Introduction and Main Results

Abstract

The free boundary problems are a special type of boundary value problems, in which the domain, where the PDE is solved, depends on the solution of the boundary value problem.

1.1 Free Boundary Problems: Classical and Variational Formulations

The free boundary problems are a special type of boundary value problems, in which the domain, where the PDE is solved, depends on the solution of the boundary value problem. A classical example of a free boundary problem is the Serrin problem:

It is well-known (see [47]) that, up to translation, the unique solution of the Serrin problem is given by the couple (B, w _B), where B is the ball of radius R = d (d is the dimension of the space) and $w_B:B\to \mathbb {R}$ is the function $w_B(x)=\frac 1{2d}\big (R^2-|x|{ }^2\big ).$

More generally, if D is a smooth bounded open set in $\mathbb {R}^d$, then we can consider the following problem. Find a couple ( Ω, u) such that:

• the domain Ω is contained in D

• while the function $u:\Omega \to \mathbb {R}$

  • solves a PDE in Ω, which in the example (1.1) below (as in the rest of these notes) is elliptic but, in general, can also involve a time variable:

    $$\displaystyle \begin{aligned} \sum_{i,j=1}^{d}a_{ij}(x)\partial_{ij} u+\sum_{i=1}^{d}b_{i}(x)\partial_{i} u+c(x)u(x)=f(x)\quad \text{in}\quad \Omega; \end{aligned} $$

    (1.1)

  • satisfies a boundary condition on the fixed boundary ∂D, that is,

    $$\displaystyle \begin{aligned} F(x,u,\nabla u)=0\quad \text{on}\quad \partial D\cap\partial\Omega; \end{aligned} $$

    (1.2)

  • satisfies an overdetermined boundary condition on the free boundary ∂ Ω ∩ D

$$\displaystyle \begin{aligned} G(x,u,\nabla u)=0\quad \text{and}\quad H(x,u,\nabla u)=0\quad \text{on}\quad \partial \Omega\cap D, \end{aligned} $$

(1.3)

where the functions $F, G, H:\mathbb {R}^{2d+1}\to \mathbb {R}$, as well as the elliptic operator and the right-hand side in (1.1), are given. The aim of the free boundary regularity theory is to describe the interaction between the free boundary ∂ Ω and the solution u of the PDE. For instance, it is well-known that, the solutions of boundary value problems (with sufficiently smooth data) inherit the regularity of the boundary ∂ Ω, that is, if ∂ Ω is C ^{1, α}, then |∇u| is Hölder continuous up to the boundary (see [35]). Conversely, one can ask the opposite question. Suppose that u is a solution of the free boundary problem (1.1)–(1.3), where the overdetermined condition (1.3) on the free boundary is given by

$$\displaystyle \begin{aligned} u=0\quad \text{and}\quad |\nabla u|{}^2=Q(x)\quad \text{on}\quad \partial \Omega\cap D, \end{aligned}$$

for some Hölder continuous function Q. Is it true that ∂ Ω is C ^{1, α}-regular? More generally, we can ask the following question:

Is it possible to obtain information on the local structure of the free boundary, just from the fact that the overdetermined boundary value problem admits a solution?

Notice that, here we do not impose any a priori regularity on the domain Ω. For an extensive introduction to the free boundary problems, with numerous concrete examples and applications, we refer to the book [33], while a more advanced reading is [15].

A free boundary problem of particular relevance for the theory is the so-called one-phase Bernoulli problem, which was the object of numerous studies in the last 40 years; it also motivated the introduction of several new tools and the development of new regularity techniques. The problem is the following. We have given:

• a smooth bounded open set D in $\mathbb {R}^d$,

• a non-negative function $g:\partial D\to \mathbb {R}$,

• a positive constant Λ,

and we search for a couple ( Ω, u), of a domain Ω ⊂ D and a function $u:\Omega \to \mathbb {R}$, such that:

$$\displaystyle \begin{aligned} \begin{cases} \Delta u=0\quad \text{in}\quad \Omega,\\ u=g\quad \text{on}\quad \partial\Omega\cap\partial D,\\ u=0\quad \text{and}\quad |\nabla u|=\sqrt{\Lambda}\quad \text{on}\quad \partial\Omega\cap D. \end{cases} \end{aligned} $$

(1.4)

We notice that a solution should depend both on the ambient domain D and the boundary value g. Thus, we cannot hope to find explicitly the domain Ω and the function u, except in some very special cases. In fact, even the existence of a couple ( Ω, u) solving (1.4) is a non-trivial question. One way to solve the existence issue is to consider the variational problem, which consists in minimizing the functional

$$\displaystyle \begin{aligned} u\mapsto \mathcal F_\Lambda(u,D)=\int_D |\nabla u|{}^2\,dx+\Lambda|\{u>0\}\cap D|, \end{aligned}$$

among all functions $u:D\to \mathbb {R}$ such that

$$\displaystyle \begin{aligned} u\in H^1(D)\qquad \text{and}\qquad u=g\quad \text{on}\quad \partial D. \end{aligned}$$

 A solution to (1.4) can be obtained in the following way (see Fig. 1.1). To any minimizer $u:D\to \mathbb {R}$, we associate the domain

$$\displaystyle \begin{aligned} \Omega_u:=\{u>0\}, \end{aligned}$$

and the free boundary ∂ Ω_u ∩ D. Then, at least formally, one can show that the couple ( Ω_u, u) is a solution to the free boundary problem (1.4).

•  First, notice that the conditions

  $$\displaystyle \begin{aligned} u=0\quad \text{on}\quad \partial\Omega_u\cap D, \end{aligned}$$
  $$\displaystyle \begin{aligned} u=g\quad \text{on}\quad \partial\Omega_u\cap \partial D, \end{aligned}$$

  are fulfilled by construction.

  Fig. 1.1

  

  A minimizer u and its free boundary; for simplicity we take D = B ₁

•  In order to show that u is harmonic in Ω_u, we suppose that Ω_u is open and that u is continuous. Let $\varphi \in C^\infty _c(\Omega _u)$ be a smooth function of compact support in Ω_u. Then, for any $t\in \mathbb {R}$ sufficiently close to zero, we have

  $$\displaystyle \begin{aligned} \{u+t\varphi>0\}=\{u>0\}, \end{aligned}$$

  and so,

  $$\displaystyle \begin{aligned} \mathcal F_\Lambda(u+t\varphi,D)=\mathcal F_\Lambda(u,D)+\int_{\Omega_u}\big(|\nabla (u+t\varphi)|{}^2-|\nabla u|{}^2\big)\,dx. \end{aligned}$$

  Now, the minimality of u gives that

  $$\displaystyle \begin{aligned} 2\int_{\Omega_u}\nabla u\cdot\nabla \varphi\,dx=\frac{\partial}{\partial t}\Big|{}_{t=0}\mathcal F_\Lambda(u+t\varphi,D)=0. \end{aligned}$$

  Integrating by parts and using the fact that φ is arbitrary, we get that

  $$\displaystyle \begin{aligned} \Delta u=0\quad \text{in}\quad \Omega_u. \end{aligned}$$

