aligned} \frac{\partial}{\partial t}\Big\vert_{t=0}\int_{D}|\nabla u_t|{}^2\,dx= \int_{D}\left(-2\nabla u\cdot D\xi \nabla u+ |\nabla u|{}^2\mathrm{div}\,\xi\,\right)dx; \end{aligned}$$

• (iii) the function $\displaystyle t\mapsto |\Omega _{u_t}\cap D|$ is differentiable at t = 0 and

  $$\displaystyle \begin{aligned} \frac{\partial}{\partial t}\Big\vert_{t=0}|\Omega_{u_t}\cap D|= \int_{\Omega_u\cap D}\mathrm{div}\,\xi\,dx. \end{aligned}$$

• (iv) if Ω _u is open, if ∂ Ω _u is a C ² regular in D and if u ∈ C ²( Ω _u), then

  $$\displaystyle \begin{aligned} \frac{\partial}{\partial t}\Big\vert_{t=0}\int_{D}|\nabla u_t|{}^2\,dx\! & =\!-\int_{\partial\Omega_u} \!\!\xi\cdot \nu\,|\nabla u|{}^2\,d\mathcal{H}^{d-1}\qquad \mathit{\text{and}}\\ \frac{\partial}{\partial t}\Big\vert_{t=0}|\Omega_{u_t}\cap D|\! & =\!\int_{\partial\Omega_u}\!\!\xi\cdot \nu\,d\mathcal{H}^{d-1}\,, \end{aligned} $$

  where ν(x) is the exterior normal to ∂ Ω at the point x ∈ ∂ Ω.

Proof

The first claim follows by the fact that ξ is smooth and compactly supported in D. Thus, we start directly by proving (ii). We use the conventions

$$\displaystyle \begin{aligned} x=\begin{pmatrix}x_1\\ \vdots\\ x_d\end{pmatrix}\ ,\quad \nabla u=\begin{pmatrix}\partial_1u\\ \vdots\\ \partial_du\end{pmatrix}\ ,\quad \Phi=\begin{pmatrix}\Phi_1\\ \vdots\\ \Phi_d\end{pmatrix}\ ,\quad D \Phi=\begin{pmatrix}\partial_1\Phi_1& \cdots & \partial_1\Phi_d\\ \vdots& \ddots& \vdots\\ \partial_d\Phi_1& \cdots & \partial_d\Phi_d\end{pmatrix},\end{aligned}$$

for general $u:\mathbb {R}^d\to \mathbb {R}$ and $\Phi :\mathbb {R}^d\to \mathbb {R}^d$, so that

$$\displaystyle \begin{aligned} \nabla(u\circ\Phi)(x)=D\Phi(x)\nabla u(\Phi(x)). \end{aligned}$$

In our case u _t = u ∘ Φ_t, by the change of variables y = Φ_t(x) (thus, x = Ψ_t(y)), we get

$$\displaystyle \begin{aligned} \int_{D}|\nabla u_t|{}^2(x)\,dx & =\int_{D}\Big(D\Phi_t(\Psi_t(y))\nabla u(y)\Big)\cdot\Big(D\Phi_t(\Psi_t(y))\nabla u(y)\Big) \left|\det D\Psi_t(y)\right|dy\\ & =\int_{D}\nabla u(y)\cdot\Big(\big[D\Phi_t(\Psi_t(y))\big]^TD\Phi_t(\Psi_t(y))\Big)\nabla u(y) \left|\det D\Psi_t(y)\right|dy\\ & =\int_{D}\nabla u(y)\cdot\Big(\big[D\Psi_t(y)\big]^{-T} \big[D\Psi_t(y)\big]^{-1}\Big)\nabla u(y)\left|\det D\Psi_t(y)\right|dy \end{aligned} $$

We now notice that

$$\displaystyle \begin{aligned} D\Psi_t\!=\!Id+tD\xi\ ,\qquad [D\Psi_t]^{-1}\!=\!Id-t D\xi+o(t)\ ,\quad \det D\Psi_t\!=\!1+t\,\text{div}\,\xi +o(t)\ , \end{aligned}$$

and we calculate

$$\displaystyle \begin{aligned} \int_{D}|\nabla u_t|{}^2\,dx=\int_{D}|\nabla u|{}^2\,dx + t\int_{D}\left(|\nabla u|{}^2\text{div}\,\xi-2\nabla u\cdot D\xi\, \nabla u\,\right)dx+o(t), \end{aligned}$$

which concludes the proof of (ii).

In order to prove (iii), we notice that

$$\displaystyle \begin{aligned} x\in\Omega_{u_t}\ \Leftrightarrow\ u_t(x)>0\ \Leftrightarrow\ \Phi_t(x)\in \Omega_u. \end{aligned}$$

This means that , and so, we can compute

which proves (iii).

