A general class of free boundary problems for fully nonlinear parabolic equations

Abstract

In this paper, we consider the fully nonlinear parabolic free boundary problem

$$\begin{aligned} \left\{ \begin{array}{ll} F(D^2u) -\partial _{t} u=1 &{} \text {a.e. in }Q_1 \cap \Omega \\ |D^2 u| + |\partial _{t} u| \le K &{} \text {a.e. in }Q_1{\setminus }\Omega , \end{array} \right. \end{aligned}$$

where $K>0$ is a positive constant, and $\Omega$ is an (unknown) open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that $W_x^{2,n} \cap W_t^{1,n}$ solutions are locally $C_x^{1,1}\cap C_t^{0,1}$ inside $Q_1$. A key starting point for this result is a new BMO-type estimate, which extends to the parabolic setting the main result in Caffarelli and Huang (Duke Math J 118(1):1–17, 2003). Once optimal regularity for $u$ is obtained, we also show regularity for the free boundary $\partial \Omega \cap Q_1$ under the extra condition that $\Omega \supset \{ u \ne 0 \}$, and a uniform thickness assumption on the coincidence set $\{ u = 0 \}$.

1 Introduction and main result

1.1 Background

This paper is the parabolic counterpart of our earlier work [10] on fully nonlinear elliptic free boundary problems of obstacle type. The problem at hand concerns very generalized version of free boundary problems that have been in focus in the last two decades.

The particular application, in the linear theory, is related to “inverse Cauchy–Kowalevskaya theory.” This amounts to showing that if a domain $\Omega \subset \mathbb {R}^{n+1}$ admits a solution to the overdetermined problem

$$\begin{aligned} \Delta u - \partial _t u=1 \quad \text {in}\quad \Omega ,\qquad u=\nabla u=0 \quad \text {on}\quad \partial \Omega , \end{aligned}$$

then both the solution and the boundary must be reasonably smooth. Notice that, by Cauchy–Kowalevskaya theory, it is well known that for smooth enough boundaries there is a solution to the above problem in a neighborhood of $\partial \Omega$, hence the question asked here is the converse.

In this paper, we shall consider a much more general version of this question, allowing fully nonlinear parabolic equations of the type $\mathcal {H}( u):= F(D^2u)- \partial _t u$, as well as a more general equation, see (1.1) below.

1.2 Setting of the problem

We will use $Q_r(X):=B_r(x)\times (t-r,t)\subset \mathbb {R}^n\times \mathbb {R}$ to denote the parabolic ball of radius $r$ centered at a point $X=(x,t) \in \mathbb {R}^{n+1}$, and we will use the notation $Q_r=Q_r(0)$.

Our starting point will be a $W_x^{2,n}(Q_1)\cap W_t^{1,n}(Q_1)$ function $u:Q_1 \rightarrow \mathbb {R}$ satisfying

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {H}(u)=1 &{} \text {a.e. in }Q_1 \cap \Omega ,\\ |{\tilde{D}}^2 u| \le K &{} \text {a.e. in }Q_1{\setminus }\Omega , \end{array} \right. \end{aligned}$$

(1.1)

where ${\tilde{D}}^2 u = (D_x^2 u, D_t u) \in \mathbb {R}^{n^2 +1},\,\mathcal {H}( u):= F(D^2u)- \partial _t u,\,K>0$, and $\Omega \subset \mathbb {R}^{n+1}$ is some unknown open set. Since, by assumption, ${\tilde{D}}^2u$ is bounded in the complement of $\Omega$, we see that $\mathcal {H}(u)$ is bounded inside the whole $Q_1$ and $u$ is a so-called strong $L^n$ solution to a fully nonlinear parabolic equation with bounded right-hand side [7]. We refer to [7, 12] as basic references to parabolic fully nonlinear equations and viscosity methods.

The above free boundary problem has a very general form and encompasses several other free boundaries of obstacle type. In the elliptic case, it has been recently studied by the authors in [10]. We also refer to several other articles concerning similar type of problems: for elliptic case see [1, 5, 11], and for parabolic case see [2, 6]. One may find applications and relevant discussions about these kinds of problems in these articles.

Since most of the results follow the same line of arguments (sometimes with obvious modifications) as that of its elliptic counterpart done in [10], here we have decided not to enter into the details of the proof as they can be worked out in a similar way as in the elliptic case. Instead, we shall give the outline of the proofs and point out all the necessary changes. For the reader unfamiliar with these techniques, we suggest first to read [10].

Going back to our problem, we observe that, if $u \in W_x^{2,n}\cap W_t^{1,n}$, then ${\tilde{D}}^2u=0$ a.e. inside $\{u=0\}$, and $D^2u=0$ a.e. inside $\{\nabla u=0\}$ (here and in the sequel, $\nabla u$ denotes only the spatial gradient of $u$). In particular, we easily deduce that (1.1) includes, as special cases, both $\mathcal {H}(u)=\chi _{\{u \ne 0\}}$ and $\mathcal {H}(u)=\chi _{\{\nabla u \ne 0\}}$.

We assume that:

• (H0) $F(0)=0$.

