On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions

In [13], Chen-Ma-Salani established the strict convexity of spacetime level sets of solutions to heat equation in convex rings, using the constant rank theorem and a deformation method. In this paper, we generalize the constant rank theorem in [13] to fully nonlinear parabolic equations, that is, establish the corresponding microscopic spacetime convexity principles for spacetime level sets. In fact, the results hold for fully nonlinear parabolic equations under a general structural condition, including the p-Laplacian parabolic equations (p > 1) and some mean curvature type parabolic equations.


1.
Introduction. This paper is devoted to the microscopic spacetime convexity principle for the second fundamental forms of the spatial and spacetime level sets of the solutions to fully nonlinear parabolic equations. In this paper, we consider the spatial convexity and the spacetime convexity of the level sets of the spacetime quasiconcave solutions to the heat equation. A continuous function u(x, t) in Ω × (0, T ] is called spacetime quasiconcave if the spacetime superlevel sets {(x, t) ∈ Ω × (0, T )|u(x, t) ≥ c} are convex for each constant c.
Spacetime convexity is a basic geometric property of the solutions of parabolic equations. In [5,6,7], Borell used the Brownian motion to study certain spacetime convexities of the solutions of diffusion equations and the level sets of the solution to a heat equations with Schrödinger potential. Ishige-Salani introduced some notions of parabolic quasiconcavity in [18,19] and parabolic power concavity in [20], which are some kinds of spacetime convexity. In [18,19,20], they studied the corresponding parabolic boundary value problems using the convex envelope method, which is a macroscopic method. At the same time, Hu-Ma [17] established a constant rank theorem for the space-time Hessian of space-time convex solutions to the heat equation, which is the microscopic method. Chen-Hu [12] generalized the microscopic spacetime convexity principle to fully nonlinear parabolic equations. Recently, Chen-Ma-Salani [13] established the strict convexity of spacetime level sets of solutions to the heat equation in convex rings, using the constant rank theorem and a deformation process. In this paper, we generalize the constant rank theorem in [13] to fully nonlinear parabolic equations. And the results hold for fully nonlinear parabolic equations under a general structural condition, including the p-Laplacian parabolic equations (p > 1) and some mean curvature type parabolic equations. As in [13], this approach can be used to establish some spacetime convexity of the solutions of some parabolic equations in convex rings, by combining a deformation process.
The convexity of the level sets of the solutions to elliptic partial differential equations has been studied extensively. For instance, Ahlfors [1] contains the wellknown result that level curves of Green function on simply connected convex domain in the plane are the convex Jordan curves. In 1956, Shiffman [29] studied the minimal annulus in R 3 whose boundary consists of two closed convex curves in parallel planes P 1 , P 2 . He proved that the intersection of the surface with any parallel plane P , between P 1 and P 2 , is a convex Jordan curve. In 1957, Gabriel [15] proved that the level sets of the Green function on a 3-dimensional bounded convex domain are strictly convex. In 1977, Lewis [23] extended Gabriel's result to pharmonic functions in higher dimensions. Caffarelli-Spruck [9] generalized the Lewis [23] results to a class of semilinear elliptic partial differential equations. Motivated by the result of Caffarelli-Friedman [8], Korevaar [22] gave a new proof on the results of Gabriel and Lewis by applying the deformation process and the constant rank theorem of the second fundamental form of the convex level sets of p-harmonic function. A survey of this subject is given by Kawohl [21]. For more recent related extensions, please see the papers by Bianchini-Longinetti-Salani [4], Bian-Guan [2], Xu [31] and Bian-Guan-Ma-Xu [3]. For the convexity of spacetime level sets, Ishige-Salani [18,19] introduced some notions of parabolic quasiconcavity, and studied the corresponding parabolic boundary value problems using the convex envelope method. Recently, Chen-Ma-Salani made a breakthrough in the strict convexity of the heat equation in [13] using the constant rank theorem method.