We now prove (iv). Assume that u is C ² in the open set Ω_u. Then, setting ξ = (ξ ₁, …, ξ _d) and using the convention for summation over the repeating indices, we compute

$$\displaystyle \begin{aligned} |\nabla u|{}^2\text{div}\,\xi-2\nabla u\, D\xi\cdot \nabla u & =\partial_i u \, \partial_i u \, \partial_j\,\xi_j-2\partial_i u\,\partial_j\xi_i\,\partial_j u\\ & =\partial_i u\, \partial_i u \,\partial_j\xi_j-2\partial_j( \partial_i u\,\xi_i\,\partial_j u)+2\partial_{ij}u\,\xi_i\partial_j u\\ & \quad +2\partial_i \xi_i\,\partial_{jj}u\\ & =\partial_i u\partial_i u\partial_j\xi_j-2\partial_j( \partial_i u\,\xi_i\,\partial_j u)+2\partial_{ij}u\,\xi_i\,\partial_j u\\ & =\partial_i u\,\partial_i u\,\partial_j\xi_j-2\partial_j( \partial_i u\,\xi_i\,\partial_j u)+\partial_{i}(\partial_j u\,\xi_i\,\partial_j u)\\ & \quad -\partial_j u\,\partial_i \xi_i\,\partial_j u\\ & =-2\partial_j( \partial_i u\,\xi_i\,\partial_j u)+\partial_{j}(\partial_i u\,\xi_j\,\partial_i u)\\ & =\text{div}\,\big(|\nabla u|{}^2\xi-2(\xi\cdot \nabla u)\nabla u\big). \end{aligned} $$

Integrating by parts we obtain

$$\displaystyle \begin{aligned} & \int_{\Omega_u}\text{div}\,\Big(|\nabla u|{}^2\xi-2(\xi\cdot \nabla u)\nabla u\Big)\,dx\\ & \quad =\int_{\partial\Omega_u}\Big(|\nabla u|{}^2(\xi\cdot\nu)-2(\xi\cdot \nabla u)(\nabla u\cdot\nu)\Big)\,d\mathcal{H}^{d-1}. \end{aligned} $$

Since u = 0 on ∂ Ω_u and positive in Ω_u, we have that ∇u = ν|∇u|. Thus,

$$\displaystyle \begin{aligned} \int_{\Omega_u}\text{div}\,\Big(|\nabla u|{}^2\xi-2(\xi\cdot \nabla u)\nabla u\Big)\,dx=-\int_{\partial\Omega_u}|\nabla u|{}^2(\xi\cdot\nu)\,d\mathcal{H}^{d-1}, \end{aligned} $$

which proves the first part of the claim (iv). The second part of (iv) follows by a simple integration by parts in Ω_u. □

As a consequence of Lemma 9.5 we obtain that for every $\Lambda \in \mathbb {R}$, u ∈ H ¹(D) and vector field $\xi \in C^\infty _c(D;\mathbb {R}^d)$ we can define the first variation of $\mathcal F_\Lambda$ at u in the direction ξ as

(9.6)

Lemma 9.6 (The Minimizers have Zero First Variation)

Let D be a bounded open set in $\mathbb {R}^d$ and let u ∈ H ¹(D) be a minimizer of $\mathcal F_\Lambda$ in D. Then,

$$\displaystyle \begin{aligned} \delta \mathcal F_\Lambda(u,D)[\xi]=0\qquad \mathit{\text{for every vector field}}\qquad \xi\in C^\infty_c(D;\mathbb{R}^d). \end{aligned}$$

If, moreover, ∂ Ω _u is C ² smooth in D, then

$$\displaystyle \begin{aligned} |\nabla u|=\sqrt{\Lambda}\qquad \mathit{\text{on}}\qquad \partial\Omega_u\cap D. \end{aligned} $$

(9.7)

Proof

The first part of the statement follows directly by Lemma 9.5. In order to prove the second part, we notice that in the case when ∂ Ω_u is smooth, we have

$$\displaystyle \begin{aligned} \delta\mathcal F_\Lambda(u,D)[\xi]=\int_{\partial\Omega_u}\!\!\big(\Lambda-|\nabla u|{}^2\big)\,\xi\cdot\nu\,d\mathcal{H}^{d-1}, \end{aligned}$$

for every vector field $\xi \in C^\infty _c(D;\mathbb {R}^d)$. This implies (9.7). □

Definition 9.7 (Stationary Free Boundaries)

Let $D\subset \mathbb {R}^d$ be a bounded open set and u ∈ H ¹(D) be a non-negative function such that

$$\displaystyle \begin{aligned} {} \delta \mathcal F_\Lambda(u,D)[\xi]=0\qquad \text{for every vector field}\qquad \xi\in C^\infty_c(D;\mathbb{R}^d). \end{aligned}$$

Then, we say that the function u and the free boundary ∂ Ω_u are stationary for $\mathcal F_\Lambda$.

As a consequence of Lemma 9.6 we obtain the following.

Lemma 9.8 (Equipartition of the Energy)

Suppose that D is a bounded open set in $\mathbb {R}^d$ and u ∈ H ¹(D) is a non-negative function which is stationary for $\mathcal F_\Lambda$ (in the sense of Definition 9.7). Then, for every x ₀ ∈ D and every 0 < r < dist(x ₀, ∂D), we have

$$\displaystyle \begin{aligned} W_\Lambda(z_{x_0,r})-W_\Lambda(u_{x_0,r})=\frac 1d\int_{\partial B_1}|x\cdot \nabla u_{x_0,r}-u_{x_0,r}|{}^2\,d\mathcal{H}^{d-1}, \end{aligned} $$

(9.8)

where we recall that $u_{x_0,r}(x)=\frac 1ru(x_0+rx)$ and that $z_{x_0,r}$ is the one-homogeneous extension of $u_{x_0,r}$ in B ₁ , that is, $z_{x_0,r}(x)=|x|u_{x_0,r}\big ({x}/{|x|}\big )$.