• (H1) $F$ is uniformly elliptic with ellipticity constants $0<\lambda _0\le \lambda _1<\infty$, that is,

  $$\begin{aligned} \fancyscript{P}^-(P_1-P_2) \le F(P_1) - F(P_2) \le \fancyscript{P}^+(P_1-P_2) \end{aligned}$$

  for any $P_1,P_2$ symmetric, where $\fancyscript{P}^-$ and $\fancyscript{P}^+$ are the extremal Pucci operators: Given a symmetric matrix $M$, one defines

  $$\begin{aligned} \fancyscript{P}^-(M):=\inf _{\lambda _0 \mathrm{Id }\le N \le \lambda _1 \mathrm{Id }} \mathrm{trace}(NM),\qquad \fancyscript{P}^+(M):=\sup _{\lambda _0 \mathrm{Id }\le N \le \lambda _1 \mathrm{Id }} \mathrm{trace}(NM), \end{aligned}$$

  where $N$ in the formula above is symmetric as well.

• (H2) $F$ is either convex or concave.

Under assumptions (H0)–(H2) above, strong $L^n$ solutions are also viscosity solutions [5], and hence, regularity results for parabolic fully nonlinear equations [12, 13] show that $u \in W_x^{2,p}(Q_\rho )\cap W_t^{1,p}(Q_\rho )$ for all $\rho \in (0,1)$ and $p <\infty$.

1.3 Main results

Our first result concerns the optimal regularity of solutions to (1.1): Once this is done, we will be able to study the regularity of the free boundary.

Theorem 1.1

(Interior $C_x^{1,1}\cap C_t^{0,1}$ regularity) Let $u:Q_1 \rightarrow \mathbb {R}$ be a $W_x^{2,n} \cap W_t^{1,n}$ solution of (1.1), and assume that $F$ satisfies (H0)–(H2). Then, there exists a constant $\bar{C} =\bar{C}(n,\lambda _0,\lambda _1,\Vert u\Vert _\infty )>0$ such that

$$\begin{aligned} |{\tilde{D}}^2u| \le \bar{C}, \quad \hbox {in}\quad Q_{1/2}. \end{aligned}$$

To state our result on the regularity of the free boundary, we need to introduce the concept of minimal diameter: for any set $E \subset \mathbb {R}^n$, let $\mathrm{MD }(E)$ denote the smallest possible distance between two parallel hyperplanes containing $E$. Then, given a point $X^0=(x^0,t^0) \in \mathbb {R}^{n+1}$, we define

$$\begin{aligned} \delta _r(u,X^0):=\inf _{t \in [t_0-r^2,t_0+r^2]}\frac{\mathrm{MD }\bigl (\Lambda \cap \bigl (B_r(x^0) \times \{t\}\bigr )\bigr )}{r},\qquad \Lambda :=Q_1{\setminus } \Omega . \end{aligned}$$

(1.2)

In other words, $\delta _r(u,X^0)$ measures the thickness of the complement of $\Omega$ at all time levels $t\in (t^0-r^2,t^0+r^2)$, around the point $x^0$. Notice that $\delta _r$ depends on $u$ since $\Omega$ does. In particular, we observe that if $u$ solves (1.1) for some set $\Omega$, then $u_r(y,\tau ):=u(x+ry, t + r^2\tau )/r^2$ solves (1.1) with

$$\begin{aligned} \Omega _r:=\left\{ (y,\tau )\,:\,(x+ry, t + r^2\tau ) \in \Omega \right\} \end{aligned}$$

in place of $\Omega$, and $\delta _r$ enjoys the scaling property $\delta _1(u_r,0)= \delta _r(u,X),\,X=(x,t)$.

Our result provides regularity for the free boundary under a uniform thickness condition. As a corollary of our result, we deduce that Lipschitz free boundaries are $C^1$, and hence smooth [8].

Theorem 1.2

(Free boundary regularity) Let $u:Q_1 \rightarrow \mathbb {R}$ be a $W_x^{2,n} \cap W_t^{1,n}$ solution of (1.1). Assume that $F$ is convex and satisfies (H0)–(H1), and that $\Omega \supset \{u \ne 0\}$. Suppose further that there exists $\varepsilon >0$ such that

$$\begin{aligned} \delta _r(u,z) > \varepsilon \qquad \forall \,r<1/4,\,\forall \,z\in \partial \Omega \cap Q_{r}(0). \end{aligned}$$

Then $\partial \Omega \cap Q_{r_0}(0)$ is a $C^1$-graph in space-time, where $r_0$ depends only on $\varepsilon$ and the data.

The paper is organized as follows:

In Sect. 2, we prove Theorem 1.1. Then in Sect. 3, we investigate the non-degeneracy of solutions and classify global solutions under a suitable thickness assumption. In Sect. 4, we show directional monotonicity for local solutions which gives Lipschitz (and then $C^1$) regularity for the free boundary, as shown in Sect. 5.

2 Proof of Theorem 1.1

The proof of this theorem follows the same line of ideas as its elliptic counterpart [10]. First one starts from a BMO-type estimate on $D^2u$, and then one shows a dichotomy that either $u$ has quadratic growth away from a free boundary point $X^0$, or the density of the set $\Lambda$ at $X^0$ vanishes fast enough to assure the quadratic bound.

In [10], the following result was a consequence of the BMO-type estimate proved in [4, Theorem A] (see [10, Lemma 2.3]). Since we could not find a reference for this estimate in the parabolic case, we prove this result in the appendix. We notice that our proof is much simpler than the one in [4] and actually gives a new proof of the results there (see Remark 6.3).

In all this section, $u$ is as in the statement of Theorem 1.1. With no loss of generality, we will carry out the proof at the origin, by letting $X^0=(0,0)$.

