AN EPIPERIMETRIC INEQUALITY APPROACH TO THE PARABOLIC SIGNORINI PROBLEM

. In this note, we use an epiperimetric inequality approach to study the regularity of the free boundary for the parabolic Signorini problem. We show that if the “vanishing order” of a solution at a free boundary point is close to 3 / 2 or an even integer, then the solution is asymptotically homoge- neous. Furthermore, one can derive a convergence rate estimate towards the asymptotic homogeneous solution. As a consequence, we obtain the regularity of the regular free boundary as well as the frequency gap.


1.
Introduction. In this note, we develop a new approach to study the asymptotics of solutions to the parabolic Signorini problem at the free boundary points. More precisely, let u be a solution to ∂ t u − ∆u = f in S + 2 u ≥ 0, ∂ n u ≤ 0, u∂ n u = 0 on S 2 . (1) Here for R > 0 and space dimension n ≥ 2 S + R := {(x, t) : x ∈ R n + , t ∈ [−R, 0]}, R n + := R n ∩ {x n > 0} S R := {(x, t) : x ∈ R n ∩ {x n = 0}, t ∈ [−R, 0]} and f ∈ L ∞ (S + 2 ) is a given inhomogeneity. Throughout the paper we will assume that (A) u is normalized such that (B) u satisfies the Sobolev regularity in the Gaussian space: there exists C depending on n and f L ∞ (S + 2 ) such that (C) u satisfies the interior Hölder estimate: there exists α ∈ (0, 1) such that for any U R n + ∪ (R n−1 × {0}), ∇u C α,α/2 (U ×[−1,0]) ≤ C for some C > 0 depending on n, α, f L ∞ (S + 2 ) and U . Here C α,α/2 is the parabolic Hölder class.
Solutions to (1) can come from solving a variational inequality for the initial value problem in the class of functions with mild growth at infinity and satisfy u ∈ L 2 ([−2, 0]; W 1,2 loc (R n + )), ∂ t u ∈ L 2 ([−2, 0]; L 2 loc (R n + )), or they can come from solutions to the Signorini problem in a bounded domain (one applies suitable cutoffs to extend them into full space solutions). In both cases, the Sobolev estimate in (B) and the interior Hölder estimate in (C) hold true, cf. [1,12]. The normalization assumption (A) is put simply to make the constants in (B) and (C) independent of u. Under our regularity assumption, the Signorini boundary condition in (1) holds in the classical sense.
1.1. Main results. The behavior of a solution around a free boundary point depends very much on how fast it vanishes towards it. The main results of the paper are the following: we prove that if the "vanishing order" of a solution at a free boundary point is close to 3/2, which is the expected lowest vanishing order, or 2m, m ∈ N + , which are the eigenvalues of the Ornstein-Uhlenbeck operator − 1 2 ∆ − x · ∇ in R n + with the vanishing Neumann boundary condition, then the solution is asymptotically a homogeneous solution. As a consequence, we derive the frequency gap around the free boundary points with vanishing order 3/2 and 2m; the openness of the regular free boundary (consisting the free boundary points with 3/2 vanishing order), as well as its space-time regularity.
More precisely, assume (0, 0) ∈ Γ u , and for r ∈ (0, 1), let u(x, t) 2 G(x, t)dxdt be the weighted L 2 space-time average at (0, 0). The first theorem is about the asymptotics of the solution around the free boundary point where the vanishing order is around 3/2: Theorem 1.1. Let u be a solution to (1) and satisfy the assumptions (A)-(C). Assume that (0, 0) ∈ Γ u . Then there exists a constant γ 0 ∈ (0, 1) depending on n and f L ∞ such that if c u r 3+γ0 ≤ H u (r) ≤ C u r 3−γ0 , for all r ∈ (0, r u ) (2) for some r u , c u , C u > 0, then there exists a unique function u 0 (x) = c 0 Re(x · e 0 + i|x n |) 3/2 with c 0 > 0 and e 0 ∈ S n−1 ∩ {x n = 0} such that for all t ∈ [−1, 0). Here C > 0 depends on n and f L ∞ (S + 2 ) . Theorem 1.1 still holds true, if we assume instead of the lower bound in (2) that the solution at t = −1 is sufficiently close to the asymptotic profile, cf. Remark 4. We remark that (2) is satisfied if (0, 0) is a free boundary point with frequency κ = 3/2, where the frequency is defined as the limit of the Almgren-Poon frequency function at (0, 0), cf. [12] for the precise definition. However, our assumption (2) is much weaker, since it does not rely on the existence of the limit of the frequency function or the optimal spacial regularity. On the other hand, if α ≥ 1−γ0 2 in the assumption (C), then we can instead derive the optimal interior regularity ∇u ∈ C 1/2,1/4 from Theorem 1.1.