•  Finally, for what concerns the overdetermined condition on the free boundary, we proceed as follows. For any compactly supported smooth vector field $\xi :D\to \mathbb {R}^d$ and any (small) t > 0, we consider the diffeomorphism Ψ_t(x) = x + tξ(x) and the test function $u_t=u\circ \Psi _t^{-1}$. Then, by the optimality of u, we obtain

  $$\displaystyle \begin{aligned} 0=\frac{\partial}{\partial t}\Big|{}_{t=0}\mathcal F_\Lambda(u_t,D). \end{aligned}$$

  On the other hand, the derivative on the right-hand side can be computed explicitly (see Lemma 9.5). Precisely, if we assume that u and ∂ Ω_u are smooth enough, we have

  $$\displaystyle \begin{aligned} \frac{\partial}{\partial t}\Big|{}_{t=0}\mathcal F_\Lambda(u_t,D)=\int_{\partial \Omega_u}\big(-|\nabla u|{}^2+\Lambda\big)\,\xi\cdot\nu\,d\mathcal{H}^{d-1}, \end{aligned}$$

  where ν is the exterior normal to ∂ Ω_u. Since ξ is arbitrary, we get that

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

In conclusion, by minimizing the function $\mathcal F_\Lambda$, we obtain at once the function u and the domain Ω solving (1.4). The function u is a minimizer of $\mathcal F_\Lambda$ and the set Ω is defined as Ω = Ω_u = {u > 0}. The equation in Ω_u and the overdetermined condition on the free boundary ∂ Ω_u ∩ D are in fact the Euler-Lagrange equations associated to the functional. Thus, instead of studying directly the free boundary problem (1.4), in these notes, we will restrict our attention to minimizers of $\mathcal F_\Lambda$. In order to fix the terminology and the notations in this section, and also for the rest of these notes, we give the following definition.

Definition 1.1 (Minimizers of $\mathcal F_\Lambda$)

Let D be a bounded open set in $\mathbb {R}^d$. We say that the function $u:D\to \mathbb {R}$ is a minimizer of $\mathcal F_\Lambda$ in D, if u ∈ H ¹(D), u ≥ 0 in D and

$$\displaystyle \begin{aligned} \mathcal F_\Lambda(u,D)\le \mathcal F_\Lambda(v,D)\qquad \text{for every}\qquad v\in H^1(D) \quad \text{such that}\quad u-v\in H^1_0(D). \end{aligned}$$

1.2 Regularity of the Free Boundary

These notes are an introduction to the free boundary regularity theory; the aim is to describe the local structure of the free boundary ∂ Ω_u (which is a geometric object) just by using the fact that u minimizes the functional $\mathcal F_\Lambda$ and solves an overdetermined boundary value problem (that is, with techniques coming from Calculus of Variations and PDEs). In fact, the free boundary regularity theory stands on the crossroad of Calculus of Variations, PDEs and Geometric Analysis, and is characterized by the interaction between geometric and analytic objects, which is precisely what makes it so fascinating (and hard) field of Analysis.

Our aim in these notes is to prove a first theorem on the local structure of the free boundary. In particular, just by using the fact that u is a minimizer of the functional $\mathcal F_\Lambda$, we will prove the following facts:

• $u:D\to \mathbb {R}$ is (locally) Lipschitz continuous;

• the set Ω_u := {u > 0} is open and the free boundary ∂ Ω_u ∩ D can be decomposed as the disjoint union of a regular part, Reg(∂ Ω_u), and a singular part, Sing(∂ Ω_u),

  $$\displaystyle \begin{aligned} \partial\Omega_u\cap D=Reg(\partial\Omega_u)\cup Sing(\partial\Omega_u)\,, \end{aligned}$$

  for instance, as on Fig. 1.2;

  Fig. 1.2

  

  A picture of a free boundary ∂ Ω_u with regular and singular points

• the regular part Reg(∂ Ω_u) is a C ^{1, α}-smooth manifold of dimension (d − 1);

• the singular part Sing(∂ Ω_u) is a closed subset of ∂ Ω_u ∩ D and its Hausdorff dimension is at most d − 3 (at the moment, the best known estimate for the Hausdorff dimension of the singular set is d − 5).

The overall approach and many of the tools that we will present are universal and have counterparts in other fields, for instance, in the regularity of area-minimizing currents, in free discontinuity problems and harmonic maps. In fact, there are several points which are common for the regularity theory in all these (and many other) variational problems:

• − the local behavior of the solution is determined through the analysis of the so-called blow-up sequences and blow-up limits;

• − the points of the free boundary are labelled regular or singular according to the structure of the so called blow-up limits at each point; this provides a decomposition of the free boundary into a regular part and a singular part;

• − at regular points, the regularity of the free boundary, which might be expressed in geometric (Theorem 7.4) or energetic (Theorem 12.1 and Lemma 12.14) terms, improves along the blow-up sequences;

• − the set of singular points can become bigger when the dimension of the ambient space is higher; the measure and the dimension of the singular set can be estimated through the so-called dimension reduction principle, which uses the fact that the blow-up limit are homogeneous functions; the homogeneity of the blow-up limits can be obtained through a monotonicity formula.

We will prove four main theorems.

In Theorem 1.2 (Sect. 1.3) we prove a regularity result for minimizers of $\mathcal F_\Lambda$. We will obtain the C ^{1, α} regularity of the regular part of the free boundary through an improvement-of-flatness approach, while we will only give a weak estimate on the measure of the singular set. The proof of this theorem is carried out through Chaps. 2–8.

In Theorem 1.4 (Sect. 1.4) we give an estimate on the dimension of the set of singular points. We will use the Weiss monotonicity formula to obtain the homogeneity of the blow-up limits and the Federer dimension reduction principle to estimate the dimension of the singular set. The proof of this theorem is contained in Chaps. 9 and 10.

In Theorem 1.9 (Sect. 1.5) we prove a regularity theorem for functions u minimizing $\mathcal F_0$ under the additional measure constraint | Ω_u| = m. In this case, we show that there is a Lagrange multiplier Λ such that u is a critical point for the functional $\mathcal F_\Lambda$. In this case, the regularity of the free boundary is a more delicate issue and the Theorems 1.2 and 1.4 cannot be applied directly. The proof requires the Chaps. 2–10, and also the specific analysis from Chap. 11.

Theorem 1.10 (Sect. 1.6) is dedicated to the epiperimetric inequality (Theorem 12.1) approach to the regularity of the free boundary, which was introduced in [49]. In particular, we give another proof of the fact that, if u is a local minimizer of $\mathcal F_\Lambda$ in dimension two, then the (entire) free boundary is C ^{1, α} regular. The proof is based on the epiperimetric inequality from Sect. 12, which replaces the improvement of flatness argument from Chap. 7, but we still use results from Chaps. 2, 3, 4, 6, 8 and 9. Finally, we notice that the fact that an epiperimetric inequality in dimension d implies the regularity of the free boundary holds in any dimension (see Sect. 12.5).

The rest of the introduction is organized as follows. Each of the Sects. 1.3, 1.4, 1.5 and 1.6 is dedicated to one of the main theorems 1.2, 1.4, 1.9 and 1.10. Finally, in Sect. 1.7, we briefly discuss some of the results, obtained or just reported in these notes, which might also be of interest for specialists in the field.

1.3 The Regularity Theorem of Alt and Caffarelli

Alt and Caffarelli pioneered the study of the one-phase free boundaries in [3], where they proved the following theorem.

