Boundedness of square functions related with fractional Schrödinger semigroups on stratified Lie groups

Dedicated to Professor Neil Trudinger on the occasion of his 80th birthday.

1.   Introduction

Let ${\mathcal M}_0$ be a smooth, closed, uniformly convex hypersurface in ${\mathbb R}^{n+1}$ enclosing the origin. In ^[20], the authors studied the following anisotropic Gauss curvature flow:

where $K(\cdot, t)$ is the Gauss curvature of the hypersurface ${\mathcal M}_t$, parametrised by $X(\cdot, t):\ {\mathbb S}^n \to {\mathbb R}^{n+1}$, $\nu(\cdot, t)$ is the unit outer normal at $X(\cdot, t)$, $f$ is a given positive smooth function on ${\mathbb S}^n$, and $r = |X(x, t)|$ is the distance from the origin to the point $X(x, t)$.

The Gauss curvature flow (1.1) was introduced to study the existence of solutions to the dual Minkowski problem proposed in ^[16]. It can be formulated as solving the following Monge-Ampère equation on the unit sphere ${\mathbb S}^n$,

where $u$ denotes the support function of a hypersurface solution ${\mathcal M}$. By establishing the a priori estimates and studying the convergence of the normalized flows of (1.1), the following results were proved in ^[20].

Theorem 1.1. (^[20]) Let $f$ be a smooth and positive function on the sphere ${\mathbb S}^n$.

$({\rm{i}})$ If $\alpha > n + 1$, there is a unique smooth, uniformly convex solution to (1.2).

$({\rm{ii}})$ If $\alpha = n + 1$, assume that $f$ satisfies the condition (1.3) below, then there is a smooth, uniformly convex solution to (1.2). The solution is unique up to dilation.

$({\rm{iii}})$ If $\alpha < n + 1$ and $f$ is even, there is an origin-symmetric, smooth and uniformly convex solution to (1.2).

$({\rm{iv}})$ If $f\equiv 1$, then the solution must be a sphere when $\alpha \ge n+1$, and the origin-symmetric solution must be a sphere when $0 \le \alpha < n + 1$.

When $\alpha = n+1$, Eq (1.2) is the Aleksandrov problem. It is known that there is a necessary and sufficient condition for the existence of solutions, namely

where $\omega^*$ is the dual set of $\omega$.

In this paper, as in ^[5,10,20] we employ a Gauss curvature flow to study the existence of solutions to the $L_p$ dual Minkowski problem introduced in ^[23]

which extends the dual Minkowski problem (1.2). Let $f$ and $g$ be positive functions on ${\mathbb S}^n$, and $p, q\in {\mathbb R}$. We study the existence of solutions to the following equation,

Equation (1.4) contains the $L_p$ dual Minkowski problem as a special case (namely when $g\equiv 1$). It extends the $L_p$-Minkowski problem (when $q = n+1$ and $g\equiv 1$) and the dual Minkowski problem (when $p = 0$ and $g\equiv 1$).

In particular, when $g\equiv 1$, $p = 1, q = n+1$, Eq (1.4) is the classical Minkowski problem, which asks for the existence of closed convex hypersurfaces with prescribed surface area measure. It is a major impetus for the development of fully nonlinear PDEs. The $L_p$-Minkowski problem, introduced in ^[22], concerns the existence of closed convex hypersurfaces with prescribed $p$-area measures. It extends the classical Minkowski problem and includes the logarithmic Minkowski problem ($p = 0$), and the centro-affine Minkowski problem ($p = -n-1$) as special cases ^[2,11]. In the last two decades, great progress has been made in the study of the $L_p$-Minkowski problem. There is a rich phenomena on the existence and multiplicity of solutions, depending on the range of the exponent $p$ (see e.g., ^{[1,9,12,15,18,19]}).

For general exponents $p$ and $q$, Eq (1.4) has been studied in ^[4,6,7,8,17]. Suppose that $g\equiv1$. The existence of smooth solutions to (1.4) was proved in ^[17] for $p > q$, and in ^[8] for $p = q$. When $p < q$, the solution may not be smooth in general. Weak solutions were obtained in ^[4] when $p > 1$ and $q > 0$, and later on in ^[8] for all $p > 0$ and $q\in{\mathbb R}$. If only origin-symmetric solutions are concerned, Eq (1.4) was solved in ^[6] when $pq\ge0$, and in ^[7] when $q > 0$ and $-q^* < p < 0$ where $q^*$ is defined as

Suppose now $g\not\equiv1$. Equation (1.4) with $p = q = 0$ characterises the Gauss image problem proposed by ^[3], which extends the classical Aleksandrov problem. It was also considered in ^[21] from the optimal transportation viewpoint.