The next theorem is about the logarithmic convergence towards the asymptotic homogeneous solution around the free boundary point with the vanishing order close to 2m. For that, we let E 2m be the 2m-eigenspace of the Ornstein-Uhlenbeck operator − 1 2 ∆ + x · ∇ on R n + with the vanishing Neumann boundary condition on {x n = 0}. Let E + 2m be the convex cone in E 2m where the restriction of p on {x n = 0} is nonnegative. We remark that givenp ∈ E + 2m , the function p(x, t) = ( , t ∈ (−∞, 0), is a 2m-parabolic homogeneous solution to (1). Theorem 1.2. Let u be a solution to (1) and satisfy the assumptions (A)-(C) with f = 0. Let m ∈ N + be an arbitrary positive integer. Assume that (0, 0) ∈ Γ u . Then there exist small constants γ m , δ 0 ∈ (0, 1) depending only on m, n, such that if H u (r) ≤ C u r 2κ−γm , κ = 2m, for each r ∈ (0, r u ) for some r u , C u > 0, and at t = −1 then there exists a unique nonzero p 0 (x, t) = ( for all t ∈ [−1, 0).
It is possible to generalize Theorem 1.2 to a nonzero inhomogeneity f , where f satisfies an additional vanishing property at (0, 0): |f (x, t)| ≤ M (|x|+ √ −t) 2(m−1+ 0 ) for some 0 > 0 and M > 0, cf. Section 5. We state and prove the theorem for f = 0 to avoid the technicalities caused by the inhomogeneity such that the proofs are neater. The assumptions of Theorem (1.2) are satisfied at the free boundary points with the frequency 2m, cf. [12], but here we do not rely on the existence of the limit of the frequency function.
Note that in Theorem 1.2 we obtain a logarithmic decay (in ln(−t)) of the L 2 norm instead of an exponential decay as in Theorem 1.1. A polynomial decay rate of this kind towards the asymptotic solutions was obtained originally in the elliptic case [9,10,11]. Moreover, in the classical obstacle problem it was shown, that there is in general no uniform exponential decay rate at singular points, cf. [15].

1.2.
Main ideas of the proof. The proofs for Theorem 1.1 and Theorem 1.2 are based on a dynamical system approach, where we establish a decay rate for the Weiss energy W κ , κ = 3/2 and κ = 2m, in the self-similar conformal coordinates. In the elliptic problems, such change of coordinates corresponds to (r, θ) → (t, θ) = (− ln r, θ) from (0, 1] × S n−1 to [0, ∞) × S n−1 , which transforms the original problem around the free boundary point at 0 to a dynamical system on S n−1 . The equilibrium of the dynamical system then corresponds to (back to the original coordinates) the blow-up limits at the origin. In the parabolic setting, we can (formally) formulate our problem in the self-similar conformal coordinates as a gradient flow of the Weiss energy under the convex constraint u ≥ 0 on {x n = 0}. The relation between the Weiss energy and the evolution of certain quantities thus becomes more transparent, cf. Lemma 2.2. Very different from the elliptic problem, in the parabolic setting the dynamical system is on the whole space R n (instead of S n−1 ). The non-compactness brings additional difficulties and we could not get the polynomial decay rate towards general blow-ups at singular points as in the elliptic setting (cf. [9,10]). It is unlikely that the above decay rate is optimal, thus it is a very interesting open problem to improve the decay rate and further explore the optimality.
The main steps in the proof are discrete decay estimates for the Weiss energy W κ , κ = 3/2 and κ = 2m, which can be viewed as parabolic epiperimetric inequalities. Epiperimetric inequalities were introduced by G. Weiss [24] to the classical obstacle problem and they continue to be a subject of intense research interest in the elliptic setting [9,10,11,16,19,22].
1.3. Known results. Existence and uniqueness of solutions for given initial and boundary data follow from the classical theory of variational inequalities, cf. for instance, [13]. For sufficiently regular inhomogeneities, solutions u satisfy the pointwise regularity ∇ x u ∈ H 1/2,1/4 loc (Hölder 1/2 in space and 1/4 in time) and ∂ t u ∈ L ∞ loc , cf. [1,2,12,21]. Such regularity is optimal at least in the space variables due to the stationary solution Re(x n−1 + i|x n |) 3/2 . Classification of free boundary points is based on the Almgren and Poon's type monotonicity formulas, cf. [12]. The regular free boundary Γ 3/2 consists of those free boundary points which have the lowest frequency 3/2. It is relatively open (possibly empty) and locally a smooth graph given by x n−1 = f (x , t) up to a rotation for smooth inhomogeneities, cf. [7,20,21]. The so-called singular set consists of those free boundary points whose frequency are even integers 2m, m ≥ 1. Singular set can be equivalently characterized by the density of the contact set, i.e. it consists of free boundary points at which the contact set Λ u has zero Lebesgue density, cf. [12]. Finer structure of the singular set is studied in [12].