Proof

Without loss of generality, we assume that x ₀ = 0. For every ε > 0, we consider a function $\phi _{\varepsilon }\in C^\infty _c(B_r)$ such that

Taking the vector field ξ _ε(x) = xϕ _ε(x) we get that

$$\displaystyle \begin{aligned} \text{div}\,\xi_{\varepsilon}(x)=d\phi_{\varepsilon}(x)+x\cdot\nabla\phi_{\varepsilon}(x), \end{aligned}$$

$$\displaystyle \begin{aligned} D\xi_{\varepsilon}(x)=\phi_{\varepsilon}(x)Id+x\otimes\nabla\phi_{\varepsilon}(x). \end{aligned}$$

Thus, the stationarity of u impies that

which passing to the limit as ε → 0 implies that

(9.9)

Since Δu = 0 on Ω_u, we have that

$$\displaystyle \begin{aligned} 2\int_{B_r}|\nabla u|{}^2\,dx=2\int_{B_r}\text{div}\,(u\nabla u)\,dx=2\int_{\partial B_r}u(\nu\cdot \nabla u)\,d\mathcal{H}^{d-1}, \end{aligned}$$

which together with (9.9) implies (9.8). □

Proposition 9.9 (Monotonicity Formula for Stationary Free Boundaries)

Suppose that D is a bounded open set in $\mathbb {R}^d$ and u ∈ H ¹(D) is a non-negative function which is stationary for $\mathcal F_\Lambda$ (in the sense of Definition 9.7). Let x ₀ ∈ D and $\delta _{x_0}=\mathrm {dist}\,(x_0,\partial D)$ . Then the function $r\mapsto W_\Lambda (u_{x_0,r})$ is non-decreasing on the interval $(0,\delta _{x_0})$ and we have

$$\displaystyle \begin{aligned} \frac{\partial}{\partial r}W_\Lambda(u_{x_0,r})\ge \frac 2r\int_{\partial B_1}|x\cdot \nabla u_{x_0,r}-u_{x_0,r}|{}^2\,d\mathcal{H}^{d-1}. \end{aligned} $$

(9.10)

Proof

By Lemmas 9.8 and 9.2 we obtain precisely (9.10). □

9.3 Homogeneity of the Blow-Up Limits

In this section, we use the Weiss’ monotonicity formula to prove that the blow-up limits of u are one-homogeneous functions. The most general result is given in Lemma 9.10. We then prove the homogeneity of the blow-up limits of stationary functions (Lemma 9.11) and the homogeneity of the blow-up limits of minimizers of $\mathcal F_\Lambda$ (Proposition 9.12).

Lemma 9.10

Suppose that D is a bounded open set in $\mathbb {R}^d$ and u ∈ H ¹(D) is a non-negative function. Let x ₀ ∈ D and $\delta _{x_0}=\mathrm {dist}\,(x_0,\partial D)$ . Let r _n → 0 be an infinitesimal sequence and let $u_n:=u_{r_n,x_0}\in H^1(B_1)$ . Suppose that

• (a) the limit

  $$\displaystyle \begin{aligned} L:=\lim_{r\to0}W_\Lambda(u_{r,x_0}), \end{aligned}$$

  exists and is finite;

• (b) u _n converges strongly in H ¹(B ₁) to a function u _∞∈ H ¹(B ₁);

• (c) converges strongly in L ¹(B ₁) to ;

• (d) u _∞ is stationary for $\mathcal F_\Lambda$ in B ₁.

Then u _∞ is one-homogeneous.

Proof

Without loss of generality, we suppose that x ₀ = 0 and we write $u_{r,x_0}=u_r$. We set for simplicity v := u _∞. By the hypothesis (a), we have that,

$$\displaystyle \begin{aligned} L=\lim_{n\to\infty}W_\Lambda(u_{s r_n})\qquad \text{for every}\qquad s<0\le 1. \end{aligned}$$

On the other hand, the strong convergence of u _n and implies that

$$\displaystyle \begin{aligned} \lim_{n\to\infty}W_\Lambda(u_{s r_n})=W_\Lambda(v_s), \end{aligned}$$

where we recall that $\displaystyle v_s(x)=\frac 1s v(sx)$. This implies that

$$\displaystyle \begin{aligned} W_\Lambda(v_s)=L\qquad \text{for every}\qquad s\in(0,1], \end{aligned}$$

and, by Proposition 9.9, we obtain that

$$\displaystyle \begin{aligned} 0=\frac{\partial}{\partial s}W_\Lambda(v_s)\ge \frac 2s\int_{\partial B_1}|x\cdot\nabla v_s-v_s|{}^2\,d\mathcal{H}^{d-1}, \end{aligned}$$

which, by Lemma 9.3, gives that v is one-homogeneous. □

Lemma 9.11 (Homogeneity of the Blow-Up Limits)

Suppose that D is a bounded open set in $\mathbb {R}^d$ and u ∈ H ¹(D) is a non-negative function which is stationary for $\mathcal F_\Lambda$ (in the sense of Definition 9.7). Let x ₀ ∈ D ∩ ∂ Ω _u , r _n → 0 be an infinitesimal sequence and $u_n:=u_{r_n,x_0}\in H^1(B_1)$ . Suppose that

• (a) u _n converges strongly in H ¹(B ₁) to a function u _∞∈ H ¹(B ₁);

• (b) converges strongly in L ¹(B ₁) to .

Then u _∞ is one-homogeneous.