The main result of this paper is the following.

Theorem 1.2. Let $f, g\in C^{1, 1}({\mathbb S}^n)$ be positive functionssatisfying $c_0^{-1}\leq f, g\leq c_0$ for some constant $c_0 > 1$. Suppose that $q > n$ and

Then there is a uniformly convex and $C^{3, \alpha}$-smooth positive solution to (1.4), where $\alpha\in (0, 1)$.

When $q = n+1$ and $g\equiv 1$, Theorem (1.2) recovers the main result in ^[13]. The range of $p$ and $q$ in Theorem 1.2 has no overlap with that in ^[7], and to the best of our knowledge, has not been studied in other papers.

To prove Theorem 1.2, we will employ the following Gauss curvature flow,

where $X(\cdot, t):{\mathbb S}^n\to {\mathbb R}^{n+1}$ is a parametrisation of the evolving convex hypersurface ${\mathcal M}_t$, $\xi = X/|X|$, $\nu$ and $K$ are respectively the unit outward normal and the Gauss curvature of ${\mathcal M}_t$. The multiplier $\eta(t)$ is given by

and as before, $u(x, t)$ and $r(\xi, t)$ are the support and radial functions of ${\mathcal M}_t$.

Denote by ${\mathcal K}_o$ the set of convex bodies $\Omega\subset {\mathbb R}^{n+1}$ containing the origin in its interior. We will show that (1.6) is a gradient flow to the following functional:

where ${\Omega}\in {\mathcal K}_o$, $u$ and $r$ are the support function and the radial function of $\Omega$, respectively. If $\Omega\in {\mathcal K}_o$ is a critical point of the functional (1.8), then the support function of $\Omega$ satisfies Eq (1.4).

For the Gauss curvature flow (1.6), a main issue is the lack of uniform estimates, namely the control of the eccentricity of ${\Omega}$ (i.e., the eccentricity of the minimum ellipsoid of ${\Omega}$). Our strategy is to use a topological method to find a special initial condition such that the evolving hypersurfaces ${\mathcal M}_t = {\partial}{\Omega}_t$ satisfy

for some positive constants $R\ge r > 0$ independent of $t$. Once the solution satisfies such a $C^0$-estimates, one can establish higher order a priori estimates for the flow (1.6). Hence by the monotonicity of the functional (1.8), the flow converges to a solution of (1.4).

To find the special initial hypersurface, we assume that $q > n$ and (1.5) such that the functional ${\mathcal J}_{p, q}(\Omega)$ will become very large if either the volume of ${\Omega}$ is sufficiently large or small, or the eccentricity of ${\Omega}$ is sufficiently large. This property enables us to find the special initial hypersurface by using the topological method and the variational structure of the equation as in ^[13], where we proved the existence of solutions to the $L_p$-Minkowski problem in the super-critical case (namely when $q = n+1$ and $p < -n-1$). Although the approach is similar to that in ^[13], equation (1.4) and the associated flow (1.6) are more complicated than the corresponding ones in ^[13]. Therefore, we need to present sufficient details of the argument for the approach.

We will consider in Section 2 the a priori estimates for the flow (1.6). In Section 3, we combine the a priori estimates and the topological method to find a special initial condition such that the flow converges to a solution of (1.4), and thus prove Theorem 1.2. Section 4 contains further remarks on some variants of Theorem 1.2.

2.   A priori estimates for the Gauss curvature flow (1.6)

For a closed convex hypersurface ${\mathcal M}\subset {\mathbb R}^{n+1}$, the support function of ${\mathcal M}$ is given by

where $\nu_{\mathcal M}:{\mathcal M}\to {\mathbb S}^n$ is the Gauss map and $\nu^{-1}_{\mathcal M}$ is its inverse. It is well known that $\nu^{-1}_{\mathcal M}(x) = u(x)x+{\nabla} u(x)$, and the Gauss curvature of ${\mathcal M}$ at $\nu_{{\mathcal M}}^{-1}(x)$ is given by

where $u_{ij}: = {\nabla}^2_{ij}u$. Assume that ${\rm{Cl}}({\mathcal M})\in\mathcal K_o$, where ${\rm{Cl}}({\mathcal M})$ denotes the convex body enclosed by ${\mathcal M}$. Recall that the radial function of ${\mathcal M}$, denoted by $r$, is given by

Denote ${\vec{\hskip1pt r}}(\xi) = r(\xi)\xi$. We also define the radial Gauss mapping by