We briefly comment about our results and the related literature. Theorem 1.1 allows to provide a simpler proof for the openness of the regular free boundary and its space-time regularity (cf. Section 4), which was shown in [2,12,20] by using a boundary Harnack approach. In particular, we do not require the optimal spacial regularity or continuity of the time derivative of the solutions. Theorem 1.2 generalizes the results about the singular set in [12] in the sense that we obtain a log-log modulus of continuity of the L 2 norm, which implies a frequency gap around the 2m-frequency points (cf. Section 4). We mention that compared with the recent progress on the fine structure of the singular set for elliptic problems, for instance [14,15,17,18,23], the parabolic counterpart is much less known. The logarithmic epiperimetric inequality was recently established for the elliptic obstacle and thin obstacle problems [9,10,11], which inspired our paper. Very different from the proofs in [9,10,19] and the existing epiperimetric inequality approach for the parabolic problem (where they reduce the parabolic problem to a stationary problem by treating the time derivative as inhomogeneity), our approach is purely dynamical and does not rely on the (almost) minimality. Hence it can possibly be generalized to study the asymptotics around critical points instead of energy minimizers for a broader class of free boundary problems. It is also possible to apply our approach to the Signorini problem for the degenerate parabolic operators considered in [2,3,5,4,6].
The rest of the paper is structured as follows: in Section 2 we introduce the conformal change of coordinates, reformulate our problem in the new coordinates, and explore the role of the Weiss energy; In Section 3 we prove discrete decay estimates for the Weiss energy W κ . For simplicity we assume that the inhomogeneity f vanishes. The idea of the proof remains the same for nonzero inhomogeneities, as one can see from Section 5. In Section 4 we show the consequences of the decay estimates, for example, we prove the C 1,α regularity of the regular free boundary and the frequency gap around 2m-frequency. In the last section we will show how to modify the proof in Section 3 to the inhomogeneous setting. In the appendix we provide a short proof for the characterization of the second Dirichlet-Neumann eigenspace for the Ornstein-Uhlenbeck operator.
2. Conformal self-similar coordinates and Weiss energy. In the sequel, it will prove convenient to work in conformal self-similar coordinates. This will simplify many of the computations, which will be carried out for the Weiss energy.
Thus, we consider the following change of variables, which should be viewed as the analogue of conformal polar coordinates: (1). We consider the change of coordinates For κ > 0 we denotẽ . Then,ũ κ is a solution to with the Signorini conditioñ Proof. The proof follows from a direct computation.
Remark 2. Let u : S + 1 → R satisfy the Sobolev regularity (A) and (B). Letũ κ be obtained from u as in Lemma 2.1. Then it holds Moreover, for any 0 ≤ τ 1 < τ 2 < ∞, there exists a constant C depending on Here and in the sequel · L 2 µ := · L 2 Now we define the Weiss energy associated to a solutionũ :=ũ κ to (4)-(5) and derive relevant quantities of the Weiss energy.
Lemma 2.2. Letũ =ũ κ be a solution to (4)-(5) and satisfy (6). We define the Weiss energy Further, Proof. On a formal level the estimate (8) can be deduced by differentiating the functional W κ (ũ(τ )). However, in order to give meaning to the arising boundary contributions, it is necessary to work in a regularized framework which is achieved by penalization. More precisely, we consider the following penalized version of (5), (4): Let β : R → R be a smooth function satisfying the following properties β (s) = 0 for s ≥ 0, We approximateũ by solutions to the penalization problem Hereũ 0 is a smooth compact supported function such that ũ 0 −ũ(·, 0) L 2 µ → 0 as → 0. There exists a unique solutionũ with a polynomial growth as |y| → ∞. The functionũ is smooth and satisfies the uniform bound in the Gaussian space (cf. Chap. 3 in [12]): there exists C = C(κ, τ 2 , ũ 0 L 2 µ , n) such that D 2ũ In particular, using the equation forũ , it is then possible to compute as follows: We test this identity with ϕ ∈ C ∞ c ((0, ∞)) which yields The a priori bounds forũ leads to space time W 1,2 µ uniform estimates. Hence it is possible to use lower semi-continuity to pass to the weak limit in the bulk integral. Using that β (ũ )∂ τũ = ∂ τ B (ũ ), where B is the primitive function of β with B (0) = 0, the boundary integral is treated as in Lemma 5.1 (3 • ) in [12] and can be shown to vanish in the limit. Hence for ϕ ≥ 0 we infer which yields the desired result. Approximating the characteristic function χ [τ1,τ2] (t) by smooth positive functions then yields the claim on the sign of the difference of the Weiss functionals.