Let us introduce some notations.
Definition 1.1. For each θ ∈ S n−1 , denote θ ⊥ the linear subspace in R n which is orthogonal to θ. Define S − n (θ) to be the class of n × n symmetric real matrices which are negative definite on θ ⊥ . Denote S 0− n (θ) the subclass of S − n (θ) of matrices that have θ as eigenvector with corresponding null eigenvalue. For any b ∈ R n with s = b, θ > 0, define where S n+1 denote the space of real symmetric (n + 1) × (n + 1) matrices, and Υ ⊂ S n be an open set.
Denote J = (I n |0) the n × (n + 1) matrix, where I n is the n × n identity matrix and 0 is the null vector in R n .
Now we state our theorems.
For the study of the spacetime level sets of fully nonlinear equation, we should consider the spatial level sets first. Suppose u is the spacetime quasiconcave solution to fully nonlinear parabolic equation (1), then u is also spatial quasiconcave, that is the spatial level sets Σ c = {x ∈ Ω|u(x, t) = c} are all convex. And we get the following constant rank theorem for the second fundamental form of the spatial level sets. Theorem 1.3. Suppose u ∈ C 3,1 (Ω × (0, T ]) is a spacetime quasiconcave to fully nonlinear parabolic equation (1), and F satisfies conditions (2)- (4). Then the second fundamental form of spatial level sets Σ c = {x ∈ Ω|u(x, t) = c} has the same constant rank in Ω for each fixed t ∈ (0, T ]. Moreover, let l(t) be the minimal rank of the second fundamental form in Ω, then l(s) ≤ l(t) for all 0 < s ≤ t ≤ T .
As it is well known, one needs to choose a suitable coordinate system to simplify the calculations in the proof of the constant rank theorem. In [13], the proof for the heat equation is based on a coordinate system such that the spatial second fundamental form a(x, t) (see (11)) is diagonalized at each point. In this paper, we generalize the constant rank theorem in [13] to fully nonlinear parabolic equations, and we give a more technical proof under the coordinate system such that the spacetime second fundamental formâ(x, t) (see (15)) is diagonalized at each point. As in [17,12] and [11], the key difficulties of two calculations are the same, and the processes of the two proofs are also the same. So the corresponding proof holds for fully nonlinear equations based on the coordinate system such that the spatial second fundamental form a(x, t) is diagonalized at each point, and the calculations must be more complicate than [13].
Remark 1. In fact, in the proof of Theorem 1.3, we just need a weaker structural condition as follows instead of (3), for fixed (θ, u) ∈ S n−1 × R. But the condition that u is spacetime quasiconcave is necessary, and it is the main difference between Theorem 1.3 and the result in [14]. That is, if u is spacetime quasiconcave, the constant rank theorem for spatial level sets holds for the parabolic equations with (5). Otherwise, if u is spatial quasiconcave, Chen-Shi [14] established the constant rank theorem for spatial level sets holds for the parabolic equations with a totally different structural condition as follows where A ∈ S n and s > 0 do not depend on A.
In fact, the p-Laplacian operator and the mean curvature operators, that is, the parabolic equations do not satisfy the structural condition (3) or (5). But we have the following theorem.
Remark 2. Theorem 1.3 can be looked as a parabolic version for Theorem 1.1 in [3].
Remark 3. The microscopic spacetime convexity principle can be used to establish the spacetime convexity of the solutions of some parabolic equations in convex rings, by combining the deformation process. For example, Chen-Ma-Salani [13] consider the heat equation in convex rings, and get the strict convexity of the spacetime level sets, with some compatible conditions on the initial data and the convex rings.
The rest of the paper is organized as follows. Section 2 contains some preliminaries. In Section 3, we prove the constant rank theorem of the spatial second fundamental form, that is Theorem 1.3 and the corresponding part of Theorem 1.4. The constant rank theorem of the spacetime second fundamental form is proved in Section 4, including Theorem 1.2 and the corresponding part of Theorem 1.4.