Theorem 1.2 (Alt-Caffarelli)

Let D be a bounded open set in $\mathbb {R}^d$ and u ∈ H ¹(D) be a non-negative minimizer of $\mathcal F_\Lambda$ in D. Then u is locally Lipschitz continuous in D, the set Ω _u = {u > 0} is open and the free boundary can be decomposed as:

$$\displaystyle \begin{aligned} \partial\Omega_u\cap D=Reg(\partial\Omega_u)\cup Sing(\partial\Omega_u), \end{aligned}$$

where Reg(∂ Ω _u) and Sing(∂ Ω _u) are disjoint sets such that:

• (i) Reg(∂ Ω _u) is a C ^{1, α} -regular (d − 1)-dimensional surface in D, for some α > 0;

• (ii) Sing(∂ Ω _u) is a closed set of zero (d − 1)-dimensional Hausdorff measure.

In these notes we will give a proof of this result, which is different from the original one (see [3]) and is based on recent methods developed in several different contexts: for instance, the two-phase problem [4, 50], almost-minimizers for the one-phase problem [19, 50], the one-phase problem for singular operators [18], the vectorial Bernoulli problems [41, 42], shape optimization problems [9, 46]. We will also use tools, which were developed after [3] as, for instance, viscosity solutions [12], [13], [14], [23], [26] and [15], monotonicity formula [52] and epiperimetric inequalities [29, 49].

In order to make these notes easier to read, we give the sketch of the proof in the introduction; for the technical details and generalizations, we refer to the results from the forthcoming chapters.

Proof

In the proof of Theorem 1.2 we will use only results from Chaps. 2–8.

Section 2 is dedicated to the existence of minimizers and also to several explicit examples and preliminary results that will be useful in the forthcoming sections. The existence of minimizers for fixed boundary datum on ∂D is obtained in Proposition 2.1. In Lemma 2.6 and Lemma 2.7 we give two different proofs of the fact that the minimizers of $\mathcal F_\Lambda$ are subharmonic functions. This result has several important applications. First of all, when we study the local behavior of u and of the free boundary ∂ Ω_u, we may assume a priori that the function u is bounded. Moreover, as for a subharmonic function, the limit

exists at every point $x_0\in \mathbb {R}$, we may also assume that u (which is a priori a Sobolev function, so defined as a class of equivalence of Lebesgue measurable functions) is defined pointwise everywhere in D. In particular, we will always work with the precise representative of u, defined by

In particular, the set Ω_u = {u > 0} and its topological boundary ∂ Ω_u are also well-defined (for all these results, we refer to Proposition 2.1). Moreover, in Lemma 2.9, we prove that the topological boundary coincides with the measure-theoretic one in the following sense:

$$\displaystyle \begin{aligned} \partial\Omega_u\cap D=\Big\{x\in D\ :\ |B_r(x)\cap\Omega_u|>0\quad \text{and}\quad |B_r(x)\cap\{u=0\}|>0,\ \forall r>0\Big\}. \end{aligned}$$

In Chap. 3 we prove that the function $u:D\to \mathbb {R}$ is locally Lipschitz continuous in D (Theorem 3.1). The main result of this section is more general (see Theorem 3.2) as for the Lipschitz continuity of u we only use that minimality of the function with respect to outwards perturbations.

We give three different proofs of the local Lipschitz continuity, inspired by three different methods, which were developed in the contexts of different free boundary problems. In Sect. 3.1, we report the original proof of Alt and Caffarelli; in Sect. 3.2, we give a proof which is inspired from the two-phase problem of Alt-Caffarelli-Friedman and already proved to be useful in several different contexts, for instance, for vectorial problems (see [9]) and for operators with drift (see [46]); in Sect. 3.3, we present the proof of Danielli and Petrosyan, which was originally introduced to deal with free boundary problems involving the p-Laplacian (see [18]); each of these subsections can be read independently.

As a consequence of the Lipschitz continuity, we obtain that the set Ω_u is open. Now, from the fact that u minimizes $\mathcal F_\Lambda$, we deduce that u is harmonic on Ω_u:

$$\displaystyle \begin{aligned} \Delta u=0\quad \text{in}\quad \Omega_u\cap D. \end{aligned}$$

In particular, u is C ^∞ regular (and analytic) in Ω_u.

In Chap. 4 (see Lemma 4.4 and/or Lemma 4.5), we prove that u is non-degenerate at the free boundary, that is, there is a constant κ > 0 such that the following claim holds:

$$\displaystyle \begin{gathered} \mathit{If}\ x_0\in \overline\Omega_u\cap D\mathit{, then }\ \|u\|{}_{L^\infty(B_r(x_0))}\ge \kappa r\mathit{,\ for\ every }\ r>0\ \mathit{such\ that}\ B_r(x_0)\subset D. \end{gathered} $$

This means that the Lipschitz estimate from Chap. 3 is optimal at the free boundary. This is a technical result, which we will use several times throughout the proof of Theorem 1.2, for instance, in Chaps. 5, 6 and 8.

In Chap. 5 we use the Lipschitz continuity and the non-degeneracy of u to obtain several results on the measure-theoretic structure of the free boundary. We will use this information in Sect. 6.4 to prove that the singular set has zero (d − 1)-Hausdorff measure. The main results of Chap. 5 are the following:

• In Sect. 5.1 (Lemma 5.1), we prove that there is a constant c ∈ (0, 1) such that, for every x ₀ ∈ D and every radius r small enough,

  $$\displaystyle \begin{aligned} c|B_r|\le |\Omega_u\cap B_r(x_0)|\le (1-c)|B_r|. \end{aligned}$$

  In particular, the free boundary cannot contain points of Lebesgue density 0 or 1.

• In Sect. 5.2 (see Proposition 5.3 and Corollary 5.4), we prove that the set Ω_u has locally finite perimeter in D. We will use this result in Sect. 6.4 in order to estimate the dimension of the singular set.

• In Sect. 5.3 (Proposition 5.7), we prove that the free boundary ∂ Ω_u ∩ D has locally finite (d − 1)-dimensional Hausdorff measure, which is slightly more general result than the one from Corollary 5.4.

Section 6 is dedicated to the convergence of the blow-up sequences and the analysis of the blow-up limits; both being essential for determining the local structure of the free boundary. The notion of a blow-up is introduced in the beginning of Chap. 6 (see Definition 6.1). For convenience of the reader, we anticipate that

$$\displaystyle \begin{aligned} \text{for every}\quad x_0\in \partial\Omega_u\cap D\quad \text{and every infinitesimal sequence}\quad (r_n)_{n\ge1}, \end{aligned}$$

the sequence of rescalings

$$\displaystyle \begin{aligned} u_{x_0,r_n}(x):=\frac 1{r_n}u(x_0+r_nx) \end{aligned}$$

is called a blow-up sequence at x ₀. The (local) Lipschitz continuity of $u:D\to \mathbb {R}$ implies that, up to a subsequence, $u_{x_0,r_n}$ converges to a globally defined Lipschitz continuous function $u_0:\mathbb {R}^d\to \mathbb {R}$. Any function u ₀ obtained in this way is called a blow-up limit of u at x ₀. Notice that the non-degeneracy of u implies that u ₀ cannot be constantly zero. In Proposition 6.2 we prove that the blow-up limit u ₀ is a global minimizer of $\mathcal F_\Lambda$ (see Sect. 6.1) and that the free boundaries $\partial \{u_{x_0,r_n}>0\}$ converge to ∂{u ₀ > 0} locally in the Hausdorff distance (Sect. 6.2).