Proof

Since u is stationary, Lemma 9.9 implies that the function $r\mapsto W_\Lambda (u_{x_0,r})$ is non-decreasing in r. Thus, the limit

$$\displaystyle \begin{aligned} L:=\lim_{r\to0}W_\Lambda(u_{x_0,r})=\inf_{r>0}W_\Lambda(u_{x_0,r}), \end{aligned}$$

does exist and so the hypothesis (a) of Lemma 9.10 is fulfilled. Now, the strong convergence of u _n and to u _∞ and in B ₁, and the definition of the first variation $\delta \mathcal F_\Lambda (\cdot ,D)$ imply that u _∞ is also stationary in B ₁. Thus, hypothesis (d) of Lemma 9.10 is also fulfilled and, so the claim follows by Lemma 9.10. □

Proposition 9.12 (Homogeneity of the Blow-Up Limits)

Suppose that D is a bounded open set in $\mathbb {R}^d$ and u ∈ H ¹(D) is a non-negative function and a local minimizer of $\mathcal F_\Lambda$ in D. Let x ₀ ∈ D. Then every blow-up limit $u_0\in \mathcal {B}\mathcal {U}_u(x_0)$ is one-homogeneous.

Proof

By Lemma 9.6, every minimizer of $\mathcal F_\Lambda$ is stationary for $\mathcal F_\Lambda$. Moreover, by Proposition 6.2, we have that the conditions (a) and (b) of Lemma 9.11 are fulfilled. This concludes the proof. □

9.4 Regularity of the Free Boundaries in Dimension Two

The main result of this section is the following.

Proposition 9.13 (One-Homogeneous Global Minimizers in Dimension Two)

Let $z:\mathbb {R}^2\to \mathbb {R}$ be a one-homogeneous global minimizer of $\mathcal F_\Lambda$ in $\mathbb {R}^2$ . Then, there is $\nu \in \mathbb {R}^2$ such that

$$\displaystyle \begin{aligned} z(x)=\sqrt{\Lambda}\,(x\cdot\nu)_+\quad \mathit{\text{for every}}\quad x\in\mathbb{R}^2. \end{aligned}$$

In particular, we obtain that the critical dimension d ^∗ is at least 3 (see Definition 1.5).

The proof of Proposition 9.13 is based on the following lemma.

Lemma 9.14

Let $z\in H^1_{loc}(\mathbb {R}^d)$ be a continuous and non-negative one-homogeneous function in $\mathbb {R}^d$ . Then,

$$\displaystyle \begin{aligned} {} \Delta z=0\quad \mathit{\text{in}}\quad \Omega_z, \end{aligned}$$

if and only if, the trace $c=z|{ }_{\partial B_1}\in H^1(\partial B_1)$ is such that

$$\displaystyle \begin{aligned} {} -\Delta_{\mathbb S}c=(d-1)c\quad \mathit{\text{in the (open) set}}\quad \Omega_c\cap\partial B_1\,. \end{aligned}$$

Proof

The proof follows simply by writing the Laplacian in polar coordinates. In fact, we have that z(r, θ) = rc(θ) and

$$\displaystyle \begin{aligned} \Delta z(r,\theta) & =\partial_{rr} z(r,\theta)+\frac{d-1}{r}\partial_r z(r,\theta)+\frac 1{r^2}\Delta_{\mathbb S} z(r,\theta)\\ & =\frac 1r\big((d-1)\,c(\theta)+\Delta_{\mathbb S} c(\theta)\big), \end{aligned} $$

which concludes the proof of Lemma 9.14. □

Proof of Proposition 9.13

Let z(r, θ) = rc(θ) and let $\Omega _c\subset \mathbb S^1$ be the set {c > 0}. Since c is continuous (see Sect. 3), we have that Ω_c is open and so it is a countable union of disjoint arcs (which we identify with segments on the real line). Notice that $\Omega _c\neq \mathbb S^1$ since z(0) = 0 and z minimizes locally $\mathcal F_\Lambda$ (the local minimizers cannot have isolated zeros, for instance, by the density estimates from Sect. 5.1). Now, Lemma 9.14 implies that on each arc $\mathcal I\subset \Omega _c$, the trace c is a solution of the PDE

$$\displaystyle \begin{aligned} -c''(\theta)=c(\theta)\quad \text{in}\quad \mathcal I\,,\qquad c>0\quad \text{in}\quad I\,,\qquad c=0\quad \text{on}\quad \partial I\,. \end{aligned}$$

Thus, up to a translation $\mathcal I=(0,\pi )$ and c(θ) is a multiple of $\sin \theta$ on $\mathcal I$. Thus, Ω_c is a union of disjoint arcs, each one of length π. Thus, these arcs can be at most two. Now, by Lemma 2.9 and the fact that 0 ∈ ∂ Ω_z, we get that | Ω_z ∩ B ₁| < |B ₁| = π and so, $\mathcal {H}^1(\Omega _c)<2\pi$. This means that Ω_c is an arc of length π and that z is of the form z(x) = a (x ⋅ ν), for some constant a > 0. Since z is a local minimizer in $\mathbb {R}^d$ and ∂ Ω_z is smooth, Lemma 6.11 implies that $a=\sqrt {\Lambda }$, which concludes the proof. □

9.5 The Optimality Condition on the Free Boundary: A Monotonicity Formula Approach

The aim of this subsection is to give an alternative proof to the fact that the (local) minimizers of $\mathcal F_\Lambda$ are viscosity solutions to the problem

$$\displaystyle \begin{aligned} {} \Delta u=0\quad \text{in}\quad \Omega_u\,,\qquad |\nabla u|=\sqrt{\Lambda}\quad \text{on}\quad \partial\Omega_u\,. \end{aligned}$$