It is easy to verify that

For any convex body $\Omega\in {\mathcal K}_o$, its polar dual $\Omega^*$ is given by

Let ${\mathcal M}_t$ be a solution to the flow (1.6) and $X(\cdot, t)$ be its parametrisation. Consider the new parametrisation

It is straightforward to compute

Since the first term on the right hand side is tangential, taking inner product with the unit outer normal of ${\mathcal M}_t$ gives that

Hence by (2.1), the flow (1.6) can be expressed as

Theorem 2.1. Suppose that both $f$ and $g$ are positive and $C^{1, 1}$-smooth. Let $u(\cdot, t)$ be a positive, smooth and uniformly convex solution to (2.3), $t\in[0, T)$. Assume that

for all $(x, t)\in{\mathbb S}^n\times[0, T)$. Then

where $C$ is a positive constant depending only on $n, p, q, C_0$, $\min_{{\mathbb S}^n}f$, $\min_{{\mathbb S}^n}g$, $\| f \|_{C^{1, 1}({\mathbb S}^n)}$, $\| g \|_{C^{1, 1}({\mathbb S}^n)}$, and the initial condition $u(\cdot, 0)$, but is independent of $T$.

Proof. We first observe that, by the convexity

It also yields the bound of $\eta(t)$ defined by (1.7):

where $C_1$ depends only on $n, p, q, \min_{{\mathbb S}^n}f, \min_{{\mathbb S}^n}g, \max_{{\mathbb S}^n}f, \max_{{\mathbb S}^n}g$ and $C_0$.

Recently in ^[14], we studied the centro-affine Minkowski problem. We established in ^[14] the a priori estimates for a more general equation of the form

so that when

the Eq (2.8) becomes (2.3).

By virtue of Lemma 6.1 and Lemma 6.2 in ^[14], and using (2.6), (2.7) and (2.8), we conclude (2.5) as desired and hence complete the proof.

By the second derivative estimates (2.5), Eq (2.3) becomes uniformly parabolic. Hence, by Krylov's regularity theory, we have the following $C^{3, \alpha}$ estimate,

for any given $\alpha\in (0, 1)$, where the constant $C$ depends only on $n, p, q, C_0$, $\min_{{\mathbb S}^n}f$, $\min_{{\mathbb S}^{n}}g$, $\| f \|_{C^{1, 1}({\mathbb S}^n)}$, $\| g \|_{C^{1, 1}({\mathbb S}^{n})}$, and the initial condition $u(\cdot, 0)$. By the a priori estimates (2.9), we have the longtime existence of solutions to the flow (1.6), provided that $u$ satisfies (2.4).

Theorem 2.2. Assume the conditions in Theorem 2.1. Let $T_{\max}$ be the maximal time such that the solution $u(\cdot, t)$ exists on $[0, T_{\max})$. Then $T_{\max} = \infty$ and $u$ satisfies the estimates(2.5) and (2.9).

3.   Proof of Theorem 1.2

In this section, we use a topological method to select an initial hypersurface ${\mathcal N}_0$, such that the flow (1.6) deforms ${\mathcal N}_0$ to a solution of (1.4).

3.1.   Monotonicity of the functional (1.8)

We first prove the monotonicity of ${\mathcal J} : = {\mathcal J}_{p, q}$ under the flow (1.6). Recall that

Lemma 3.1. Suppose ${\mathcal M}_t$, $t\in [0, T)$, is a solution to the flow (1.6) in ${\mathcal K}_o$. Then

where ${\Omega}_t = {\mathit{\rm{Cl}}}({\mathcal M}_t)$. Equality holds if and only if the support function of ${\mathcal M}_t$ satisfies

where $\eta$ is given in (1.7).

Proof. The following formulas can be found in ^[20]:

where $\text{Jac}\mathscr A$ is the Jacobian of the radial Gauss mapping.

By virtue of (2.1), (2.3) and (3.2), we obtain

Clearly, equality $\frac{d}{dt}{\mathcal J}(\Omega_t) = 0$ holds if and only if $u(\cdot, t)$ satisfies (3.1).

The proof of Lemma 3.1 also verifies that (1.4) is the Euler-Lagrange equation of the functional (1.8) up to a constant. Note that once we have a solution to (3.1), by a proper rescaling, we can obtain a solution to (1.4).

In the following, we aim to find an initial condition $u(\cdot, 0)$ such that (2.4) is satisfied.