In Sect. 6.4, we decompose the free boundary into regular and singular parts (see Definition 6.10), Reg(∂ Ω_u) and . Precisely, we say that a point x ₀ ∈ ∂ Ω_u ∩ D is regular, if there is a blow-up limit u ₀, of u at x ₀, of the form

$$\displaystyle \begin{aligned} u_0(x)=\sqrt{\Lambda}\,(x\cdot\nu)_+ \end{aligned} $$

(1.5)

for some unit vector ν. We then prove (see Lemma 6.11) that the regular part Reg(∂ Ω_u) contains the reduced boundary ∂ ^∗ Ω_u ∩ D. This is a consequence to the following two facts: first, at points of the reduced boundary x ₀ ∈ ∂ ^∗ Ω_u ∩ D, the support of the blow-up limits is precisely a half-space {x  :  x ⋅ ν > 0}; second, if u ₀ is a global solution supported on a half-space, then it has the form (1.5). This implies that $\mathcal {H}^{d-1}\big (Sing(\partial \Omega _u)\big )=0$. In fact, this is an immediate consequence of the inclusion Reg(∂ Ω_u) ⊂ ∂ ^∗ Ω_u and a well-known theorem of Federer, which states that if Ω is a set of finite perimeter, then

and of the fact that ∂ Ω ∩( Ω⁽¹⁾ ∪ Ω⁽⁰⁾) = ∅ (see Sect. 5.1). In particular, this completes the proof of claim (ii) of Theorem 1.2.

Chapters 7 and 8 are dedicated to the regularity of Reg(∂ Ω_u) (Theorem 1.2 (i)). We will use the theory presented in this sections both for Theorem 1.2 and Theorem 1.9.

In Sect. 7.1 (Proposition 7.1) we use the examples of radial solutions from Sect. 2.4 (Propositions 2.15 and 2.16) as test functions to prove that the minimizer u satisfies the following optimality condition in viscosity sense:

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

The Sects. 7.2, 7.3 and 7.4 are dedicated to the proof of the improvement-of-flatness theorem of De Silva [23] (Theorem 7.4), which holds for viscosity solutions. We notice that in the two-dimensional case (Theorem 1.10) all the result from this section will be replaced by the epiperimetric inequality approach from Chap. 12.

In Chap. 8 we show how the improvement of flatness implies the regularity of the free boundary. Precisely, in Sect. 8.1 we prove that the improvement of flatness (Condition 8.3) implies the uniqueness of the blow-up limit $u_{x_0}$ at every point x ₀ of the free boundary. Moreover, it provides us with a rate of convergence of the blow-up sequence (Lemma 8.4). Finally, in Sect. 8.2, we show how the uniqueness of the blow-up limit and the rate of convergence of the blow-up sequence imply the C ^{1, α} regularity of the free boundary (Proposition 8.6), which concludes the proof of Theorem 1.2. □

Remark 1.3

The proof of the regularity of Reg(∂ Ω_u) is based on an improvement-of-flatness argument and is due to De Silva (see [23]). Just as the original proof of Alt and Caffarelli it is based on comparison arguments and does not make use of any type of monotonicity formula. In order to keep the original spirit of [3], we do not use monotonicity formulas in the proof of Theorem 1.2 (Chaps. 2–8). On the other hand, without a monotonicity formula, one can prove that the singular set has zero (d − 1)-dimensional Hausdorff measure. Notice that, in [3] it was also shown that the singular set is empty in dimension two. We postpone this result to Sect. 9.4 since it is a trivial consequence of the monotonicity formula of Weiss. We also notice that the proof of Theorem 1.2 is essentially self-contained and requires only basic knowledge on Sobolev spaces and elliptic PDEs.

1.4 The Dimension of the Singular Set

In Theorem 1.2, we show that the singular part of the free boundary Sing(∂ Ω_u) has the following properties:

• it is a closed subset of the free boundary ∂ Ω_u ∩ D;

• it has zero Hausdorff measure, that is, $\mathcal {H}^{d-1}\big (Sing(\partial \Omega _u)\big )=0$; in particular, this implies that the (Hausdorff) dimension of Sing(∂ Ω_u) is at most d − 1.

In [52], using a monotonicity formula and the Federer’s dimension reduction principle, Weiss proved the following result.

Theorem 1.4 (Weiss)

Let D be a bounded open set in $\mathbb {R}^d$ and u ∈ H ¹(D) be a non-negative minimizer of $\mathcal F_\Lambda$ in D. Let Reg(∂ Ω _u) and Sing(∂ Ω _u) be the regular and singular sets from Theorem 1.2 . There exists a critical dimension d ^∗ (see Definition 1.5) such that the following holds.

• (i) If d < d ^∗ , then Sing(∂ Ω _u) is empty.

• (ii) If d = d ^∗ , then the singular set Sing(∂ Ω _u) is a discrete (locally finite) set of isolated points in D.

• (iii) If d > d ^∗ , then the singular set Sing(∂ Ω _u) is a closed set of Hausdorff dimension d − d ^∗ , that is,

  $$\displaystyle \begin{aligned} \mathcal{H}^{d-d^\ast+{\varepsilon}}(\partial\Omega_u\cap D)=0\quad \mathit{\text{for every}}\quad {\varepsilon}\in(0,1). \end{aligned}$$

Definition 1.5 (Definition of d ^∗)

We will denote by d ^∗ the smallest dimension d such that there exists a function $z:\mathbb {R}^d\to \mathbb {R}$ with the following properties:

• z is non-negative and one-homogeneous;

• z is a local minimizer of $\mathcal F_\Lambda$ in $\mathbb {R}^d$;

• the free boundary ∂ Ω_z is not a (d − 1)-dimensional C ¹-regular manifold in $\mathbb {R}^d$.

Remark 1.6

The value of d ^∗ does not depend on Λ > 0. Without loss of generality, we may take Λ = 1.

Remark 1.7 (On the Critical Dimension d ^∗)

In this notes, we prove that d ^∗≥ 3 (see Sect. 9.4). Already this is a better estimate (on the dimension of the singular set) with respect to the one from Theorem 1.2 as it means that

$$\displaystyle \begin{aligned} \mathcal{H}^{d-3+{\varepsilon}}(\partial\Omega_u\cap D)=0\quad \text{for every}\quad {\varepsilon}\in(0,1). \end{aligned}$$

In fact, it is now known that

$$\displaystyle \begin{aligned} d^\ast=5,6,\text{ or }7. \end{aligned}$$

Precisely, Caffarelli, Jerison and Kenig [16] proved that there are no singular one-homogeneous global minimizers in $\mathbb {R}^3$ (thus, d ^∗≥ 4). Later, Jerison and Savin [37] proved the same result in $\mathbb {R}^4$ (so, d ^∗≥ 5). On the other hand, De Silva and Jerison [24] gave an explicit example (see Fig. 1.3) of a singular free boundary in dimension seven (which means that d ^∗≤ 7).