We say that $P$ is a “parabolic (second-order) polynomial” if it is of the form

$$\begin{aligned} P(x,t)=a_0 +\langle b_0,x\rangle + \langle M_0 x,x\rangle + c_0 t, \qquad a_0,c_0 \in \mathbb {R},\,b_0 \in \mathbb {R}^n,\,M_0 \in \mathbb {R}^{n\times n}. \end{aligned}$$

Lemma 2.1

There exist a constant $C=C(n,\lambda _0,\lambda _1,\Vert u\Vert _\infty )$, and a family of parabolic polynomials $\{P_r\}_{r \in (0,1)}$ solving $\mathcal {H}(P_r)=0$, such that

$$\begin{aligned} \sup _{Q_r(0)}|u -P_r| \le C r^2, \qquad \forall \ r \in (0,1). \end{aligned}$$

(2.1)

Consequently

$$\begin{aligned} \sup _{Q_r(0)}|u| \le \left( C r^2+| \tilde{D}^2 P_r|\right) , \qquad \forall \ r \in (0,1). \end{aligned}$$

(2.2)

The first statement in the Lemma is proven in “Appendix” [see (6.1) and Lemma 6.2 there], while the second estimate is a straightforward consequence of the first one. It should be remarked that these polynomials $P_r$ need not to be unique.

Define

$$\begin{aligned} A_r:=\bigl \{(x,t): \ (rx,r^2t) \in Q_r{\setminus } \Omega \bigr \} \subset Q_1 \qquad \forall \ r<1/4. \end{aligned}$$

(2.3)

We shall prove that if $|P_r|$ is sufficiently large, then the Lebesgue measure of $A_r$ has to decay geometrically.

Proposition 2.2

Let $P_r$ be as in Lemma 2.1 and set $\tilde{P}_r:= \tilde{D}^2 P_r$. There exists $M>0$ universal such that, for any $r \in (0,1/8)$, if $|\tilde{P}_r| \ge M$ then

$$\begin{aligned} |A_{r/2}| \le \frac{|A_r|}{2^{n+1}}. \end{aligned}$$

The proof of the proposition follows the same lines of ideas as that of [10, Proposition 2.4]. However, since the changes are not completely straightforward, for the reader’s convenience we present the proof here.

Proof

Set $u_r(y,t):=u(ry,r^2)/r^2$ and let

$$\begin{aligned} u_r(y,t)=P_r (y,t) + v_r(y,t)+w_r(y,t), \end{aligned}$$

(2.4)

where $v_r$ is defined as the solution of

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {H}(P_r+v_r)-1=0 &{} \quad \text {in} \quad Q_1,\\ v_r(y,t)=u_r(y,t)-P_r ( y,t) &{} \quad \text {on} \quad \partial _p Q_1, \end{array} \right. \end{aligned}$$

(2.5)

where $\partial _pQ_1$ denotes the parabolic boundary of $Q_1$, and by definition $w_r:=u_r-P_r -v_r$.

Set $f_r:=\mathcal {H}(u_r) \in L^\infty (B_1)$ (recall that $|{\tilde{D}}^2 u_r|\le K$ a.e. inside $A_r$). Notice that, since $f_r=1$ outside $A_r$,

$$\begin{aligned} \mathcal {H}(u_r)-\mathcal {H}(P_r+v_r) =(f_r-1)\chi _{A_r}, \end{aligned}$$

so it follows by (H1) that $w_r$ solves

$$\begin{aligned} \left\{ \begin{array}{ll} \fancyscript{P}^-(D^2w_r) - \partial _t w_r \le (f_r-1)\chi _{A_r} \le \fancyscript{P}^+(D^2w_r) -\partial _t w_r &{} \text {in }Q_1,\\ w_r=0&{} \text {on }\partial _p Q_1. \end{array} \right. \end{aligned}$$

(2.6)

Hence, we can apply the ABP estimate [12, Theorem 3.14] to deduce that

$$\begin{aligned} \sup _{Q_1}|w_r| \le C \Vert \chi _{A_r}\Vert _{L^{n+1}(Q_1)} = C|A_r|^{1/(n+1)}. \end{aligned}$$

(2.7)

Also, since $\mathcal {H}(P_r)=0$ and $v_r$ is universally bounded on $\partial _p Q_1$ [see (2.1) and (2.5)], by the parabolic Evans–Krylov’s theorem [9] applied to (2.5) we have

$$\begin{aligned} \Vert \tilde{D}^2 v_r\Vert _{C^{0,\alpha }(Q_{3/4})} \le C. \end{aligned}$$

(2.8)

This implies that $w_r$ solves the fully nonlinear equation with Hölder coefficients

$$\begin{aligned}&G(D^2w_r,X)-\partial _t w_r - \partial _t ( v_r + P_r)=(f_r-1)\chi _{A_r} \quad \text {in }Q_{3/4},\nonumber \\&G(M,X):=F\left( D^2P_r +D^2 v_r(x) +M\right) -1. \end{aligned}$$

Since $G(0,X)=0$, we can apply [12, Theorem 5.6] with $p=n+2$ and (2.7) to obtain

$$\begin{aligned} \int \limits _{Q_{1/2}}|\tilde{D}^2 w_r|^{n+2} \le C\left( \Vert w_r\Vert _{L^\infty (Q_{3/4})} + \Vert \chi _{A_r}\Vert _{L^{2+n}(Q_{3/4})} \right) ^{n+2} \le C\, |A_r| \end{aligned}$$

(2.9)

(recall that $|A_r|\le |Q_1|$).