3.2.   An estimate for the functional (1.8)

For any convex body ${\Omega}$ in ${\mathbb R}^{n+1}$, let $E({\Omega})$ denote John's minimum ellipsoid of ${\Omega}$. We have

Let $a_1({\Omega})\le a_2({\Omega})\le\cdots\le a_{n+1}({\Omega})$ be the lengths of semi-axes of $E({\Omega})$. Denote $e_{\mathcal M} = e_{\Omega} = \frac{a_{n+1}({\Omega})}{a_1({\Omega})}$ the eccentricity of ${\mathcal M}: = {\partial} \Omega$ (or the eccentricity of ${\Omega}$).

In this subsection, we will show Proposition 3.2 by assuming that Lemma 3.3 and Lemma 3.4 hold. The proofs of Lemma 3.3 and Lemma 3.4 will be presented after the proof of Proposition 3.2.

Proposition 3.2. Suppose that $q > n$ and $p$ satisfies (1.5). Suppose that $f, g$ satisfy $c_0^{-1}\leq f, g\leq c_0$ for some constant $c_0 > 1$. For any given constant $A > {\mathcal J}(B_1(0))$, if one of the quantities $e_{\Omega}$, ${\rm{Vol}}({\Omega})$, $[{\rm{Vol}}({\Omega})]^{-1}$, and $[\operatorname{dist}(O, {\partial}{\Omega})]^{-1}$ is sufficiently large, then ${\mathcal J}(\Omega)\geq A$.

Proof. We divide the proof into three steps.

Step 1: If $e_{\Omega}\ge e$ for a large constant $e > 1$, we have ${\mathcal J}(\Omega) > A$.

By Lemmas 3.3 and 3.4, we have

From (3.3) and (3.4), there exists a constant $C > 0$ depending only on $n, p, q, c_0$ such that

That is

Condition (1.5) yields that

To see this, if $q\geq n+1$, then $|p| > \frac{nq}{q-1}$ and

While if $n < q < n+1$, then $|p| > \frac{q}{q-n}$ and

Note that $a_1\ge d$. If $e_{\Omega}$ is large, there is a $j$ such that $a_{j+1}/a_j$ is large. We see from (3.5) that ${\mathcal J}({\Omega})\ge A$.

Step 2: If either ${\rm{Vol}}(\Omega)\leq v_0$ or ${\rm{Vol}}(\Omega)\geq v_0^{-1}$, for a small constant $v_0 > 0$, then ${\mathcal J}(\Omega) > A$.

We have

and

If $a_1/d$ is large, then (3.5) implies that ${\mathcal J}({\Omega})\ge A$. Therefore we may assume $a_1 \le C d$ for some $C$, and so $a_1/a_{n+1} \le Cd/a_{n+1}$. Hence if either $d/a_{n+1}$ or $a_1/a_{n+1}$ is sufficiently close to $0$, then (3.5) again shows that ${\mathcal J}({\Omega})\ge A$. Hence we assume that $d/a_{n+1}$ or $a_1/a_{n+1}$ are away from $0$. By (3.6) and (3.7), if either ${\rm{Vol}}(\Omega)$ or $[{\rm{Vol}}(\Omega)]^{-1}$ is large, then ${\mathcal J}({\Omega})\ge A$.

Step 3: If $\text{dist}(O, {\partial} \Omega)\le d_0$ for a sufficiently small $d_0 > 0$, then ${\mathcal J}(\Omega) > A$.

In this case, we may assume that $a_j\le Cd$ for all $1\le j\le n+1$ for some $C\ge1$. As discussed in Step 2, $a_1\le Cd$, otherwise we are done. If $a_j/d$ is sufficiently large for some $j$, then $e_{\Omega}$ is also huge as $e_{\Omega} \ge a_j/a_1 \ge a_j/(Cd)$. Step 1 shows that ${\mathcal J}({\Omega})\ge A$.

Under the above assumption, if $d$ is sufficiently small, then ${\rm{Vol}}({\Omega})$ becomes very small. By Step 2, we have ${\mathcal J}({\Omega}) > A$.

Lemma 3.3. Let $\Omega\in {\mathcal K}_o$. Suppose $q > 0$. There exists a constant $C > 0$ depending only on $n$ and $q$ such that

Here, $r$ is the radial function of $\Omega$, and $a_1\leq \cdots\leq a_{n+1}$ are the lengths of semi-axes of $E(\Omega)$.

Proof. By a proper rotation of coordinates, we assume that $E = E({\Omega})$ is given by

where $\zeta_E = (\zeta_1, \cdots, \zeta_{n+1})$ is the center of $E$. We can further assume that $\zeta_{n+1}\geq 0$.