Fig. 1.3

The free boundary (in red) of the one-homogeneous global solution $u: \mathbb {R}^7\to \mathbb {R}$ of De Silva and Jerison

In order to prove Theorem 1.4 we will need most of the theory developed for the proof of Theorem 1.2. For instance, the Lipschitz continuity and the non-degeneracy of the minimizers (Chaps. 3 and 4), the convergence of the blow-up sequences (Chap. 6) and the epsilon regularity theorem (Theorem 8.1 from Chap. 8). On the other hand, we will not need the results from Chap. 5.

The main results that we will need for the proof of Theorem 1.4 are contained in Chaps. 9 and 10. Chapter 9 is dedicated to the Weiss monotonicity formula from [52], which we prove both for minimizing and stationary free boundaries. Chapter 10 is dedicated to the Federer’s dimension reduction principle (see [32]). Even if the results of this section concern the one-phase free boundaries, the underlying principle is universal and can be applied to numerous other problems; for instance, in geometric analysis (see [32] and [48]) or to other free boundary problems [42].

Proof of Theorem 1.4

We will first prove that all the blow-up limits of u (at any point of the free boundary) are one-homogeneous global minimizers of $\mathcal F_\Lambda$. The global minimality (see Definition 2.12) of the blow-up limits follows from Proposition 6.2. In order to prove the one-homogeneity of the blow-up limits (Proposition 9.12) we will use the Weiss’ boundary adjusted energy, defined for any function φ ∈ H ¹(B ₁) as

$$\displaystyle \begin{aligned} W_\Lambda(\varphi):=\int_{B_1}|\nabla \varphi|{}^2\,dx-\int_{\partial B_1}\varphi^2\,d\mathcal{H}^{d-1}+\Lambda \big|\{\varphi>0\}\cap B_1\big|. \end{aligned}$$

Let now x ₀ ∈ ∂ Ω_u ∩ D and $u_{x_0,r}$ be the usual rescaling (blow-up sequence)

$$\displaystyle \begin{aligned} u_{r,x_0}(x)=\frac 1ru(x_0+rx). \end{aligned}$$

If we choose r > 0 small enough, then the function $u_{x_0,r}$ is defined on B ₁ and so, we can compute the Weiss energy $W_\Lambda (u_{x_0,r})$. In Lemma 9.2 we compute the derivative of $W_\Lambda (u_{x_0,r})$ with respect to r, from which we deduce that (see Proposition 9.4):

• the function $r\mapsto W_\Lambda (u_{x_0,r})$ is monotone increasing in r;

• and is constant on an interval of the form (0, R), if and only if, u is one-homogeneous in the ball B _R(x ₀).

In particular, the monotonicity of $r\mapsto W_\Lambda (u_{x_0,r})$ and the Lipschitz continuity of u (which gives a lower bound on $W_\Lambda (u_{x_0,r})$) imply that the limit

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

exists and is finite.

Let now v be a blow-up limit of u at x ₀ and (r _n)_n be an infinitesimal sequence such that

$$\displaystyle \begin{aligned} v=\lim_{n\to\infty} u_{x_0,r_n}. \end{aligned}$$

Let s > 0 be fixed. Then, the blow-up sequence $u_{x_0,sr_n}=\frac {1}{sr_n}u(x_0+sr_nx)$ converges locally uniformly to the rescaling $v_s(x):=\frac 1sv(sx)$ of the blow-up v. Now, Proposition 6.2 implies that:

• the sequence $u_{x_0,sr_n}$ converges to v _s strongly in H ¹(B ₁);

• the sequence of characteristic functions converges to the characteristic function in L ¹(B ₁).

Thus, for every s > 0, we have

$$\displaystyle \begin{aligned} L=\lim_{r\to0}W_\Lambda(u_{x_0,r})=\lim_{n\to\infty}W_\Lambda(u_{x_0,sr_n})=W_\Lambda(v_s), \end{aligned}$$

and so the function s↦W _Λ(v _s) is constant in s. Applying again Proposition 9.4, we get that v is one-homogeneous.

Theorem 1.4 now follows by the more general result proved in Proposition 10.13, which can be applied to u since we have the epsilon regularity theorem (Theorem 8.1), the non-degeneracy of u (see Chap. 4), the strong convergence of the blow-up sequences (Proposition 6.2) and the homogeneity of the blow-up limits, which we proved above. □

Remark 1.8

Finally, we notice that an even better result was recently obtained by Edelen and Engelstein (see [27]). Using the powerful method of Naber and Valtorta (see [44]), they proved that the singular set Sing(∂ Ω_u) has locally finite (d − d ^∗)—Hausdorff measure, which in particular implies claim (ii) of Theorem 1.4.

1.5 Regularity of the Free Boundary for Measure Constrained Minimizers

Let $D\subset \mathbb {R}^d$ be a smooth and connected bounded open set, m ∈ (0, |D|) and $g:D\to \mathbb {R}$ be a given non-negative function in H ¹(D). This section is dedicated to the following variational minimization problem with measure constraint

$$\displaystyle \begin{aligned} \min\big\{\mathcal F_0(v,D)\ :\ v\in H^1(D),\ v-g\in H^1_0(D),\ |\Omega_v|=m\big\}, \end{aligned} $$

(1.6)

which means

$$\displaystyle \begin{gathered} \mathit{Find}\ u\in H^1(D)\ \mathit{such\ that}\ u-g\in H^1_0(D),\ |\Omega_u|=m\ \mathit{and}\\ \mathcal F_0(u,D)\le \mathcal F_0(v,D),\ \mathit{for\ every}\ v\in H^1(D)\ \mathit{such\ that}\ v-g\in H^1_0(D)\ \mathit{and}\ |\Omega_v|=m. \end{gathered} $$

This is the constrained version of the variational problem from Theorems 1.2 and 1.4. We notice that if u is a minimizer of $\mathcal F_\Lambda$ in D, for some Λ > 0, then u is (obviously) a solution to the minimization problem (1.6) with m := | Ω_u|. Conversely, if u is a solution to the variational problem (1.6), then (as we will show in Proposition 11.2) there is a Lagrange multiplier Λ > 0, depending on u, such that u formally satisfies the optimality condition

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

(1.7)

in the sense that u is stationary for $\mathcal F_\Lambda$ in D (see Definition 9.7). Unfortunately, this does not imply that u is a minimizer of $\mathcal F_\Lambda$ in D. The free boundary regularity theory for the solutions to (1.6) is more involved since the competitors used to prove the Lipschitz continuity (Chap. 3), non-degeneracy (Chap. 4), improvement of flatness (Chap. 7) and the monotonicity formula (Proposition 9.4) do not satisfy the measure constraint in (1.6).

The free boundary regularity for solutions of (1.6) was first obtained by Aguilera, Alt and Caffarelli in [1]. Our approach is different and strongly relies on the Weiss’ monotonicity formula, from which we will deduce both:

• the optimality condition in (1.7) in viscosity sense, which in turn allows to apply the De Silva epsilon regularity theorem (Theorem 8.1) and thus to obtain the C ^{1, α}-regularity of Reg(∂ Ω_u) (see Chap. 8);

• the estimate of the dimension of the singular set, which is a consequence of the homogeneity of the blow-up limits and the Federer’s dimension reduction (Chap. 10).