2.
Preliminaries. In this section, we will give some preliminaries.
First, we introduce the definitions of spatial quasiconcave and spacetime quasiconcave.
The second fundamental form II of the spatial level sets of function u with respect to the upward normal direction (9) is Note that if Σ c = {x ∈ Ω|u(x, t) = c} is locally convex, then the second fundamental form of Σ c is semipositive definite with respect to the upward normal direction (9). Let a(x, t) = (a ij (x, t)) be the symmetric Weingarten tensor of Σ c = {x ∈ Ω|u(x, t) = c}, then a is semipositive definite. As computed in [3], if u n = 0, and the Weingarten tensor is where and Einstein summation convention is used. With the above notations, at the point (x, t) where u n (x, t) = |∇u(x, t)| > 0, u i (x, t) = 0, i = 1, · · · , n − 1, a ij,k is commutative, that is, they satisfy the Codazzi property a ij,k = a ik,j , ∀i, j, k ≤ n−1.

2.2.
Spacetime level sets and the spacetime second fundamental form. Suppose u(x, t) ∈ C 2 (Ω × (0, T ]), and u t = 0 for any fixed (x, t) ∈ Ω × (0, T ]. It follows that the upward inner normal direction of the spatial level setsΣ where Du = (u 1 , u 2 , · · · , u n−1 , u n , u t ) is the spacetime gradient of u. The second fundamental form II of the spacetime level sets of function u with respect to the upward normal direction (13)  Note that ifΣ c = {(x, t) ∈ Ω × (0, T )|u(x, t) = c} is locally convex, then the second fundamental form of Σ c,t is semipositive definite with respect to the upward normal direction (13). Letâ(x, t) = (â ij (x, t)) be the symmetric Weingarten tensor of Σ c = {(x, t) ∈ Ω × (0, T )|u(x, t) = c}, thenâ is semipositive definite. If u t = 0, and the Weingarten tensor iŝ wherê With the above notations, at the point ( Sô Also, at any point (x, t), we can translate the spacetime coordinate systems. When we choose the coordinates y = (y 1 , · · · , y n , y n+1 ) as a new spacetime coordinates, such that u yn+1 > 0, the Weingarten tensor is wherē for 1 ≤ α, β ≤ n. With the above notations, at the point (x, t) with the new coordinates y such that u yi = 0 for any 1 ≤ i ≤ n and u yn+1 = |Du| > 0, we get 2.3. Elementary symmetric functions. In this subsection, we recall the definition and some basic properties of elementary symmetric functions, which could be found in [24].
We need the following standard formulas of elementary symmetric functions.
Properties of A − θ , B − θ and their relationship have been studied in [4]. In particular, if θ = (0, · · · , 0, 1), then where the (n − 1) × (n − 1) matrix (a ij ) is negative definite and can be assumed diagonal, (a ij ) is the inverse matrix of (a ij ), s = B n+1,n = 1 µ > 0. The values at the positions denoted by "×" which are not important in the calculations.
For any given V = ((X ij ), Y, (Z i ), D) ∈ S n × R × R n × R, we define a quadratic form where Einstein summation convention is used, the derivative functions of F are evaluated at (s −1 A, s −1 θ, u, x, t), T := {1, 2, · · · , n − 1}, and Through direct calculations, we can get where the derivative functions of F are evaluated at (s −1 A, s −1 θ, u, x, t), and Q * is defined in (31).