In the sequel, we will in particular exploit the second observation frequently. When κ = 3/2 and κ = 2m, m ∈ N + , by Liouville type theorems, we can characterize stationary solutions.
Proof. (i) Case κ = 3/2. We will prove thatũ is two dimensional. Given any tangential direction e with |e| = 1 and e · e n = 0, v := ∂ eũ solves the Dirichlet eigenvalue problem for L 0 := − 1 2 ∆ + y · ∇ on W 1,2 0 (R n \ Λũ; dµ) ⊂ L 2 µ (R n ): where Λũ := {(y , 0) :ũ(y , 0) = 0}. We claim that v does not change the sign in R n . Let 0 < λ 1 ≤ λ 2 ≤ · · · denote the Dirichlet eigenvalues. Assume that v changes the sign, then necessarily λ 2 ≤ 1 2 . By the min-max theorem, We consider the 2d subspaces M spanned by the (oddly reflected) Dirichlet eigenfunctions on R n + with w = 0 on the whole R n−1 × {0}. Since w(y) = y n is the first Dirichlet eigenfunction, whose Rayleigh quotient is equal to 1, then necessarily λ 2 ≥ 1. This is a contradiction. Therefore we conclude that v = ∂ eũ is nonpositive or nonnegative in the whole space R n . Since this holds for any tangential direction, it follows thatũ is of the formũ(y) =ũ(y · e, y n ) for some tangential direction e. In other words,ũ is two dimensional. Direct computation shows that the function Re(y ·e+i|y n |) 1/2 is an eigenfunction. The uniqueness of the principal eigenfunction implies that actually ∂ eũ = c Re(y · e + i|y n |) 1/2 for some c ∈ R. Thusũ ∈ E 3/2 .
At the end of this section we compare the Weiss energy in the original coordinates and the conformal coordinates. Firstly, ifũ κ is associated with a solution u : S + 2 → R to the parabolic Signorini problem (1) as in Lemma 2.1, then the Weiss energy ofũ κ in (7) can be rewritten in terms of u as Next, for λ > 0, let be the (parabolic) κ-homogeneous scaling, and let i.e. the homogeneous κ scaling for u(x, t) corresponds to the time shift forũ κ (y, τ ) by −2 ln λ. The Weiss energy in the original coordinates is well-behaved with respect to the parabolic rescaling, i.e. W κ (u λ (t)) = W κ (u(λ 2 t)). In the conformal coordinates this leads to 3. Parabolic epiperimetric inequality. We describe a dynamical system approach for deriving the decay of the Weiss energy W κ (ũ(τ )) along solutions to (4)-(5) with κ = 3/2 or κ = 2m, m ∈ N + .
3.1. The case κ = 3 2 . In this case, stationary solutions to (4)-(5) are in E 3/2 by Proposition 1. We will project our solutionũ(τ ) : and study the evolution of dist L 2 µ (ũ(τ, ·), E 3/2 ). Due to the non-convexity of E 3/2 the projection λh e ofũ(τ, ·) onto E 3/2 is not necessarily unique, hence the regularity of the parameters λ, e in dependence of τ is in question. In particular this implies that we have to take care in our dynamical systems argument and can not directly work with the evolution equations for the parameters λ(τ ) and e(τ ). Instead, we rely on robust (energy type) identities for the Weiss energy.
To this end, we splitũ into its leading order profile and an error: Here λ(τ )h e(τ ) (y) is chosen such as to minimize the L 2 µ distance ofũ to the set E 3/2 . We stress again, that this decomposition is a priori not necessarily unique. From the minimality of ũ(y, τ ) − λ(τ )h e(τ ) (y) L 2 µ we infer the following orthogonality conditions Here we have used that c Re( The next lemma concerns about the Weiss energy in terms of the error termṽ: can be written in terms ofṽ as In particular, ifũ ≥ 0 on {y n = 0}, then W (ṽ) ≤ W (ũ).
As our main auxiliary result, we deduce the following contraction argument, which is of the flavour of an epiperimetric inequality. Proposition 2. Letũ : R n × [0, ∞) → R be a solution of the parabolic thin obstacle problem (4)-(5) and satisfy (6). Then there exists a constant c 0 ∈ (0, 1) depending only on n, such that Proof. We argue by contradiction and use the contradiction assumption in combination with (17) to derive enough compactness.