Our approach is inspired by the theory developed in [46] and contains several ideas from [41] and the work of Briançon [5] and Briançon-Lamboley [6]. Our main result is the following.

Theorem 1.9 (Regularity of the Measure Constrained Minimizers)

Let D be a connected smooth bounded open set in $\mathbb {R}^d$ , m ∈ (0, |D|) be a positive real constant and $g:D\to \mathbb {R}$ be a given non-negative function in H ¹(D). Then, there is a solution to the problem (1.6). Moreover, every solution u is non-negative and locally Lipschitz continuous in D, the set Ω _u = {u > 0} is open and the free boundary can be decomposed as:

$$\displaystyle \begin{aligned} \partial\Omega_u\cap D=Reg(\partial\Omega_u)\cup Sing(\partial\Omega_u), \end{aligned}$$

where Reg(∂ Ω _u) and Sing(∂ Ω _u) are disjoint sets such that:

• (i) the regular part Reg(∂ Ω _u) is a C ^{1, α} -regular (d − 1)-dimensional manifold in D, for some α > 0;

• (ii) the singular part Sing(∂ Ω _u) is a closed set of Hausdorff dimension d − d ^∗ (where the critical dimension d ^∗ is again given by Definition 1.5), that is,

  $$\displaystyle \begin{aligned} \mathcal{H}^{d-d^\ast+{\varepsilon}}(\partial\Omega_u\cap D)=0\quad \mathit{\text{for every}}\quad {\varepsilon}\in(0,1). \end{aligned}$$

Moreover, if d < d ^∗ , then Sing(∂ Ω _u) is empty, and if d = d ^∗ , then Sing(∂ Ω _u) is a countable discrete (locally finite) set of points in D.

Proof of Theorem 1.4

We prove the existence of a solution $u:D\to \mathbb {R}$ in Sect. 11.1, where we also show that u is harmonic in Ω_u in the following sense

In particular, applying Lemma 2.7, we get that u is subharmonic in D. Thus, we can suppose that u is defined at every point of D and that

Moreover, the subharmonicity of u implies that it is locally bounded so, from now on, without loss of generality, we will assume that u ∈ L ^∞(D). Finally, we notice that the set Ω_u is defined everywhere in D (not just up to a set of zero Lebesgue measure) and its topological boundary coincides with the measure-theoretic one (see Lemma 2.9). Precisely, this means that

$$\displaystyle \begin{aligned} x_0\in\partial\Omega_u\qquad \text{if and only if}\qquad 0<|\Omega_u\cap B_r(x_0)|\le |B_r|\quad \text{for every}\quad r>0. \end{aligned}$$

In order to prove the Lipschitz continuity of u and the regularity of the free boundary ∂ Ω_u ∩ D we proceed in several steps. Notice that we cannot apply directly the results from Chaps. 3–10 since it is not a priori known if the solution u is a local minimizer of $\mathcal F_\Lambda$ for some Λ > 0, that is, one cannot remove the constraint in (1.6) by adding a Lagrange multiplier Λ directly in the functional. In fact, it is only possible to prove the existence of Λ for which the solution u of (1.6) is stationary but not minimal for $\mathcal F_\Lambda$. From this, we will deduce that u satisfies a quasi-minimality condition, which will allow to proceed as in the proof of Theorems 1.2 and 1.4.

Step 1 Existence of a Lagrange multiplier. In Sect. 11.2, we show that there exists Λ > 0 such that u is stationary for the functional $\mathcal F_\Lambda$, that is,

$$\displaystyle \begin{aligned} \partial\mathcal F_\Lambda(u,D)[\xi]=0\quad \text{for every}\quad \xi\in C^{\infty}_c(D;\mathbb{R}^d), \end{aligned}$$

where the first variation $\partial \mathcal F_\Lambda (u,D)[\xi ]$ of $\mathcal F_\Lambda$ in the direction of the (compactly supported) vector field ξ is defined in (9.6). The existence of a non-negative Lagrange multiplier can be obtained by a standard variational argument (see Proposition 11.2 and its proof in Sect. 11.2, after Lemma 11.3). The strict positivity of Λ is a non-trivial question which requires some fine analysis of the functions, which are stationary for $\mathcal F_0$; we prove it in Sect. 11.3 using the Almgren’s frequency function and following the proof of an analogous result from [46], which is a (small with respect to the original result) improvement of the unique continuation principle of Garofalo-Lin [34].

Step 2 Almost-minimality of u. Let x ₀ ∈ ∂ Ω_u ∩ D. In Sect. 11.5 (Proposition 11.10), we prove that u is an almost minimizer of $\mathcal F_\Lambda$ (Λ is the Lagrange multiplier from the previous step) in a neighborhood of x ₀ in the following sense. There exists a ball B, centered in x ₀, in which u satisfies the following almost-minimality condition:

For every ε > 0, there is r > 0 such that, for every ball B _r(y ₀) ⊂ B, u satisfies the following optimality conditions in B _r(y ₀):

$$\displaystyle \begin{aligned} \mathcal F_{\Lambda+{\varepsilon}}(u,D)\le \mathcal F_{\Lambda+{\varepsilon}}(v,D)\ \,\text{for every}\ \, v\in H^1(D)\ \,\text{such that}\ \, \begin{cases}v-u\in H^1_0(B_r(y_0)),\\ |\Omega_u|\le|\Omega_v|. \end{cases} \end{aligned} $$

(1.8)

$$\displaystyle \begin{aligned} \mathcal F_{\Lambda-{\varepsilon}}(u,D)\le \mathcal F_{\Lambda-{\varepsilon}}(v,D)\ \,\text{for every}\ \, v\in H^1(D)\ \,\text{such that}\ \, \begin{cases}v-u\in H^1_0(B_r(y_0)),\\ |\Omega_u|\ge|\Omega_v|. \end{cases} \end{aligned} $$

(1.9)

The proof of Proposition 11.10 follows step-by-step the proof of the analogous result from [46] and is based on the method of Briançon [5]. Once we have Proposition 11.10, we can proceed as in Theorems 1.2 and 1.4.

Step 3 Lipschitz continuity and non-degeneracy of u. In order to prove the (local) Lipschitz continuity of u, we notice that (1.8) leads to an outwards optimality condition. In fact, fixed ε > 0 and x ₀ ∈ D, there is a ball B _r(x ₀) such that:

$$\displaystyle \begin{aligned} \mathcal F_{\Lambda+{\varepsilon}}(u,D)\le \mathcal F_{\Lambda+{\varepsilon}}(v,D)\ \,\text{for every}\ \, v\in H^1(D)\ \,\text{such that}\ \, \begin{cases}v-u\in H^1_0(B_r(x_0)),\\ \Omega_u\subset\Omega_v. \end{cases} \end{aligned} $$

(1.10)

Now, the Lipschitz continuity of u follows by (1.10) and Theorem 3.2.

On the other hand, for the non-degeneracy of u, we notice that, (1.9) implies the following inwards optimality condition:

Fixed ε > 0 and x ₀ ∈ D, there is a ball B _r(x ₀) such that:

$$\displaystyle \begin{aligned} \mathcal F_{\Lambda-{\varepsilon}}(u,D)\le \mathcal F_{\Lambda-{\varepsilon}}(v,D)\ \,\text{for every}\ \, v\in H^1(D)\ \,\text{such that}\ \, \begin{cases}v-u\in H^1_0(B_r(x_0)),\\ \Omega_u\supset\Omega_v. \end{cases} \end{aligned} $$

(1.11)

The non-degeneracy of u follows by (1.11) and the results from Chap. 4 (one can apply both Lemma 4.4 and 4.5).