CHUANQIANG CHEN
The proof of Lemma 2.5 is similar to the discussion in [2], and we omit it.
Remark 4. F satisfies the condition (5) if and only if for each fixed p ∈ R n , and for anyṼ = ((X ij ), Y, where the derivative functions of F are evaluated at (s −1 A, s −1 θ, u, x, t), and Q * is defined in (31). Obviously, the condition (5) is weaker than the condition (3).
Proof. The same arguments as in the proof of Lemma 2.5 in [2] carry through with a small modification since W is a general matrix instead of a Hessian of a convex function.
Since W (x) ≥ 0, so we choose h(x) = W ii (x) ≥ 0. Then we can get from the above argument Similarly, for i = j, we choose h = W ii W jj ≥ 0, then we get And So from (35) and (36), we get So (34) holds for i = j.
3. Constant rank theorem of the spatial second fundamental form. In this section, we consider the spatial level sets Σ c = {x ∈ Ω|u(x, t) = c}. Since u is the spacetime quasiconcave solution to fully nonlinear parabolic equation (1), u is also spatial quasiconcave, that is the spatial level sets Σ c are all convex for t ∈ (0, T ], that is the spatial second fundamental form a ≥ 0. We will establish the constant rank theorem for the spatial second fundamental form a under the structural condition (5) as follows.
is a spacetime quasiconcave to fully nonlinear parabolic equation (1), and F satisfies conditions (2), (4) and (5). Then the second fundamental form of spatial level sets Σ c = {x ∈ Ω|u(x, t) = c} has the same constant rank in Ω for any fixed t ∈ (0, T ]. Moreover, let l(t) be the minimal rank of the second fundamental form in Ω, then l(s) ≤ l(t) for all 0 < s ≤ t ≤ T .
From the discussion in Section 2, the structural condition (5) is weaker than the structural condition (3), then Theorem 1.3 holds directly from Theorem 3.1.
In the following of this section, we will prove Theorem 3.1, and discuss some constant rank properties of the spatial second fundamental form a. And we will prove the constant rank theorem of the spatial fundamental form of the spacetime quasiconcave solutions to the parabolic equations (6)-(8).

CHUANQIANG CHEN
Note that for any δ > 0, we may choose O × (t 0 − δ, t 0 ] small enough such that For each c, let a = (a ij ) be the symmetric Weingarten tensor of Σ c . Set Since we are dealing with general fully nonlinear equation (1), as in the case for the convexity of solutions in [2], there are technical difficulties to deal with p(a) alone.
A key idea in [2] is the introduction of function q as in (41) and explore some crucial concavity properties of q. We consider function where p and q as in (41). We will prove the differential inequality n α,β=1 where C is a positive constant independent of φ. Combining with the conditions we can get by the strong maximum principle By the method of continuity, Theorem 3.1 holds. In the following, we prove the differential inequality (43).
To get around σ l+1 (a) = 0 in q(a), for ε > 0 sufficiently small, we instead consider where a ε = a + εI. We will also denote G To simplify the notations, we will drop subindex ε with the understanding that all the estimates will be independent of ε. In this setting, In the following, we denote , we choose a coordinate system as in (39) so that |∇u| = u n > 0 and the matrix (a ij (x, t)) is diagonal for 1 ≤ i, j ≤ n − 1 and semipositive definite. From the definition of φ, we get And from (10) -(12), we get From the definition of a ij , and u k = 0 for k = 1, · · · , n − 1, we can get so for i, j ∈ B, we get In fact, from (14)- (16), and from the spacetime convexity, we can get Following the proof of Lemma 2.1 in [14], we can get and n α,β=1 where the second "=" holds from (51). For each j ∈ B, differentiating equation (1) in e j direction at x, Set and denote then we can get So it yields where From the structural condition (5) (i.e. Remark 4), it implies so for j ∈ B, we get Condition (2) implies Set Combining (59), (60) and (61), ].
By Lemma 3.3 in [2], for each M ≥ 1, for any M ≥ |γ i | ≥ 1 M , there is a constant C depending only on n and M such that, ∀α, Taking for each i ∈ B, the Newton-MacLaurine inequality implies Therefore we conclude from (63) and (64) that i,j∈B |∇a ij | can be controlled by the rest terms on the right hand side in (62) and φ + |∇φ|. So (43) holds, and the proof of Theorem 3.1 is complete.