We are now ready to conclude the proof: since $|\tilde{D}^2 u_r|\le K$ a.e. inside $A_r$ [by (1.1)], recalling (2.4) we have

$$\begin{aligned} \int \limits _{A_r\cap Q_{1/2}}|\tilde{D}^2 v_r+ \tilde{D}^2 w_r + \tilde{P}_r|^{n+2} =\int \limits _{A_r \cap Q_{1/2}} |\tilde{D}^2 u_r|^{n+2} \le K^{n+2}|A_r|. \end{aligned}$$

Therefore, by (2.8) and (2.9),

$$\begin{aligned} |A_r\cap Q_{1/2}|\,|\tilde{P}_r|^{n+2}&= \int \limits _{A_r\cap Q_{1/2}}|\tilde{P}_r|^{n+2}\\&\le 3^{2n}\biggl ( \int \limits _{A_r\cap Q_{1/2}}|\tilde{D}^2 v_r|^{n+2} + \int \limits _{A_r\cap Q_{1/2}}|\tilde{D}^2 w_r|^{n+2} +K^{n+2}|A_r|\biggr )\\&\le 3^{n+2} \biggl (|A_r\cap Q_{1/2}|\,\Vert \tilde{D}^2 v_r\Vert _{L^\infty (Q_{1/2})} \!+\! \int \limits _{Q_{1/2}}|\tilde{D}^2 w_r|^{n+2}+K^{n+2}|A_r|\biggr )\\&\le C\,|A_r\cap Q_{1/2}| + C\, |A_r|, \end{aligned}$$

which gives

$$\begin{aligned} |A_r\cap Q_{1/2}(0)|\,|\tilde{P}_r|^{n+2} \le C|A_r|. \end{aligned}$$

Hence, if $|\tilde{P}_r|$ is sufficiently large so that $C \le \frac{1}{4^{n+1}}|\tilde{P}_r|^{n+2}$ we get

$$\begin{aligned} |A_r\cap Q_{1/2}(0)|\,|\tilde{P}_r|^{n+2} \le \frac{1}{4^{n+1}}|\tilde{P}_r|^{n+2}|A_r|. \end{aligned}$$

Since $|A_{r/2}|= 2^{n+1}|A_r\cap Q_{1/2}(0)|$, this gives the desired result. $\square$

2.1 Proof of Theorem 1.1

Taking $M >0$ as in Proposition 2.2, we have that one of the following hold:

• (i) $\liminf _{k\rightarrow \infty }|P_{2^{-k}}| \le 3M$,

• (ii) $\liminf _{k\rightarrow \infty }|P_{2^{-k}}| \ge 3M$.

Then, one consider the two cases separately and, arguing exactly as in the proof of Theorem 1.2 in [10], one obtains the desired result. (We notice that the reference  [3, Theorem 3] in that proof is to be replaced by [13, Theorem 1.1].)

3 Non-degeneracy and global solutions

3.1 Local non-degeneracy

The $C_x^{1,1}\cap C_t^{0,1}$-regularity proved in the previous section implies that $u$ cannot grow more than quadratically in space and linearly in time away from the free boundary. Nevertheless, this is not sufficient for the blow-up method to work out, as the limit function may degenerate to zero.

As shown in [10, Section 3], non-degeneracy fails in general for the elliptic case, hence for our problem as well. Nevertheless, the non-degeneracy does hold for the case $\Omega \supset \{\nabla u \ne 0\}$, see [10, Lemma 3.1]. We now show that this non-degeneracy result still holds in the parabolic case:

Lemma 3.1

Let $u:Q_1 \rightarrow \mathbb {R}$ be a $W_x^{2,n}\cap W_t^{1,n}$ solution of (1.1), assume that $F$ satisfies (H0)-(H2), and that $\Omega \supset \{\nabla u \ne 0\}$. Then, for any $X^0=(x^0,t^0) \in \overline{\Omega }\cap Q_{1/2}$,

$$\begin{aligned} \max _{\partial _p Q_r(X^0)} u \ge u(X^0)+\frac{r^2}{2n\lambda _1 + 1} \qquad \forall \, r \in (0,1/4). \end{aligned}$$

Proof

For

$$\begin{aligned} v(x):=u(x)-\frac{|x-x^0|^2- (t-t^0)}{2n\lambda _1+1}, \qquad X^0 \in \Omega \cap Q_{1/2}, \end{aligned}$$

one readily verifies that $\mathcal {H}(v) \ge 0$ in $Q_r(X^0)$. Then, by the very same argument as in the proof of [10, Lemma 3.1] we deduce that¹

$$\begin{aligned} \max _{\partial _p Q_r(X^0)}v =\sup _{Q_r(X^0)}v, \end{aligned}$$

and the result follows easily. By continuity the lemma holds for $X^0 \in \overline{\Omega }\cap Q_{1/2}$ $\square$

3.2 Classification of global solutions

As already discussed in the previous section, to have non-degeneracy of solutions we need to assume that $\Omega \supset \{\nabla u \ne 0\}$. In the elliptic case, this assumption is also sufficient to classify global solutions with a “thick free boundary” (see [10, Proposition 3.2]). However, in the parabolic case, the situation is much more complicated: Indeed, while global solutions of the elliptic problem with “thick free boundary” are convex and one-dimensional, in the parabolic case we have non-convex solutions. For instance, the function

$$\begin{aligned} u(x,t)= \left\{ \begin{array}{ll} -2t-x_1^2/2 &{} \quad \text {if} \quad x_1>0,\\ -2t &{} \quad \text {if} \quad x_1\le 0,\\ \end{array} \right. \end{aligned}$$

is a solution of (1.1) on the whole $\mathbb {R}^{n+1}$ with $F(D^2u)=\Delta u$ and $\Omega :=\{x_1>0\}=\{\nabla u\ne 0\}$. In order to avoid this kind of examples, here we shall only consider the case $\Omega \supset \{u \ne 0\}$.