As a consequence of the Lipschitz continuity and the non-degeneracy of u, we obtain the following results:

• Ω_u satisfies interior and exterior density estimates (Lemma 5.1);

• Ω_u has locally finite perimeter in D (Proposition 5.3);

• ∂ Ω_u has locally finite (d − 1)-dimensional Hausdorff measure in D (Proposition 5.7).

Step 4. Convergence of the blow-up sequences and analysis of the blow-up limits. We recall that, for any x ₀ ∈ D and any r > 0, the function

$$\displaystyle \begin{aligned} u_{x_0,r}(x):=\frac 1ru(x_0+rx), \end{aligned}$$

is well-defined on the set $\frac 1r(-x_0+D)$ and, in particular, on the ball of radius $\frac 1r\text{dist}\,(x_0,\partial D)$ centered in zero. By the Lipschitz continuity of u, we notice that for any x ₀ ∈ ∂ Ω_u ∩ D and any R > 0 the family of functions

$$\displaystyle \begin{aligned} \Big\{u_{x_0,r}\ :\ 0<r<\frac 1R\text{dist}\,(x_0,\partial D)\Big\}, \end{aligned}$$

is equicontinuous and uniformly bounded on the ball $\overline B_R\subset \mathbb {R}^d$. This implies that for every sequence $u_{x_0,r_n}$, with x ₀ ∈ ∂ Ω_u ∩ D and lim_n→∞ r _n = 0, there are a subsequence (still denoted by $(u_{x_0,r_n})_{n\in \mathbb {N}}$) and a (Lipschitz) function $u_0:\mathbb {R}^d\to \mathbb {R}$ such that, for every fixed R > 0, the sequence $u_{x_0,r_n}$ converges uniformly to u ₀ in the ball B _R. We say that u ₀ is a blow-up limit of u at x ₀ and $u_{x_0,r_n}$ is a blow-up sequence. Recall that u is Lipschitz continuous, non-degenerate, harmonic in Ω_u and satisfies the following quasi-minimality condition, which is a direct consequence of (1.8) and (1.9). For every x ₀ ∈ ∂ Ω_u ∩ D, there are r ₀ > 0 and a continuous non-negative function ${\varepsilon }:[0,r]\to \mathbb {R}$, vanishing in zero and such that

$$\displaystyle \begin{aligned} \mathcal F_\Lambda(u,D)\le & \mathcal F_{\Lambda}(v,D)+{\varepsilon}(r)|B_r|\quad \text{for every}\quad 0<r\le r_0\\ & \quad \text{and every}\quad v\in H^{1}(D)\quad \text{such that}\quad u-v\in H^1_0(B_r(x_0)).\qquad {} \end{aligned} $$

Let $u_{x_0,r_n}$ be a blow-up sequence converging locally uniformly to the blow-up limit u ₀. By Proposition 6.2 and the results of Sect. 6.2 we have that, for every R > 0,

• (i) the sequence $u_{x_0,r_n}$ converges to u ₀ strongly in H ¹(B _R);

• (ii) the sequence of characteristic functions converges to in L ¹(B _R), where

  $$\displaystyle \begin{aligned} \Omega_n:=\{u_{x_0,r_n}>0\}\qquad \text{and}\qquad \Omega_0:=\{u_0>0\}\,; \end{aligned}$$

• (iii) the sequence of sets $\overline \Omega _n$ converges locally Hausdorff in B _R to $\overline \Omega _0$;

Moreover, using again Proposition 6.2, we get that every blow-up limit u ₀ of u is a global minimizer of $\mathcal F_\Lambda$. Next, since u is a critical point of $\mathcal F_\Lambda$, we can apply Lemma 9.11 obtaining that every blow-up limit of u ₀ is one-homogeneous. We summarize this in the following statement, with which we conclude this step of the proof:

$$\displaystyle \begin{aligned} \text{Every blow-up of}\ u\ \text{is a one-homogeneous global minimizer of}\ \mathcal F_\Lambda.\end{aligned} $$

(1.12)

Step 5. Optimality condition on the free boundary. Using the convergence of the blow-up sequences (proved in the previous step) and the structure of the blow-up limits (claim (1.12)), we can apply Proposition 9.18. Thus, u is a viscosity solution of

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

(1.13)

in viscosity sense (see Definition 7.6).

Step 6. Decomposition of the free boundary into a regular and a singular parts. As in the proof of Theorem 1.2, we say that x ₀ ∈ Reg(∂ Ω_u) if x ₀ ∈ ∂ Ω_u ∩ D and there exists a blow-up limit u ₀ of u (at x ₀), for which there is a unit vector $\nu \in \mathbb {R}^d$ such that

$$\displaystyle \begin{aligned} u_0(x)=\sqrt{\Lambda}\,(x\cdot \nu)_+\quad \text{for every}\quad x\in\mathbb{R}^d. \end{aligned}$$

The singular part of the free boundary is defined as . The C ^{1, α}-regularity of Reg(∂ Ω_u) now follows by Theorem 8.1 and the fact that u is a solution of (1.13). The estimate on the dimension of the singular set (Theorem 1.9 (ii)) now follows directly from Proposition 10.13. □

1.6 An Epiperimetric Inequality Approach to the Regularity of the Free Boundary in Dimension Two

This section is dedicated to a recent alternative approach to the regularity of the free boundaries based on the so-called epiperimetric inequality, which was first introduced by Reifenberg in the contact of area-minimizing surfaces, but in the context of the one-phase problem, it was first proved in [49]. We restrict our attention to the two-dimensional case since the epiperimetric inequality is (for now) known to hold only in dimension two (see Theorem 12.1 and Theorem 12.3). Precisely, we will give an alternative proof to the following result.

Theorem 1.10 (Regularity of the Free Boundary in Dimension Two)

Let D be a bounded open set in $\mathbb {R}^2$ . Let $u:D\to \mathbb {R}$ be a non-negative function and a minimizer of $\mathcal F_\Lambda$ in D. Then:

• (i) u is locally Lipschitz continuous in D and the set Ω _u = {u > 0} is open;

• (ii) the free boundary ∂ Ω _u ∩ D is C ^{1, α} -regular.

Proof of Theorem 1.4

We first notice that the Lipschitz continuity of u follows by Theorem 3.1. In what follows, without loss of generality we assume that Λ = 1. By the non-degeneracy of the solutions (Chap. 4) and the convergence of the blow-up sequences (Chap. 6, Proposition 6.2), we have that, for every free boundary point x ₀ ∈ ∂ Ω_u and every infinitesimal sequence r _n → 0, there exists a subsequence of r _n (still denoted by r _n) such that $u_{x_0,r_n}$ converges locally uniformly to a non-trivial blow-up limit $u_0:\mathbb {R}^2\to \mathbb {R}$. Moreover,

• the sequence $u_{x_0,r_n}$ converges to u ₀ strongly in H ¹(B ₁);

• the sequence of characteristic functions converge to in L ¹(B ₁).