The main result of the subsection is Proposition 9.18, which can be applied not only to minimizers, but also to measure constrained minimizers (see Theorem 1.9 and Sect. 11). It can also be applied to a large class of problems in which a monotonicity formula does hold. In fact, the proof is quite robust and can be applied to almost-minimizers (see for instance [46]) and to vectorial problems (see [41]), for which the construction of competitors is typically more involved. The proof of Proposition 9.18 is based on the following two lemmas. Before we give the two statements, we recall that, for any d ≥ 2, we identify the (d − 1)-dimensional sphere $\mathbb S^{d-1}$ with the boundary of the unit ball ∂B ₁ in $\mathbb {R}^d$. In particular, we will use the notation

$$\displaystyle \begin{aligned} \mathbb S^{d-1}_+=\big\{x:=(x_1,\dots,x_d)\in \partial B_1\subset\mathbb{R}^d\ :\ x_d>0\big\}. \end{aligned}$$

Lemma 9.15

Suppose that $c\in H^1(\mathbb S^{d-1})$ is a continuous non-negative and non-constantly-vanishing function, satisfying the following conditions:

• (a) $\Omega _c\subset \mathbb S^{d-1}_+$ , where as usual Ω _c := {c > 0};

• (b) $\Delta _{\mathbb S} c+(d-1)c=0$ in Ω _c.

Then, $\Omega _c=\mathbb S^{d-1}_+$ and there is a constant α > 0 such that

$$\displaystyle \begin{aligned} c(x)=\alpha (x\cdot e_d)_+\qquad \mathit{\text{for every}}\qquad x\in \partial B_1. \end{aligned}$$

Lemma 9.16

Suppose that $c\in H^1(\mathbb S^{d-1})$ is a continuous non-negative function, satisfying the following conditions:

• (a) $\mathbb S^{d-1}_+\subset \Omega _c=\{c>0\}$;

• (b) $\Delta _{\mathbb S} c+(d-1)c=0$ in Ω _c.

Then, c is given by one of the following functions:

• (i) c(x) = α(x ⋅ e _d)₊ , where α > 0 is a positive constant;

• (ii) c(x) = α(x ⋅ e _d)₊ + β(x ⋅ e _d)₋ , where α > 0 and β > 0.

In the proofs of Lemmas 9.15 and 9.16 we will use the following well-known result, whose proof we the leave to the reader.

Lemma 9.17 (Variational Characterization of the Principal Eigenvalue)

Let $\Omega \subset \mathbb S^{d-1}$ be a connected open subset of the unit sphere. Let $\phi \in H^1_0(\Omega )$ be a given non-zero function. Then, the following are equivalent:

• (i) ϕ > 0 in Ω, ∫_Ω ϕ ² dθ = 1 , and there is λ ≥ 0 for which ϕ solves the PDE

  $$\displaystyle \begin{aligned} -\Delta_{\mathbb S}\phi=\lambda\phi\quad \mathit{\text{in}}\quad \Omega\; \end{aligned}$$

  in the usual weak sense:

  $$\displaystyle \begin{aligned} \int_\Omega\nabla_\theta\phi\cdot\nabla_\theta \eta\,d\theta=\lambda \int_\Omega\phi \eta\,d\theta\quad \mathit{\text{for every}}\quad \eta\in H^1_0(\Omega); \end{aligned}$$

• (ii) ϕ is the unique (up to a sign) solution of the variational problem

  $$\displaystyle \begin{aligned} {} \min\Big\{\int_\Omega|\nabla_\theta \psi|{}^2\,d\theta\ :\ \psi\in H^1_0(\Omega),\ \int_\Omega \psi^2\,d\theta=1\Big\}. \end{aligned}$$

Proof of Lemma 9.15

Since the linear functions are one-homogeneous and harmonic in $\mathbb {R}^d$, we have that the function

$$\displaystyle \begin{aligned} \phi_1(\theta)=(\theta\cdot e_d)_+, \end{aligned}$$

defined on the sphere solves the equation

$$\displaystyle \begin{aligned} -\Delta_{\mathbb S}\phi_1=(d-1)\phi_1\quad \text{in}\quad \mathbb S^{d-1}_+. \end{aligned}$$

In particular, setting $\displaystyle \alpha _d:=\left (\int _{\mathbb S^{d-1}}\phi _1^2\,d\theta \right )^{-1}$, we get that α _d ϕ ₁ is the unique minimizer of

$$\displaystyle \begin{aligned} {} d-1=\min\Big\{\int_{\mathbb S^{d-1}_+}|\nabla_\theta \psi|{}^2\,d\theta\ :\ \psi\in H^1_0(\mathbb S^{d-1}_+),\ \int_{\mathbb S^{d-1}_+} \psi^2\,d\theta=1\Big\}. \end{aligned}$$

On the other hand, $c\in H^1_0(\mathbb S^{d-1}_+)$ and solves the equation $-\Delta _{\mathbb S}c=(d-1)c$ in Ω_c. Thus,

$$\displaystyle \begin{aligned} \int_{\mathbb S^{d-1}_+}|\nabla_\theta c|{}^2\,d\theta=\int_{\Omega_c}|\nabla_\theta c|{}^2\,d\theta=(d-1)\int_{\Omega_c}c^2\,d\theta=(d-1)\int_{\mathbb S^{d-1}_+}c^2\,d\theta, \end{aligned}$$

which means that (up to a multiplicative constant) c is a solution of the same problem. Thus, the uniqueness of ϕ ₁ gives the claim. □

Proof of Lemma 9.16

Let $\widetilde \Omega _c$ be the connected component of Ω_c containing $\mathbb S^{d-1}_+$; and let $\widetilde c$ be the restriction of c to $\widetilde \Omega _c$. Thus, $\widetilde \Omega _c=\{\widetilde c>0\}$ and $\widetilde c$ solves the PDE

$$\displaystyle \begin{aligned} -\Delta_{\mathbb S}\widetilde c=(d-1)\,\widetilde c\quad \text{in}\quad \widetilde\Omega_c. \end{aligned}$$