3.2.
Constant rank properties of a. In the proof of Theorem 3.1, we can get for any ( and by the strong maximum principle, So it must have for any ( In fact, we can get more information from the differential inequality, and the constant rank properties is as follows with the suitable coordinate (39), we have from (11) and (67) u jj = 0, for j ∈ B.
and from (66) and (65), we get for j ∈ B, then
So the proof is complete.
3.3. Constant rank theorem of the spatial second fundamental form for the equation (6). In this subsection, we consider the p-Laplacian parabolic equation, that is where It is easy to know the equation (73) is parabolic when p > 1 and |∇u| > 0 in Ω × [0, T ]. We will establish the constant rank theorem for the spatial second fundamental form a as follows.
is a spacetime quasiconcave to the parabolic equation (73) and satisfies (4). Then the second fundamental form of spatial level sets Σ c = {x ∈ Ω|u(x, t) = c} has the same constant rank in Ω for any fixed t ∈ (0, T ]. Moreover, let l(t) be the minimal rank of the second fundamental form in Ω, then l(s) ≤ l(t) for all 0 < s ≤ t ≤ T .
Proof. The proof is similar to the the proof of Theorem 3.1, with some modifications. Suppose the spatial second fundamental form a(x, t) attains minimal rank l at some point (x 0 , t 0 ) ∈ Ω × (0, T ]. We may assume l ≤ n − 2, otherwise there is nothing to prove. So there is a small neighborhood O × (t 0 − δ, t 0 ] of (x 0 , t 0 ), such that there are l "good" eigenvalues of (a ij ) which are bounded below by a positive constant, and the other n − 1 − l "bad" eigenvalues of (a ij ) are very small. Denote G be the index set of these "good" eigenvalues and B be the index set of "bad" eigenvalues. We will prove the differential inequality n α,β=1 where φ is defined in (42) and C is a positive constant independent of φ. Then by the strong maximum principle and the method of continuity, Theorem 3.2 holds.
For any fixed point (x, t) ∈ O × (t 0 − δ, t 0 ], we may express (a ij ) in a form of (11), by choosing e 1 , · · · , e n−1 , e n such that Following the proof of Theorem 3.1, we get from (59) where Under the coordinate (76), we get L kl = 0, k = l; L kk = u p−2 n , k < n; L nn = (p − 1)u p−2 n ; ∂L kl ∂u n = 0, k = l; and From the equation (73), we know And from (51), we getĥ So So we can get Following the proof of Theorem 3.1, we get (75).
Remark 6. The constant rank properties ( that is, Corollary 1) still holds for the equation (6).

3.4.
Constant rank theorem of the spatial second fundamental form for the equation (7). In this subsection, we consider the mean curvature parabolic equation, that is where We will establish the constant rank theorem for the spatial second fundamental form a as follows. Proof. The proof is similar to the the proof of Theorem 3.1 and Theorem 3.2, with some modifications. Suppose the spatial second fundamental form a(x, t) attains minimal rank l at some point (x 0 , t 0 ) ∈ Ω × (0, T ]. We may assume l ≤ n − 2, otherwise there is nothing to prove. So there is a small neighborhood O × (t 0 − δ, t 0 ] of (x 0 , t 0 ), such that there are l "good" eigenvalues of (a ij ) which are bounded below by a positive constant, and the other n − 1 − l "bad" eigenvalues of (a ij ) are very small. Denote G be the index set of these "good" eigenvalues and B be the index set of "bad" eigenvalues. We will prove the differential inequality n α,β=1 where φ is defined in (42) and C is a positive constant independent of φ. Then by the strong maximum principle and the method of continuity, Theorem 3.3 holds.