Since we will use minimal diameter to measure sets [recall (1.2)], we need some classical facts about their stability properties. First of all we recall that, for polynomial global solutions $P_2=\sum _j a_j x_j^2 + bt$ (with $A=\mathrm{diag}(a_j)$, and $b$ such that $F(A)-b=1$), one has

$$\begin{aligned} \delta _r(P_2,0)=0. \end{aligned}$$

(3.1)

Also, the scaling and stability estimate

$$\begin{aligned} \delta _r(u,X)= \delta _1(u_r,0), \qquad \limsup _{r \rightarrow 0} \delta _r(u,X^0)\le \delta _1(u_0,0) \end{aligned}$$

(3.2)

holds whenever $u_r(y,\tau )=u(x+ry, t + r^2\tau )/r^2$ converges in $C^1$ to some function $u_0$.

In the next proposition, we shall prove that global solutions with a “thick free boundary,” must be time-independent and hence by elliptic results they must be one-dimensional solutions.

We notice that assumption (3.3) below allows us to exclude the family of global solutions $u_\sigma (t,x)=-(t-\sigma )_+,\,\sigma \in \mathbb {R}$.

Proposition 3.2

Let $u:\mathbb {R}^{n+1} \rightarrow \mathbb {R}$ be a $W^{2,n}$ solution of (1.1) on the whole $\mathbb {R}^{n+1}$, assume that $F$ is convex and satisfies (H0)–(H1), and that $\Omega \supset \{u \ne 0\}$. Furthermore, assume that there exists $\epsilon _0>0$ such that

$$\begin{aligned} \delta _r(u,X^0) \ge \epsilon _0 \qquad \forall \,r>0,\,\forall \,X^0 \in \partial \Omega . \end{aligned}$$

(3.3)

Then, $u$ is time-independent. In particular, by the elliptic case [10, Proposition 3.2], $u$ is a half-space solution, i.e., up to a rotation, $u(x)=\gamma [(x_1)_+]^2/2$, where $\gamma \in (1/\lambda _1,1/\lambda _0)$ is such that $F(\gamma e_1\otimes e_1 )=1$.

Proof

Let $m:=\sup _{\mathbb {R}^{n+1}} \partial _t u$ (notice that $m$ is finite by Theorem 1.1) and consider a sequence $m_j = \partial _t u(X^j)$ such that $m_j\rightarrow m$.

We now perform the scaling

$$\begin{aligned} u_j(x,t):=\frac{u\left( d_jx+x^j,d_j^2t+t^j\right) }{d_j^2}, \end{aligned}$$

where $X^j=(x^j,t^j)$ and $d_j:=\mathrm{dist}(X^j,\partial \Omega )$.

The functions $u_j$ still satisfy (1.1). Also, since $u=0$ on $\partial \Omega$ it follows by the $C^{1,1}_x\cap C^{0,1}_t$ regularity of $u$ that $u_j$ are uniformly bounded, hence, up to subsequences, they converge to another global solution $u_\infty$ which satisfies $\partial _{t}u_\infty (0)=m$. By (3.2) and the assumption (3.3), we obtain

$$\begin{aligned} \delta _r(u_\infty ,X^0) \ge \epsilon _0 \qquad \forall \,r>0,\,\forall \,X^0 \in \partial \Omega _\infty , \end{aligned}$$

(3.4)

where $\Omega _\infty$ is the limit, as $j \rightarrow \infty$, of the family of open sets

$$\begin{aligned} \Omega _j:=\bigl \{(x,t)\,:\,\left( d_jx+x^j,d_j^2t+t^j\right) \in \Omega \bigr \}. \end{aligned}$$

Let us observe that, by the condition $\Omega \supset \{u \ne 0\}$ we get $u_\infty (t,x)=0$ on $\partial \Omega _\infty$.

In addition, $\partial _{t}u_\infty$ is a solution of the uniformly parabolic linear operator $F_{ij}(D^2u_\infty )\partial _{ij} - \partial _t$ inside $\Omega _\infty$. Hence, since $\partial _t u_\infty \le m$ and $\partial _{t}u_\infty (0)=m$, by the strong maximum principle we deduce that $\partial _{t}u_\infty$ is constant inside the connected component of $\Omega _\infty$ containing $0$ (call it $\Omega _0$).

Therefore, integrating $u_\infty$ in the direction $t$ gives

$$\begin{aligned} u_\infty (t,x)= mt + U(x) \quad \text {inside}\quad \Omega _0, \qquad u_\infty =0\quad \text {on}\quad \partial \Omega _0. \end{aligned}$$

(3.5)

We claim that $m=0$. Indeed, suppose by contradiction that $m \ne 0$. Then, for any point $(\bar{x},\bar{t}) \in \Omega _0$, it follows by (3.5) that: (a) either there exists $t' \in \mathbb {R}$ such that $(\bar{x},t') \in \partial \Omega _0$; (b) or $\{\bar{x}\}\times \mathbb {R}\subset \Omega _0$. Thanks to the thickness assumption (3.4) we see that $\nabla u_\infty =0$ on $\partial \Omega _0$, so in case (a) we obtain that $\nabla U(\bar{x})=\nabla u_\infty (\bar{x},t')=0$. Hence, by the arbitrariness of $\bar{x}$, we can write

$$\begin{aligned} \Omega _0=\Omega _1 \cup \Omega _2, \end{aligned}$$

where $\nabla u_\infty \equiv 0$ in $\Omega _1$, and $\Omega _2$ is a cylinder of the form $V\times \mathbb {R}$ with $V\subset \mathbb {R}^n$. So, it follows from (3.5) that $u_\infty =0$ on $\partial \Omega _2$, which is incompatible with the fact that $u_\infty (t,x)= mt + U(x)$ inside $\Omega _2$ (and so, by continuity, also on $\partial \Omega _2$) unless $m=0$. This proves the claim, showing that $\sup _{\mathbb {R}^{n+1}}\partial _t u = 0$.