Thus, $\tilde c$ is the unique minimizer of

$$\displaystyle \begin{aligned} {} d-1=\min\Big\{\int_{\widetilde\Omega_c}|\nabla_\theta \psi|{}^2\,d\theta\ :\ \psi\in H^1_0(\widetilde\Omega_c),\ \int_{\widetilde\Omega_c} \psi^2\,d\theta=1\Big\}. \end{aligned}$$

Thus, reasoning as in the proof of Lemma 9.15, we get that $\widetilde \Omega _c=\mathbb S^{d-1}_+$ and that there is a constant α > 0 such that

$$\displaystyle \begin{aligned} \widetilde c(\theta)=\alpha(\theta\cdot e_d)_+. \end{aligned}$$

We now consider two cases. If Ω_c has only one connected component, then $\Omega _c=\widetilde \Omega _c$ and $c=\widetilde c$, which concludes the proof. If Ω_c has more than one connected components, then is non-empty and is contained in the half-sphere

$$\displaystyle \begin{aligned} \mathbb S^{d-1}_-=\{x:=(x_1,\dots,x_d)\in \partial B_1\subset\mathbb{R}^d\ :\ x_d<0\}. \end{aligned}$$

Thus, applying Lemma 9.15, we get that the restriction of c on should be of the form β(θ ⋅ e _d)₋, for some positive constant β, which concludes the proof. □

Proposition 9.18

Suppose that $D\subset \mathbb {R}^d$ is a bounded open set and that u ∈ H ¹(D) is a continuous non-negative function such that:

• (a) u is harmonic in Ω _u = {u > 0}.

• (b) Ω _u satisfies the upper density bound

  $$\displaystyle \begin{aligned} \limsup_{r\to0}\frac{|\Omega_u\cap B_r(x_0)|}{|B_r|}<1\qquad \mathit{\text{for every}}\qquad x_0\in\partial\Omega_u\cap D. \end{aligned}$$

• (c) For every x ₀ ∈ D ∩ ∂ Ω _u and every infinitesimal sequence r _n → 0, there is a subsequence (that we still denote by r _n ) such that the blow-up sequence $u_{r_n,x_0}$ converges uniformly in B ₁ to a blow-up limit $u_0:B_1\to \mathbb {R}$ ( $u_0\in \mathcal {B}\mathcal {U}_u(x_0)$ ).

• (d) Every blow-up limit $\mathcal {B}\mathcal {U}_u(x_0)\ni u_0:B_1\to \mathbb {R}$ is a one-homogeneous non-identically-zero function, which is stationary for the functional $\mathcal F_\Lambda$.

Then u satisfies the optimality condition

$$\displaystyle \begin{aligned} {} |\nabla u|=\sqrt{\Lambda}\quad \mathit{\text{on}}\quad \partial\Omega_u\cap D\,, \end{aligned}$$

in viscosity sense.

Proof

Suppose first that the function φ touches u from below in x ₀ ∈ ∂ Ω_u and assume that x ₀ = 0. Consider the blow-up sequences

$$\displaystyle \begin{aligned} u_n(x)=\frac 1{r_n}u(r_nx)\qquad \text{and}\qquad \varphi_n(x)=\frac 1{r_n}\varphi(r_nx), \end{aligned}$$

as r _n → 0, the condition (c) implies that, up to a subsequence, we have

$$\displaystyle \begin{aligned} u_0=\lim_{n\to\infty}u_n(x)\qquad \text{and}\qquad \varphi_0=\lim_{n\to\infty}\varphi_n(x), \end{aligned} $$

(9.11)

the convergence being uniform in B ₁. In particular, since u _n are harmonic in $\Omega _{u_n}$, the uniform convergence of u _n to u ₀ implies that also u ₀ is harmonic on $\Omega _{u_0}$.

Notice that, as φ is smooth, we have φ ₀(x) = ξ ⋅ x, where the vector $\xi \in \mathbb {R}^d$ is precisely the gradient ∇φ(0). Without loss of generality we may assume that ξ = Ae _d for some constant A ≥ 0, thus

$$\displaystyle \begin{aligned} |\nabla\varphi(0)|=|\nabla\varphi_0(0)|=A\qquad \text{and}\qquad \varphi_0(x)=A x_d. \end{aligned} $$

(9.12)

Moreover, we can assume that A > 0 since otherwise the inequality $|\nabla \varphi |\le \sqrt {\Lambda }$ holds trivially.

Now, since u ₀ ≥ φ ₀, we obtain that u ₀ > 0 on the set {x _d > 0}. Thus, u ₀ is a 1-homogeneous harmonic function on the cone {u ₀ > 0}⊃{x _d > 0}. By Lemma 9.16, there are only two possibilities:

$$\displaystyle \begin{aligned} u_0(x)=\alpha x_d^+\qquad \text{or}\qquad u_0(x)=\alpha x_d^++\beta x_d^-\,. \end{aligned}$$

The second case is ruled out since it contradicts (b). Thus,

$$\displaystyle \begin{aligned} u_0(x)=\alpha x_d^+\quad \text{for every}\quad x\in B_1. \end{aligned} $$

(9.13)

Now, the stationarity of u ₀ (condition (d)) and Lemma 9.5 imply that $\alpha =\sqrt {\Lambda }$. By the inequality u ₀ ≥ φ ₀, we get that $\sqrt {\Lambda }\ge A$.