Remark 7.
The constant rank properties ( that is, Corollary 1) still holds for the equation (7).
We will establish the constant rank theorem for the spatial second fundamental form a as follows.
Theorem 3.4. Suppose u ∈ C 3,1 (Ω × (0, T ]) is a spacetime quasiconcave to the parabolic equation (96) and satisfies (4). Then the second fundamental form of spatial level sets Σ c = {x ∈ Ω|u(x, t) = c} has the same constant rank in Ω for any fixed t ∈ (0, T ]. Moreover, let l(t) be the minimal rank of the second fundamental form in Ω, then l(s) ≤ l(t) for all 0 < s ≤ t ≤ T .
Proof. The proof is similar to the the proof of Theorem 3.1, with some modifications. Suppose the spatial second fundamental form a(x, t) attains minimal rank l at some point (x 0 , t 0 ) ∈ Ω × (0, T ]. We may assume l ≤ n − 2, otherwise there is nothing to prove. So there is a small neighborhood O × (t 0 − δ, t 0 ] of (x 0 , t 0 ), such that there are l "good" eigenvalues of (a ij ) which are bounded below by a positive constant, and the other n − 1 − l "bad" eigenvalues of (a ij ) are very small. Denote G be the index set of these "good" eigenvalues and B be the index set of "bad" eigenvalues. We will prove the differential inequality n α,β=1 where φ is defined in (42) and C is a positive constant independent of φ. Then by the strong maximum principle and the method of continuity, Theorem 3.4 holds. For any fixed point (x, t) ∈ O × (t 0 − δ, t 0 ], we may express (a ij ) in a form of (11), by choosing e 1 , · · · , e n−1 , e n such that Following the proof of Theorem 3.1, we get from (59) where Under the coordinate (99), we get M kl = 0, k = l; M kk = 1 + u 2 n , k < n; M nn = 1; ∂M kl ∂u n = 0, k = l; ∂M kk ∂u n = 2u n , k < n; ∂M nn ∂u n = 0; and From (47), (49) From (51), we getĥ From the equation (96), we know and by u kk ≤ 0 for k < n, it yields Hence we can get Hence from (101), where 4u n k∈G u kkj · u jn u 2 n + 2(1 + u 2 n ) So we can get Following the proof of Theorem 3.1, we get (98).
Remark 8. The constant rank properties ( that is, Corollary 1) still holds for the equation (8).
(110) So the spatial second fundamental form a = (a ij ) (n−1)×(n−1) attains the minimal rank l − 1 at (x 0 , t 0 ). From Theorem 3.1, the constant rank theorem holds for the spatial second fundamental form a. So we can get a ii = 0, ∀i ∈ B for any (x, t) ∈ O × (t 0 − δ, t 0 ]. Furthermore, We denote M = (â ij ) 1≤i,j≤n−1 , so Then we have (114) By the continuity method, Theorem 1.2 holds under the Case 1.
Proof. Taking the second derivatives of φ in y coordinates, we have So we have For γ ∈ B, we havē From (163) and (164), (157) holds.
and n i,j=1 Proof. Similarly with (147), taking the first derivatives of φ in t, we have In the following, we prove (166). In fact, the calculation is similar as in [3] and [16].

(188)
Denote Hence, (183) holds from (184) and (191). For the general case, the CLAIM also holds following the above proof.

4.3.
Constant rank theorem of the spacetime second fundamental form for the equations (6)- (8). Following the proof of Theorem 1.2, we establish the constant rank theorem for the spacetime fundamental form for the equations (6)- (8) in this subsection as follows.
Proof. The proof is following the proof of Theorem 1.2.
For the Case 1, the theorem holds from Subsection 4.1 and Theorem 3.3, Theorem 3.5, Theorem 3.7.
For the Case 2, we need to prove the differential inequality (192), which is similar to the proof of Theorem 3.3, Theorem 3.5, and Theorem 3.7, with some modifications. In the following, we just prove (192) for the equation (6). And for the equation (7) and (8), the proofs follow from the proofs of Theorem 3.5 and Theorem 3.7 with the same modifications.