By a completely symmetric argument, we obtain $\inf _{\mathbb {R}^{n+1}} \partial _t u= 0$. Thus $\partial _tu=0$, which implies that $u$ is time-independent and therefore, by [10, Proposition 3.2], up to a rotation $u$ is of the form $u(x)=\gamma [(x_1)_+]^2/2+c$ $\gamma \in (1/\lambda _1,1/\lambda _0)$ is such that $F(\gamma e_1\otimes e_1 )=1$ and $c \in \mathbb {R}$. Since $\Omega \supset \{u \ne 0\}$, we see that $c=0$, which proves the result. $\square$

4 Local solutions and directional monotonicity

In this section, we shall prove a directional monotonicity for solutions to our equations. In the next section, we will use Lemma 4.2 below to show that, if $u$ is close enough to a half-space solution $\gamma [(x_1)_+]^2$ in a ball $B_r$, then for any $e=(e_x,e_t) \in \mathbb S^{n}$ with $e \cdot (e_1,0)\ge s>0$ we have $C_0\partial _e u - u \ge 0$ inside $B_{r/2}$.

Lemma 4.1

Let $u:Q_1 \rightarrow \mathbb {R}$ be a $W_x^{2,n}\cap W_t^{1,n}$ solution of (1.1) with $\Omega \supset \{u \ne 0\}$. Then, under the conditions of Theorem 1.2, we have

$$\begin{aligned} \lim _{\Omega \ni X \rightarrow \partial \Omega } \partial _t u (X)=0. \end{aligned}$$

Proof

The proof of this lemma follows easily by a contradiction argument, along with scaling and blow-up. Indeed, given a sequence $X^j \rightarrow \partial \Omega$ such that $|\partial _tu(X^j)| \ge c>0$, then one may scale at $X^j$ with $d_j=\hbox {dist} (X^j,\partial \Omega )$ and define $u_j(X):=\left[ u(d_jx + x^j,d_j^2t + t^j) -u(X^j)\right] /d_j^2$ to end up with a global solution $u_\infty$ with the property $\partial _t u_\infty (0) \ne 0$, contradicting Proposition 3.2. $\square$

The proof of the following result is a minor modification of the one of [10, Lemma 4.1], so we just give a sketch of the proof.

Lemma 4.2

Let $u:Q_1 \rightarrow \mathbb {R}$ be a $W_x^{2,n}\cap W_t^{1,n}$ solution of (1.1) with $\Omega \supset \{ u \ne 0\}$. Assume that for some space-time direction $e=(e_x,e_t)$ with $|e|=1$ we have $C_0\partial _eu-u \ge -\varepsilon _0$ in $Q_1$ for some $C_0,\varepsilon _0 \ge 0$, and that $F$ is convex and satisfies (H0)-(H1). Then $C_0\partial _eu-u \ge 0$ in $Q_{1/2}$ provided $\varepsilon _0 \le \frac{1}{4(2n\lambda _1+1)}$.

Proof

Since $F$ is convex, for any matrix $M$ we can choose an element $P^M$ inside $\partial F(M)$ (the subdifferential of $F$ at $M$) in such a way that the map $M \mapsto P^M$ is measurable, and we define the measurable uniformly elliptic coefficients

$$\begin{aligned} a_{ij}(x,t):=\left( P^{D^2u(x,t)}\right) _{ij} \in \partial F(D^2u(x,t)). \end{aligned}$$

As in the proof of [10, Lemma 4.1], by the convexity of $F$ if follows that, in the viscosity sense,

$$\begin{aligned} a_{ij}\partial _{ij} (\partial _e u) -\partial _{t} (\partial _e u) \le 0\quad \text {in}\quad \Omega \end{aligned}$$

(4.1)

and

$$\begin{aligned} a_{ij}\partial _{ij}u -\partial _t u \ge 1\quad \text {in}\quad \Omega . \end{aligned}$$

(4.2)

Now, let us assume by contradiction that there exists $X^0=(x^0,t^0)\in Q_{1/2}$ such that $C_0\partial _eu(X^0)-u(X^0)<0$, and consider the function

$$\begin{aligned} w(X):=C_0\partial _eu(X)- u(X)+\frac{|x-x^0|^2-(t-t_0)}{2n\lambda _1+1}. \end{aligned}$$

Thanks to (4.1), (4.2), and assumption (H1) (which implies that $\lambda _0 \mathrm{Id }\le a_{ij}\le \lambda _1 \mathrm{Id }$) we deduce that $w$ is a supersolution of the linear operator $\fancyscript{L}:=a_{ij}\partial _{ij} -\partial _t$, hence, by the minimum principle,

$$\begin{aligned} \min _{\partial _p \left( \Omega \cap Q_{1}(Y^0)\right) } w = \min _{\Omega \cap Q_{1}(Y^0)} w \le w(Y^0)<0. \end{aligned}$$