Suppose now that φ touches u from above at a point x ₀ and assume that x ₀ = 0. Again, we consider the blow-up limits U ₀ and φ ₀ defined in (9.11) and we assume that φ ₀ is given by (9.12). Since u ₀ is not identically zero (assumption (d)), we get that a > 0. Since u ₀ ≤ φ ₀ we have that the set {u ₀ > 0} is contained in the half-space {x _d > 0}. By the one-homogeneity of u ₀ and Lemma 9.15 we obtain that necessarily {u ₀ > 0} = {x _d > 0}. Thus, u ₀ is of the form (9.13) for some α > 0. Now, the stationarity of u ₀ implies that necessarily $\alpha =\sqrt {\Lambda }$ and, since u ₀ ≤ φ ₀, we get that $|\nabla \varphi (0)|=A\ge \sqrt {\Lambda }$, which concludes the proof. □

9.6 Energy and Lebesgue Densities

In this section, we prove that if u is a (local) minimizer of $\mathcal F_\Lambda$, then at every boundary point x ₀ ∈ ∂ Ω_u the Lebesgue density of the set Ω_u is well-defined. Moreover, we characterize the regular part of the free boundary in terms of the Lebesgue density. Most of the ideas in this section come from [41], where we used a similar characterization of the regular part of the vectorial free boundaries. In the case of the one-phase problem, we will not use this result in the proofs of neither of the Theorems 1.2, 1.4, 1.9 nor 1.10; we give it here only for the sake of completeness. The precise statement is the following:

Proposition 9.19

Suppose that $D\subset \mathbb {R}^d$ is a bounded open set and that u ∈ H ¹(D) is a non-negative function, a local minimizer of $\mathcal F_\Lambda$ in D. Then, the limit

$$\displaystyle \begin{aligned} \lim_{r\to0}\frac{|\Omega_u\cap B_r(x_0)|}{|B_r|}\quad \mathit{\text{exists, for every}}\quad x_0\in\partial\Omega_u\cap D\,. \end{aligned} $$

(9.14)

Thus, we can write

$$\displaystyle \begin{aligned} \partial\Omega_u \cap D= \bigcup_{\frac 12\le \gamma<1}\Omega_u^{(\gamma)}\cap D\,. \end{aligned} $$

(9.15)

The regular and the singular parts of the free boundary are given by

$$\displaystyle \begin{aligned} \mathrm{Reg}\,(\partial\Omega_u)\cap D=\Omega_u^{({1}/{2})}\cap D\qquad \mathit{\text{and}}\qquad \mathrm{Sing}\,(\partial\Omega_u) \cap D= \bigcup_{\frac 12< \gamma<1}\Omega_u^{(\gamma)}\cap D\, .\end{aligned} $$

(9.16)

Moreover, for every γ ∈ [1∕2, 1), we have

$$\displaystyle \begin{aligned} \!\!\!\Omega_u^{(\gamma)}\cap D=\Big\{x\in\partial\Omega_u\cap D\ :\ |\Omega_{u_0}\cap B_1|=\omega_d\gamma\,,\ \mathit{\text{ for every }}\ \,u_0\in\mathcal{B}\mathcal{U}_u(x)\Big\}. \end{aligned} $$

(9.17)

Proof

The claims (9.14), (9.15) and (9.17) follow directly by Lemma 9.20 below. The claim (9.16), follows by Lemma 9.22. □

Lemma 9.20 (Energy and Lebesgue Densities)

Suppose that $D\subset \mathbb {R}^d$ is a bounded open set and that u ∈ H ¹(D) is a continuous non-negative function such that:

• (a) For every x ₀ ∈ D and every infinitesimal sequence r _n → 0, there is a subsequence (that we still denote by r _n ) such that:

  • $u_n:=u_{r_n, x_0}$ converges strongly in H ¹(B ₁) to a function $u_0:B_1\to \mathbb {R}$;

  • converges in L ²(B ₁) to .

  (As usual, we say that u ₀ is a blow-up limit of u, and we note $u_0\in \mathcal {B}\mathcal {U}_u(x_0)$ .)

• (b) Every blow-up limit $\mathcal {B}\mathcal {U}_u(x_0)\ni u_0:B_1\to \mathbb {R}$ is a one-homogeneous non-identically-zero function such that Δu ₀ = 0 in $\Omega _{u_0}\cap B_1$.

• (c) For every x ₀ ∈ ∂ Ω _u ∩ D, the limit

  $$\displaystyle \begin{aligned} \Theta(u, x_0):=\lim_{r\to0}W_\Lambda(u_{r,x_0})\,, \end{aligned}$$

  does exist.

Then, for every x ₀ ∈ ∂ Ω _u ∩ D, we have that

$$\displaystyle \begin{aligned} \frac 1{\Lambda\omega_d}\Theta(u,x_0)=\lim_{r\to0}\frac{|\Omega_u\cap B_r(x_0)|}{|B_r|}\,. \end{aligned}$$

Moreover, for every $u_0\in \mathcal {B}\mathcal {U}_{u}(x_0)$ , we have that

$$\displaystyle \begin{aligned} \frac 1{\Lambda\omega_d}\Theta(u,x_0)=\frac{|\Omega_{u_0}\cap B_1|}{|B_1|}=\frac 1{\Lambda\omega_d}W_\Lambda(u_0). \end{aligned}$$

Proof

We first notice that (b) implies that

$$\displaystyle \begin{aligned} W_\Lambda(u_0)=\Lambda |\Omega_{u_0}\cap B_1|. \end{aligned}$$