(i). Assume that the statement were not true. Then there exists a sequence {c j } with c j ∈ (0, 1/2), c j → 0, solutionsũ j and times τ j such that The contradiction assumption (18) implies that Using (8) and the monotone decreasing property of τ → W (ũ(τ )) we infer (ii). We projectũ j (τ ) to E 3/2 for each τ > 0, and splitũ j into its leading order profile and an error term as in (14): In this step we aim to derive the following upper bound for W (ṽ j ) from the contradiction assumption (19): First we observe that for any solutionũ and any time interval and an integration by parts we have Next we apply (21) toũ j with I j := [τ j , τ j + 1]. For the bulk integral we use Hölder and (19) to get Combining (21) and (22) and using Young's inequality, we obtain 3 4 Recalling the relation between W (ũ) and W (ṽ) in (17), we infer Since, by the Signorini conditions, the second integral on the right hand side is less or equal to zero, we obtain the upper bound in (20). We remark that by rearrangement (23) also entails that In the next steps (iii)-(iv) we will use a compactness argument to arrive at a contradiction. The main idea is that one can find sequencesτ j ∈ I j andv j (y) := v j (τ j , y)/ ṽ j (τ j ) L 2 µ , such thatv j converges to a nonzero blow-up profile in E 3/2 . This leads to a contradiction. The bounds on the Weiss energy forṽ in step (ii) are used to derive the desired compactness properties.
(iv). Letũ j be the sequence from step (iii) such thatû j →û 0 ∈ E 3/2 . Letτ j ∈ I j be such that ṽ j (τ j ) 2 We will prove that up to a subsequenceŵ j converge in C([0, 1]; L 2 µ ) to a nonzero functionŵ 0 = λ n h up to a rotation of coordinates, where λ n = 0. This gives a contradiction. In fact, if λ n > 0, we get a contradiction because at each time step we have projected out E 3/2 fromũ j . If λ n < 0, then necessarily λ j (τ j ) = 0 for each j (because otherwise h ej (τj )ŵj dµ = 0 by (15), which leads to a contradiction in the limit j → ∞). However, it is a contradiction to Step (iii).
by the interior estimateŝ w j converges locally smoothly in R n + . This together with (28) implies thatŵ 0 is stationary and it solves Lŵ 0 = 0 in R n + . We claim that the limiting functionŵ 0 satisfies the Dirichlet-Neumann boundary condition where e 0 is the tangential direction from step (iii). With this at hand,ŵ 0 ∈ W 1,2 µ solves the eigenvalue problem for − 1 2 ∆+y ·∇ with the Dirichlet-Neumann boundary condition: By the characterization of the eigenfunctions for the second Dirichlet-Neumann eigenvalue (cf. Appendix 5), and after a rotation of coordinate (such that e 0 = e n−1 ),ŵ Here h 1/2 (y n−1 , y n ) := c n Re(y n−1 + i|y n |) 1/2 . The orthogonality condition (16) implies that λ i = 0 for i = 1, · · · , n − 2. Thus λ n = 0. This leads to a contradiction as argued at the beginning of Step (iv).
In the end, we verify (29). This is a consequence of the complementary boundary conditions satisfied byû j and the uniform convergence. Indeed, given U Ω 0 , using c n h e0 > c > 0 in U and the uniform convergence ofû j to c n h e0 in U × [0, 1] we have, for sufficiently large j depending on U ,û j > 0 in U × [0, 1]. By the complementary condition in terms ofũ j we have ∂ nûj = 0 in U × [0, 1]. Next since e j (τ j ) converges to e 0 , which follows from the convergence ofû j to c n h e0 in C 0 ([0, 1]; L 2 µ ), one has ∂ n h ej (τ ) = 0 in U for j sufficiently large. Thus ∂ nŵj = 0 in U × [0, 1] for sufficiently large j. Therefore, after extendingŵ j evenly about 1]). This implies that in the limit ∂ nŵ0 = 0 on U . Since U is arbitrary we have ∂ nŵ0 = 0 on Ω 0 . Using the fact that c n ∂ n h e0 < −c < 0 in U int(Λ 0 ) and arguing similarly we can conclude thatŵ 0 = 0 on int(Λ 0 ).