By Lemma 4.1 and the assumption $\Omega \supset \{u\ne 0\}$ we have $\partial _t u =u=|\nabla u|= 0$ on $\partial \Omega$, therefore $w\ge 0$ on $\partial \Omega$. Thus, since $|x-x^0|^2 - (t-t^0)\ge 1/4$ on $\partial _p Q^-_{1}$ it follows that

$$\begin{aligned} 0>\min _{\partial _p Q^-_{1/2}(X^0)} w \ge -\varepsilon _0+ \frac{1}{4(2n\lambda _1+1)}, \end{aligned}$$

a contradiction if $\varepsilon _0 \le \frac{1}{4(2n\lambda _1+1)}$. $\square$

5 Proof of Theorem 1.2

The proof of this theorem is very similar to the proof of [10, Theorem 1.3]. Indeed, take $X^0=(x^0,t^0) \in \partial \Omega \cap Q_{1/8}$, and rescale the solution around $X^0$, that is $u_r(x,t):=[u(rx+x^0,r^2t+ t^0)-u(x^0,t^0) - r\nabla u(x^0,t^0)\cdot x]/r^2$.

Because of the uniform $C^{1,1}_x\cap C_t^{0,1}$ estimate provided by Theorem 1.1 and the thickness assumption on the free boundary of $u$, we can find a sequence $r_j\rightarrow 0$ such that $u_{r_j}$ converges locally uniformly to a global solution $u_\infty$ of the form $u_\infty (x)=\gamma [(x\cdot e_{_{X^0}})_+]^2/2$ with $\gamma \in [1/\lambda _1,1/\lambda _0]$ and $e_{_{X^0}} \in \mathbb S^{n-1}$ (see Proposition 3.2).

Notice now that, for any $s \in (0,1)$, we can find a large constant $C_s$ such that

$$\begin{aligned} C_s\partial _e u_\infty - u_\infty \ge 0\quad \text {inside}\quad B_1 \end{aligned}$$

for all directions $e=(e_x,e_t) \in \mathbb S^{n}$ such that $e\cdot (e_{_{X^0}},0) \ge s$, hence by the $C^1_x$ convergence of $u_{r_j}$ to $u_\infty$ and Lemmas 4.1 and‘4.2 we deduce that

$$\begin{aligned} C_s\partial _e u_{r_j}- u_{r_j} \ge 0\quad \text {in}\quad Q_{1/2}, \end{aligned}$$

(5.1)

i.e., $\partial _e e^{-C_s e\cdot x }u_{r_j} \ge 0$. Since $u_{r_j}=0$ in $Q_{1/2} \cap \{x\cdot e < -1/2 \}$ for $r_j$ small enough, we deduce $u_{r_j} \ge 0$ in $Q_{1/4}$.

Using (5.1) again, this implies that $\partial _e u_{r_j}$ inside $Q_{1/4}$, and so in terms of $u$ we deduce that there exists $r=r(s)>0$ such that

$$\begin{aligned} \partial _e u \ge 0 \quad \text {inside}\quad Q_{r}(X^0) \end{aligned}$$

for all $e \in \mathbb S^{n}$ such that $e \cdot (e_{_{X^0}},0)\ge s$.

A simple compactness argument shows that $r$ is independent of the point $X^0$, which implies that the free boundary is $s$-Lipschitz. Since $s$ can be taken arbitrarily small (provided one reduces the size of $r$), this actually proves that the free boundary is $C^1$. Higher regularity is then classical.

Appendix: Parabolic BMO estimates

Let $u:Q_1 \rightarrow \mathbb {R}$ satisfy $|u|\le 1$ and $|\mathcal {H}(u)|\le M$. Up to replacing $u$ by $u(x/R, t/R^2)$ and $\mathcal {H}$ by $(F(R^2\ \cdot )/R^2 -\partial _t )$ with $R$ a large fixed constant, we can assume that $|\mathcal {H}(u)|\le \delta$ with $\delta$ a small constant to be fixed later. Observe that, with this scaling, the ellipticity remains the same.

Let us first state a standard stability result.

Lemma 6.1

(Compactness) Let $\varepsilon >0$, and $u$ be such that $u:Q_1\rightarrow \mathbb {R}$ satisfy $|u|\le 1$. Let further $v:Q_{1/2}\rightarrow \mathbb {R}$ solve

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {H}(v)=0&{}\quad \text {in}\quad Q_{1/2},\\ v=u&{} \quad \text {on}\quad \partial _p Q_{1/2}. \end{array} \right. \end{aligned}$$

Then there exists $\delta =\delta (\varepsilon )>0$ such that

$$\begin{aligned} |u-v| \le \varepsilon \quad \text {in}\quad Q_{1/2}, \end{aligned}$$

provided $|\mathcal {H}(u)|\le \delta$.

The proof of the lemma is based on a standard compactness argument, using that both $u$ and $v$ are uniformly Hölder continuous (in $(x,t)$-variables) inside $Q_{1/2}$; see [12], Lemma 5.1.

Recall that $P$ is a parabolic polynomial if it is of the form

$$\begin{aligned} P(x,t)=a_0 +\langle b_0,x\rangle + \langle M_0 x,x\rangle + c_0 t, \qquad a_0,c_0 \in \mathbb {R},\,b_0 \in \mathbb {R}^n,\,M_0 \in \mathbb {R}^{n\times n}. \end{aligned}$$

We now prove by induction the following result:

Lemma 6.2

Let $u:Q_1\rightarrow \mathbb {R}$ be a solution to our problem (1.1), with $|u|\le 1$. Then there exists $\rho >0$ universal such that

$$\begin{aligned} \left| u(X) - P_k(X) \right| \le \rho ^{2k} \quad \text {inside}\quad Q_{\rho ^k} \quad \forall \,k \in \mathbb {N}, \end{aligned}$$

where $P_k$ is a parabolic polynomial such that $\mathcal {H}(P_k)=0$.