Let x ₀ ∈ ∂ Ω_u ∩ D and the infinitesimal sequence r _n → 0 be given. Then, by (a), up to a subsequence, $u_{r_n,x_0}$ converges to a blow-up limit u ₀. Using (c) and then again (a), we get

$$\displaystyle \begin{aligned} \lim_{r\to0}W_\Lambda(u_{r,x_0})=\lim_{n\to\infty}W_\Lambda(u_{r_n,x_0})=W_\Lambda(u_0). \end{aligned}$$

On the other hand, the strong H ¹(B ₁) convergence of $u_{r_n,x_0}$ to u ₀ implies that

$$\displaystyle \begin{aligned} \lim_{n\to\infty}W_0(u_{r_n,x_0})=W_0(u_0)=0. \end{aligned}$$

Then, we have

$$\displaystyle \begin{aligned} |\Omega_{u_0}\cap B_1|=\frac 1\Lambda\lim_{n\to\infty}W_\Lambda(u_{r_n,x_0})=\lim_{n\to\infty}\big|\{u_{r_n,x_0}>0\}\cap B_1\big|=\lim_{n\to\infty}\frac{|\Omega_u\cap B_{r_n}(x_0)|}{r_n^d}\, \end{aligned}$$

which concludes the proof. □

In the proof of Lemma 9.22, we will use the following result.

Theorem 9.21 (The Spherical Caps Minimize λ ₁ on the Sphere)

For any (quasi-)open spherical set $\Omega \subset \mathbb S^{d-1}$ we define the first eigenvalue λ ₁( Ω) as

$$\displaystyle \begin{aligned} \lambda _1(\Omega):=\inf\Big\{\int_\Omega|\nabla_\theta c|{}^2\,d\theta\ :\ \int_\Omega c^2(\theta)\,d\theta=1,\ c\in H^1_0(\Omega)\Big\}. \end{aligned}$$

For every open set $\Omega \subset \mathbb S^{d-1}$ such that $\mathcal {H}^{d-1}(\Omega )\le \frac 12 d\omega _d$ we have that

$$\displaystyle \begin{aligned} \lambda_1(\Omega)\ge \lambda_1(\mathbb S^{d-1}_+), \end{aligned}$$

with equality if and only if, up to a rotation, $\Omega =\mathbb S^{d-1}_+$.

Lemma 9.22 (Characterization of the Regular Part of the Free Boundary)

Suppose that $D\subset \mathbb {R}^d$ is a bounded open set and that u ∈ H ¹(D) is as in Lemma 9.20 . Then,

$$\displaystyle \begin{aligned} \lim_{r\to0}\frac{|\Omega_u\cap B_r(x_0)|}{|B_r|}\ge\frac 12\,\qquad \mathit{\text{for every}}\qquad x_0\in\partial\Omega_u\cap D\,. \end{aligned} $$

(9.18)

Moreover,

$$\displaystyle \begin{aligned} \lim_{r\to0}\frac{|\Omega_u\cap B_r(x_0)|}{|B_r|}=\frac 12\,, \end{aligned}$$

if and only if, every blow-up limit $u_0\in \mathcal {B}\mathcal {U}_u(x_0)$ is of the form

$$\displaystyle \begin{aligned} u_0(x)=(\nu\cdot x)_+\qquad \mathit{\text{for some}}\qquad \nu\in\mathbb{R}^d. \end{aligned} $$

(9.19)

In particular, if u is a minimizer of $\mathcal F_\Lambda$ in D, then $Reg(\partial \Omega _u)=\Omega _u^{({1}/{2})}$ in D.

Proof

Suppose that x ₀ ∈ ∂ Ω_u ∩ D and let

$$\displaystyle \begin{aligned} \gamma:=\lim_{r\to0}\frac{|\Omega_u\cap B_r(x_0)|}{|B_r|}. \end{aligned}$$

Let r _n → 0 be an infinitesimal sequence. Then, by the assumption Lemma 5.1 (a), up to extracting a subsequence, we can suppose that $u_{r_n,x_0}$ converges to a blow-up limit $u_0:\mathbb {R}^d\to \mathbb {R}$. By the hypothesis Lemma 5.1 (b), we get that u ₀ is one-homogeneous and harmonic in $\Omega _{u_0}\cap B_1$. This implies that, on the sphere ∂B ₁, u ₀ solves the PDE

$$\displaystyle \begin{aligned} \Delta_{\mathbb S}u_0=(d-1)u_0\quad \text{in}\quad \Omega_{u_0}\cap\partial B_1. \end{aligned}$$

Thus, Theorem 9.21 implies that

$$\displaystyle \begin{aligned} \mathcal{H}^{d-1}(\Omega_u\cap\partial B_1)\ge \frac{d\omega_d}{2}, \end{aligned}$$

which by the homogeneity of u ₀ gives that

$$\displaystyle \begin{aligned} |\Omega_{u_0}\cap B_1|\ge \frac{\omega_d}{2}. \end{aligned}$$

Now, the convergence of $\Omega _{u_{r_n,x_0}}$ to $\Omega _{u_0}$ implies that

$$\displaystyle \begin{aligned} \gamma=\lim_{n\to\infty}\frac{\big|\Omega_u\cap B_{r_n}(x_0)\big|}{|B_{r_n}|}=\lim_{n\to\infty}\frac{\big|\Omega_{u_{r_n,x_0}}\cap B_1\big|}{|B_1|}=\frac{|\Omega_{u_0}\cap B_1|}{|B_1|}\ge \frac 12\,, \end{aligned}$$

which concludes the proof of the lower bound (9.18). In the case of equality γ = 1∕2, we have that $u_0\Big |{ }_{\partial B_1}$ is precisely the first eigenvalue on the half-sphere $\mathbb S_{d-1}^+$, whose one-homogeneous extension is precisely (9.19). □