Remark 3. At this stage it is possible thatũ(∞) is zero. However, if at the initial time for some small δ n > 0, then in the limit λ(∞) > 0. To see this, we note that the bound on the Weiss energy together with the exponential convergence of λ(τ ) 2 from Corollary 2 (i) implies µ > 0 if δ n is chosen sufficiently small. We also note that (30) is satisfied by requiring that the solution stays close to 3.2. The case κ = 2m. In this section we derive the decay estimate of the Weiss energy along solutionsũ :=ũ 2m to the Signorini problem (4)- (5). Recall that when the frequency κ = 2m, stationary solutions are in E + 2m by Proposition 1, where E + 2m is a subset of zero eigenspace of the Ornstein-Uhlenbeck operator L 2m := 1 4 ∆− y 2 ·∇+m. The strategy is similar as for the case κ = 3/2. For each τ we project our solution to the finite dimensional linear space E 2m : Then we argue by contradiction that, if the associated Weiss energy W 2m (τ ) does not decay fast enough, then after renormalizationṽ(τ ) would converge to a nonzero element in the eigenspace E 2m as τ goes to infinity. Different from the case κ = 3/2, where the zero Dirichlet-Neumann eigenspace for L 3/2 is the tangent space to the manifold generated by the unique blow-up profile Re(x + iy) 3/2 together with rotation symmetry (cf. Appendix), in the case κ = 2m the zero eigenspace is not associated with a unique blow-up profile. That is one of the main reasons we choose to project out the whole linear space E 2m instead of the cone generated by a blow-up limit. Since functions in E 2m can change the sign on R n−1 × {0}, the estimates of the nonlinearity (which is concentrated on the unknown contact set {ũ = 0} at the boundary R n−1 × {0}) thus become more complicated in the case κ = 2m.
Multiplyingṽ on both sides of the equation, using the Signorini condition and the orthogonality (32) we obtain (33). Multiplying p α on both sides of the equation, using the orthogonality condition (which gives R n ∂ τṽ p α dµ = 0 for a.e. τ ) we get the evolution equation for λ α in (34).
In the sequel, we will frequently use the following auxiliary function. Let h 2m denote the 0-eigenfunction of L 2m , which has the expression where C m,n ∼ c n 2 2m 2 m m!, c n > 0, is a normalization factor such that h 2m L 2 µ = 1. Note that h 2m (y , 0) = C −1 m,n (2 2m |y | 2m + 1). In the sequel we will denote λ 2m := R n +ũ h 2m dµ.

Note that
W 2m (p <2m ) ≤ 0, W 2m (w) ≥ 0. Thus the contradiction assumption implies that This together with (35) and the monotone decreasing property of τ → W 2m (ũ(τ )) implies that 2 Ij α:|α|≤2mλ We will show that (38) implies for some C = C(m, n) > 0 However, from the spectral gap we have, for any τ > 0 This is a contradiction to (39) if j is sufficiently small. It remains to prove (39). Multiplying the equation ofũ by p <2m (τ ) and an integration by parts in space yield By (38) the first integral can be estimated from below as To estimate the boundary integral we observe that for each τ > 0 sup y ∈R n−1 |p <2m (y , 0, τ )| h 2m (y ) ≤ c n,m p <2m (y, τ ) L 2 µ .
Using ∂ nũ (y , 0) ≤ 0 and recalling the expression ofλ 2m in (34), we can estimate the boundary term from below by Invoking ( Combining together we obtain (39), and the proof is complete.
In the next proposition we derive a discrete logarithmic decay of the Weiss energy under the assumption that W 2m (ũ(τ )) > 0 for all τ .
(iii). Assuming (41) we can estimate the Weiss energy forṽ j from above:
In the end, we prove (41). We will divide the proof into three parts (a)-(c). Since the estimate is trivial if (p j ) − = 0, in the sequel we assume that (p j ) − is not identically zero on {y n = 0}. For a.e. τ we denote where h 2m (y ) ∼ n,m |y | 2m + 1 by (37). Noticing that and thatλ 2m is related to the Weiss energy via (40), we mainly need to estimate c * .
(a). We show that if , then there exists a positive constant C = C(m, n) such that Indeed, let y 0 ∈ B R0 a point which realizes the maximum. Since p j (τ ) 2 Let r 0 := M/(c n L) > 0. In B r0 (y 0 ) ∩ B R0 we have (p j ) − /h 2m ≥ M/2. Therefore, there exists a constant C > 0 depending only on m, n such that Since h 2m is uniformly bounded away from zero and recalling our choice of R 0 , we thus have Using the Signorini conditionũ j = p j +ṽ j ≥ 0 on {y n = 0} we have where the second inequality follows from the trace lemma. Combining the above two inequalities, using the definition of R 0 as well as the relation c * ≤ 2M , we complete the proof for (a).
(b). We show that if c * > 2M , then there exists C = C(m, n) > 0 such that .
If δ 0 = δ 0 (m, n) is sufficiently small, Similarly, using the monotonicity of W 2m (ṽ(τ )), Hölder's inequality and (40) we have Thus, combining the above estimates and using again the monotonic property of the Weiss energy we arrive at to estimate the first and the third term, and applying Young's inequality to the second term, we obtain (41).

Consequences of the epiperimetric inequality.
4.1. The case κ = 3/2: Uniqueness of blow-ups and regularity of the regular free boundary. In this section we apply the decay estimates for the Weiss energy in Section 3 to our original Signorini problem to derive the regularity of the free boundary.