A straight forward implication of this result is that there is a universal constant $C=1/\rho ^2$ such that

$$\begin{aligned} \left| u(X) - P_r(X) \right| \le Cr^2 \qquad \text {inside}\, Q_r \quad \forall \, 0<r<1, \end{aligned}$$

(6.1)

where $P_r$ is a parabolic second-order polynomial such that $\mathcal {H}(P_r)=0$. This in turn implies an $L^p$-BMO-type result, see the corollary below.

Proof of Lemma 6.2

Since the result is obviously true for $k=0$ (just take $P_0=0$), we prove the inductive step. So, let us assume that the result is true for $k$ and we prove it for $k+1$.

Define $u_k(X):=\frac{u(\rho ^kx,\rho ^{2k} t)-P_k(\rho ^kx,\rho ^{2k} t)}{\rho ^{2k}}$. Then, by the inductive hypothesis $|u_k|\le 1$ inside $Q_1$. In addition

$$\begin{aligned} |\mathcal {H}_k(u_k)|\le \delta ,\qquad \mathcal {H}_k(v):=F(D^2v + D^2P_{k}) - \partial _t P_k- \partial _t v. \end{aligned}$$

Observe that $\mathcal {H}_k$ keeps the same ellipticity as $\mathcal {H}$. Hence, we can apply the lemma above to deduce that

$$\begin{aligned} |u_k-v_k| \le \varepsilon \quad \text {in}\quad Q_{1/2}, \end{aligned}$$

where $v_k$ solves

$$\begin{aligned} \left\{ \begin{array}{ll} \mathcal {H}_k(v_k)=0&{}\quad \text {in }Q_{1/2},\\ v_k=u_k&{}\quad \text {on}\quad \partial _p Q_{1/2}. \end{array} \right. \end{aligned}$$

Since $\Vert v_k\Vert _{L^\infty (Q_{1/2})} \le \Vert u_k\Vert _{L^\infty (Q_{1})} \le 1$, by interior $C_{\alpha }^{2,1}$ estimates we get

$$\begin{aligned} \Vert v_k\Vert _{C_\alpha ^{2,1} (Q_{1/4})} \le C_0. \end{aligned}$$

Let $\hat{P}_{k}$ be the “parabolic” second-order Taylor expansion of $v_k$ at $(0,0)$, and notice that $\mathcal {H}_k(\hat{P}_k)=\mathcal {H}_k(v_k(0,0))=0$. Then

$$\begin{aligned} |v_k - \hat{P}_k| \le C_0\rho ^{2+\alpha } \quad \text {inside}\quad Q_\rho , \end{aligned}$$

which gives

$$\begin{aligned} |u_k - \hat{P}_k| \le C_0\rho ^{2+\alpha }+\varepsilon \quad \text {inside}\quad \, Q_\rho . \end{aligned}$$

In particular, if we choose $\rho$ sufficiently small so that $C_0\rho ^\alpha \le 1/2$ and then $\varepsilon \le \rho ^2/2$ we arrive at

$$\begin{aligned} |u_k(X) - \hat{P}_k(X)| \le \rho ^2 \quad \text {inside}\quad Q_\rho , \end{aligned}$$

or equivalently (recalling the definition of $u_k$)

$$\begin{aligned} |u(X) - P_{k+1}(X)| \le \rho ^{2(k+1)}, \qquad P_{k+1}(X):=P_k(X) +\rho ^{2k}\hat{P}_k\left( x/\rho ^k,t/\rho ^{2k}\right) . \end{aligned}$$

Also, since $\mathcal {H}_k(\hat{P}_k)=0$ we will have

$$\begin{aligned} \mathcal {H}(P_{k+1})= F(D^2 P_{k+1}) - \partial _t P_{k+1} = F\left( D^2 P_k + D^2 \hat{P}_k\right) -\partial _t P_k - \partial _t \hat{P}_k = \mathcal {H}_k(\hat{P}_{k})=0 \end{aligned}$$

which concludes the proof of the inductive step. $\square$

Remark 6.3

As a corollary of our result we deduce $L^p$-BMO estimates on ${\tilde{D}}^2u$ ($p \in (1,\infty )$) for solutions to general elliptic/parabolic operators of type $F=F({\tilde{D}}^2u, \nabla u,u, X)$ provided $F$ is Hölder continuous and $u \in C^{1,\alpha }_x$.

Indeed, if $F=F(D^2u,X)$, since $u_k(X):=\frac{u(\rho ^k x,\rho ^2t)-P_k(\rho ^kx,\rho ^2t)}{\rho ^{2k}}$ satisfies $|u_k|\le 1$ and $|\mathcal {H}_k(u_k)|\le \delta$ inside $Q_1$, by interior $W_p^{2,1}$ estimates we get

$$\begin{aligned} \Vert {\tilde{D}}^2u_k\Vert _{L^p(Q_{1/2})} \le C, \end{aligned}$$

that is

$$\begin{aligned} \frac{1}{|Q_{\rho ^k/2}|} \int \limits _{Q_{\rho ^k/2}} |{\tilde{D}}^2 u -{\tilde{D}}^2 P_k|^p \le C \qquad \forall \,k \in \mathbb {N}. \end{aligned}$$

For general operators $F$, it suffices to apply the above argument to $G(M,X):=F(M,Du(X),u(X),X)$.