Next, we notice that by the Weiss monotonicity formula (Proposition 9.4) the function $r\mapsto W_1(u_{x_0,r})$ is monotone increasing in r and the blow-up limit u ₀ is one-homogeneous global minimizer of $\mathcal F_1$ in $\mathbb {R}^2$ (see Lemma 9.10). Thus, by Proposition 9.13, we obtain that u ₀ is a half-plane solution, that is

$$\displaystyle \begin{aligned} u_0(x)=(x\cdot\nu)_+\,, \end{aligned}$$

for some unit vector $\nu \in \mathbb {R}^2$. Now, the strong convergence of the blow-up sequence and the monotonicity formula (Proposition 9.4) imply that

$$\displaystyle \begin{aligned} \inf_{r>0}W_1(u_{x_0,r})=\lim_{r\to0}W_1(u_{x_0,r})=\lim_{n\to\infty}W_1(u_{x_0,r_n})=W_1(u_0)=\frac\pi2. \end{aligned}$$

In conclusion, we have that:

• the energy

  $$\displaystyle \begin{aligned} \mathcal E(u):=W_1(u)-\frac\pi2, \end{aligned}$$

  is non-negative along any blow-up sequence $u_{x_0,r}$ with x ₀ ∈ ∂ Ω_u ∩ D,

  $$\displaystyle \begin{aligned} \mathcal E(u_{x_0,r}):=W_1(u_{x_0,r})-\frac\pi2\ge 0\quad \text{for every}\quad r>0; \end{aligned}$$

• the free boundary is flat, that is, for every x ₀ ∈ ∂ Ω_u ∩ D and every ε > 0, there exists r > 0 and ν ∈ ∂B ₁, such that:

  $$\displaystyle \begin{aligned} (x\cdot\nu-{\varepsilon})_+\le u_{x_0,r}(x)\le (x\cdot\nu+{\varepsilon})_+\quad \text{for every}\quad x\in B_1. \end{aligned}$$

Now, by the epiperimetric inequality (Theorem 12.1) and Proposition 12.13, we obtain that, in a neighborhood of x ₀, ∂ Ω_u is the graph of a C ^{1, α} regular function. □

1.7 Further Results

The main objective of these notes is to introduce the reader to the free boundary regularity theory and to provide a complete and self-contained proof of the regularity of the one-phase free boundaries. In this perspective, our main results are Theorems 1.2, 1.4, 1.9 and 1.10. On the other hand, in these notes, we also prove several other results, which might be interesting for specialists and non. Here is a list of results, by section, which are worth to be mentioned in this context.

Chapter 2

In Proposition 2.10 we give a direct proof to the fact that the half-plane solutions are global minimizers of $\mathcal F_\Lambda$. This is well-known, as the result can be obtained from the following facts:

• − the blow-up limits of a solution u at points of the reduced boundary ∂ ^∗ Ω_u are half-plane solutions (Lemma 6.11);

• − the reduced boundary ∂ ^∗ Ω_u is non-empty as Ω_u is a set of finite perimeter (Proposition 5.3) and for sets of finite perimeter we have the identity $Per(\Omega _u)=\mathcal {H}^{d-1}(\partial ^\ast \Omega _u)$ (see [43]).

In Lemmas 2.15 and 2.16 we prove the existence and the uniqueness of two one-phase free boundary problems. Moreover, we prove that the solutions are radially symmetric and we write them explicitly.

Chapters 3 and 4

In Proposition 3.15 and Lemma 4.5, we present the methods of Danielli-Petrosyan ([18], for the Lipschitz continuity) and David-Toro ([19], for the non-degeneracy) in the simplified context of the classical one-phase Bernoulli problem. Both methods are very robust and can be applied to more general free boundary problems.

Chapter 5

In Proposition 5.3 we prove that if u is a minimizer of $\mathcal F_\Lambda$ in a set D, then Ω_u has locally finite De Giorgi perimeter in D. The method is a localized version of a global estimate by Bucur (see [8]), on the perimeter of the optimal shapes for the eigenvalues of the Dirichlet Laplacian.

In Proposition 5.7 we prove that, if u is a minimizer of $\mathcal F_\Lambda$ in a set D, then the $\mathcal {H}^{d-1}$ Hausdorff measure of the free boundary ∂ Ω_u is locally finite in D. The method is very general and can be applied to many different free boundary problems, for instance, to the vectorial problem (see [42]).

Chapter 6

In Proposition 6.2 we give the detailed proof of the strong convergence of the blow-up sequences, which is often omitted in the literature. Moreover, we state and prove a general result (Lemma 6.3) which can be applied to different free boundary and shape optimization problems.

Chapter 7

In Proposition 7.1 we prove that if u is a minimizer of $\mathcal F_\Lambda$ in D, then it is satisfies the optimality condition

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

in viscosity sense (Definition 7.6). This result is well-known, but in the literature the proof is usually omitted. Our proof of Proposition 7.1 is based on a comparison with the radial solutions constructed in Lemmas 2.15 and 2.16. We give another proof of this fact in Chap. 9.

Chapter 8

In this section we give a detailed proof of the fact that the improvement of flatness (Condition 8.3) implies the C ^{1, α} regularity of the free boundary (see Lemma 8.4 and Proposition 8.6). In particular, in Sect. 8.2, we explain the relation between the uniqueness of the blow-up limits, the rate of convergence of the blow-up sequences, and the regularity of the free boundary (Proposition 8.6).

Chapter 9

In Sect. 9.5, we give another proof of the fact that, if u is a local minimizer of $\mathcal F_\Lambda$ in D, then it satisfies the optimality condition

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

in viscosity sense (see also Proposition 7.1). The method that we propose is based on the Weiss monotonicity formula and is very robust, for instance, it applies to general operators (see [46]) and to vectorial problems (see [41]). This method was first introduced in [41].

Chapter 10

This section is an introduction to the Federer’s Dimension Reduction Principle in the context of free boundary problems. Our main result (Proposition 10.13) is an estimate on the dimension of the singular set under general conditions.

Chapter 11

In Sect. 11.3 we combine the unique continuation principle of Garofalo-Lin [34] with the Faber-Krahn-type inequality from [10] to prove a strong unique continuation result for stationary functions of the Dirichlet energy $\mathcal F_0$ (see Proposition 9.19 and [46]).

Chapter 12

This section is dedicated to the epiperimetric inequality (Theorem 12.1) that first appeared in [49]. We give here a different proof that inspired the approach to the epiperimetric inequality at the singular points in higher dimension (see [29]).

In Lemma 12.14 we prove that the epiperimetric inequality at the flat free boundary points in any dimension (Condition 12.12) implies the regularity of the free boundary. The proof is similar to the one in [49], but has to deal with the closeness condition in the epiperimetric inequality (see Condition 12.12), precisely as in [29] and [28].

In Sect. 12.6 we prove comparison results for minimizers of $\mathcal F_\Lambda$ (Proposition 12.19 and Lemma 12.22) and for viscosity solutions (Lemma 12.21).

In Theorem 12.3 we prove an epiperimetric inequality in dimension two without any specific assumption on the trace on the sphere. This results covers both Theorem 12.1 and the main theorem of [49]. Both Theorem 12.3 and Theorem 12.1 are new results.