In the end we will show that the lower bound in (2) implies that u 0 = 0. Indeed, if u 0 vanishes identically, then it holds The above estimate yields that H u (r) ≤ Cr 3+2γ0 . This is a contradiction to the lower bound in (2).

Remark 4.
Rewriting (30) in Remark 3 into the original variable, we see that instead of the lower bound assumption in (2), we can assume the solution is close to E 3/2 at t = −1 to guarantee the non-triviality of the 3/2-blowup limit. More precisely, assume that at t = −1 where W 3/2 (u 0 ) is the Weiss energy in the original variable as in (11), and inf Then if δ 0 is sufficiently small depending on n and f L ∞ , there is a unique u 0 = c 0 Re(x · e 0 + i|x n |) 3/2 ∈ E 3/2 with c 0 ≥ c n > 0 such that (48) holds true. We note that conditions (49)-(50) are satisfied if We also note that under the assumptions of Theorem 1.1, (49)-(50) are satisfied for u λ for sufficiently small λ > 0 depending on u 0 .
Proof. We note that by rotation invariance c (x0,t0) = c n u (x0,t0) L 2 µ for c n > 0. Hence, for (x 0 , t 0 ), (y 0 , s 0 ) ∈ Γ 3/2 (u) ∩ Q 1 and λ > 0 Here d((x 0 , t 0 ), (y 0 , s 0 )) = |x 0 − y 0 | + |t 0 − s 0 | 1/2 is the parabolic distance, and to estimate the three terms coming from the triangle inequality, we have used (48) to bound the first two integrals and the interior Hölder C α,α/2 estimate of the solution to bound the third integral. Balancing the above two bounds we get Next we note that Using similar estimate as above and combining it with the estimate for c (x0,t0) then yields the claimed Hölder continuity of e (x0,t0) .
With the previous results at hand, we can prove the regularity of the regular free boundary.
for any (x 0 , t 0 ), (y 0 , s 0 ) ∈ Γ u ∩ Q r (x 0 , t 0 ), and C > 0 independent of λ. Thus we find a parameter family of hypersurfaces Γ λ u , the normals of which are spacial and equal to e (x0,t0),λ at each (x 0 , t 0 ) ∈ Γ λ u , and they are uniformly C θ regular with respect to the parabolic distance. Passing to the limit as λ → 0 we thus obtain that the limiting hypersurface, which is the free boundary Γ u ∩ Q r (x 0 , t 0 ), is a C θ hypersurface. Thus up to a rotation of the spacial cooridnates, it can be represented as the graph x n−1 = g(x , t) for some function g, where ∇ g ∈ C θ,θ/2 .

4.2.
The case κ = 2m. 4.2.1. Uniqueness and nondegeneracy. We first prove Theorem 1.2 by using decay estimate of the Weiss energy in Corollary 3.
Proof for Theorem 1.2. Letũ =ũ 2m be the 2m conformal normalized solution as in Lemma 2.1. The upper bound on H u (r) implies that W 2m (ũ(τ )) ≥ 0 for all τ ∈ (0, ∞). In fact, if W 2m (ũ(τ 0 )) < 0 for some τ 0 > 0, then by Corollary 3, there exist γ m ∈ (0, 1) and C 0 > 0 such that ũ(τ ) 2 Back in the original coordinates we have for some C > 0 depending on τ 0 and for each t ∈ (−e −τ0 , 0). This is however a contradiction to our assumption on H u when |t| is sufficiently small. Thus one can apply Corollary 3 (ii) toũ(τ )/ ũ(0) L 2 µ and conclude that there is a unique non-zero p(y) ∈ E + 2m such that . Writing the above inequality by the original variables and by (12), we obtain the desired estimate.
(ii). Assume that (0, 0) is a free boundary point with frequency 2m − , > 0. Then > γ m , where γ m ∈ (0, 1) is the constant from Corollary 3 (i). Indeed, similar as in (i) we consider a nontrivial blow-up limit at (0, 0). After 2m-normalization and in the conformal coordinates this leads toũ 2m (y, τ ) = e τ /2 v(y), where v solves Thus one has , which implies that ≥ γ m > 0. 5. Perturbation. In this section we show how to modify our proof in Section 3 to the nonzero inhomogeneity setting. We consider global solutions which satisfies (A)-(C) to where f = f (x, t) ∈ L ∞ (S + 2 ). By chapter 4 of [12], the study of local solutions with nonzero obstacles can be reduced to the study of global solutions to the above inhomogeneous equations by subtracting the obstacle and applying suitable cut-offs.
Proof of Proposition 10. (i). Assume not, then there exists c j ∈ (0, 1/4), c j → 0, solutionsũ j to (52) with inhomogeneityf j and times τ j such that In the sequel for notational simplicity we writeũ andf instead ofũ j andf j . Then, using (53), one has By the almost monotonicity (53), for any τ ∈ I j := [τ j , τ j + 1], Therefore, we obtain (ii). We seek to estimate the Weiss energy for the error termṽ j . We remark that the relation (17) still holds for the inhomogeneous problem. First, we can write Here as in the zero homogeneity case, L := 1 4 ∆ − y 2 · ∇ + 3 4 . Applying (54) to the first term on the RHS we obtain By Cauchy-Schwartz and applying (17) to replace W (ũ) by W (ṽ) we get, Noting that the term involving integral on {y n = 0} is non-positive by the Signorini boundary condition, we thus obtain Here in (55) and (56), the constant C > 0 is an absolute constant. With (54), (55) and (56) at hand, we argue as step (iii) and (iv) in Proposition 2 and reach a contradiction.
From this and arguing similarly as in Corollary 2 we conclude that if W (ũ(τ )) ≥ 0 for all τ > 0, then for all 0 < τ 1 < τ 2 < ∞, and if W (ũ(τ 0 )) < 0 for some τ 0 > 0, then For the case κ = 2m, m ∈ N + in (52) we assume further that the inhomogeneity in the conformal coordinates satisfies: for some M > 0 and 0 ∈ (0, 1), Note that (57) is satisfied if in the original coordinates f (x, t) has the vanishing property at (0, 0) that |f (x, t)| ≤ M (|x| + √ −t) 2(m−1+ 0 ) . Under such assumption, the inhomogeneity only contributes as a higher order term in our estimates. In particular, similar as (53) in the κ = 3/2 case, we have the almost monotone decreasing property for the Weiss energy: for 0 < τ a < τ b < ∞, Thus with slight modification as in the case κ = 3/2, one can generalize Proposition 4 and Proposition 5 to the nonzero inhomogeneity case, and we do not repeat the proof here.
Proof. Direct computation shows that Re(y 1 +i|y 2 |) 3/2 ∈ E is an eigenfunction with λ = 3/2. Therefore, by the characterization of the second eigenvalue: we have that λ 2 ≤ 3/2. The proof for λ 2 = 3/2 and characterization of the eigenspace is mainly based on the following property, which we are going to prove: if u ∈ E is an eigenfunction with eigenvalue λ ∈ (1/2, 5/2), then its tangential derivative ∂ u, = 1, · · · , n − 1, is a ground state. We also note that eigenfunctions which are L 2 integrable against the Gaussian measure dµ = e −|y| 2 dy have at most power growth at infinity (cf. for instance, [8]).
(i) Case n = 2. The proof is divided into two steps. In the first step, we will show that if u is an eigenfunction with λ > 1/2, then it is in C 1,1/2 loc (R 2 + ) up to the slit. This implies that its derivative ∂ 1 u is in the space E. In the second step we apply the characterization of the ground state to show that, if u is an eigenfunction associated to λ 2 , then up to a constant u(y) = Re(y 1 + i|y 2 |) 3/2 .
We can further extend u to a solution in the full space R 2 by odd reflection about z 1 = 0. Since the operator is hypoelliptic, solution u is real-analytic, thus has the power series expansion: u(z) = k H k (z), where H k (z) is the kth-order homogeneous polynomial. Using the even/odd symmetry and the fact that u is orthogonal to the ground state z 1 , we have H 0 = H 1 = H 2 = 0. Observing that the right hand side of the equation is of order o(|z| 2 ), then necessarily ∆H 3 (z) = 0, which together with the symmetry gives H 3 (z) = bz 1 (z 2 2 − 1 3 z 2 1 ) for some constant b. Therefore, Transforming back to the original variables, we obtain u(y) =b Re(y 1 + i|y 2 |) 3/2 + O(|y| 2 ).
In particular, this implies that u ∈ C 1,1/2 loc up to the slit.
This implies g = 0 (one can see this from, for instance, the classical result on spectrum of the 1d Ornstein-Uhlenbeck operator). Thus the eigenfunction for λ 2 is u(y) = c Re(y 1 + i|y 2 |) 3/2 .
Similarly as in (i), using λ 2 ≤ 3/2 and the characterization of the ground state we can conclude that w = c Re(y n−1 + i|y n |) 1/2 and λ 2 = 3/2. This implies that u(y) = n−2 i=1 c i y i Re(y n−1 + |y n |) 1/2 + h(y n−1 , y n ) for some function h only depending on y n−1 and y n . From the equation and the boundary conditions for u, we infer that h is the second Dirichlet-Neumann eigenfunction for L. Thus using the 2d result from (i) we conclude that h = c Re(y n−1 + i|y n |) 3/2 .