Since $\frac{1}{n+1}E\subset \Omega$, we have

Hence, there exists a point $p_0\in \Omega$ such that

Consider the hyperplane $L$ which is orthogonal to ${\bf e}_{n+1}$ and passes through $\zeta_E$:

Let $P = L\cap \frac{1}{n+1}E$ be the intersection of $L$ with the ellipsoid $\frac{1}{n+1}E$, and $V$ be the cone in $\mathbb{R}^{n+1}$ with base $P$ and the vertex $p_0$. Clearly $V\subset \Omega$.

Case 1: $q > n+1$. Let us consider the following subset of $V$:

This together with (3.8) implies that

Using $V'\subset \Omega$, $q > n+1$ and $(3.9)$, we have

where $C$ is a constant depending only on $n$ and $q$.

It is easy to see that

Combining (3.10) and (3.11), we conclude that

Case 2: $0 < q\leq n+1$. Since $\Omega$ contains the origin and $\Omega\subset E$, we have

Using $\frac{1}{n+1}E\subset \Omega$, $n < q\leq n+1$ and (3.12), we derive that

This completes the proof.

Lemma 3.4. Let $\Omega\in {\mathcal K}_o$. Suppose $p < 0$. There exists a constant $C > 0$ depending only on $n$ and $p$ such that

Here, $u$ is the support function of $\Omega$, $d = \operatorname{dist}(O, {\partial}{\Omega})$ and $a_1\leq \cdots\leq a_{n+1}$ are the lengths of semi-axes of $E(\Omega)$.

Proof. By a proper rotation of coordinates, we assume that $E = E({\Omega})$ is given by

where $\zeta_E = (\zeta_1, \cdots, \zeta_{n+1})$ is the center of $E$. Let $x_0\in{\mathbb S}^n$ be a point such that $u(x_0) = \min_{{\mathbb S}^n}u = d$, where $u$ is the support function of $\Omega$ and $d = \operatorname{dist}(O, {\partial}{\Omega})$. We choose $i_{\#}$ and switch ${{\bf e}}_{i_\#}$ and $-{{\bf e}}_{i_\#}$ if necessary such that

This implies that $x_0\cdot {{\bf e}}_{i_\#}\geq c_n$. We use $c_n$ to denote a constant which depends only on $n$ but may change from line to line.

Let $w(x) = u(x)+u(-x)$, $x\in{\mathbb S}^n$, be the width of ${\Omega}$ in $x$. Since $\frac{1}{n+1}E({\Omega})\subset \Omega\subset E(\Omega)$, we have

and

By switching ${{\bf e}}_i$ and $-{{\bf e}}_i$ if necessary, we assume that $u({{\bf e}}_i)\leq c_na_i$ for all $i = 1, \cdots, n+1$.

Consider the cone $\mathcal V$ in ${\mathbb R}^{n+1}$ with the vertex $p_0 = r^*(x_0)x_0$ and the base

Here, $r^*$ is the radial function of the polar dual $\Omega^*$ of $\Omega$:

Let $\mathcal V'$ be a subset of $\mathcal{V}$:

Recall that $r^* = 1/u$. So $r^*(x_0) = \frac 1d$. Since $x_0\cdot {{\bf e}}_{i_\#}\geq c_n$, we see that

The second inequality above follows by $\mathcal V' \subset {\Omega}^*\subset B_{1/d}(0)$ (as $B_d(0)\subset{\Omega}$). Therefore

Since $r^*({{\bf e}}_k) = \frac{1}{u({{\bf e}}_k)}\ge \frac{c_n}{a_k}$ for all $k\geq 1$, we obtain

Plugging this into (3.13), we obtain

Remark 3.5. Let ${\mathcal M}_t$, $t\in[0, T_{\max})$, be a solution to (1.6). By Proposition 3.2, if ${\mathcal J}({\mathcal M}_t) < A$ for a constant $A$ independent of $t$, then there exist positive constants $e_0, v_0, d_0$ depending on $A$, but independent of $t$, such that

where ${\Omega}_t$ is the convex body enclosed by ${\mathcal M}_t$. Note that (3.14) implies (2.4). Hence, the a priori estimates (2.5) and (2.9) hold, and one has the long-time existence of solution (Theorem 2.2). Therefore, all we need is to establish the condition ${\mathcal J}({\mathcal M}_t) < A$ for some constant $A$.

3.3.   A modified flow of (1.6)

We introduce a modified flow (as in ^[13]) such that for any initial condition, the solution exists for all time $t\ge 0$. It is more convenient to work with a flow which exists for all $t\ge 0$.

Let us fix a constant

If the minimum ellipsoid of ${\Omega}$ is $B_1(0)$, then $\frac{1}{n+1}B_1(0)\subset{\Omega}\subset B_1(0)$ and hence

For a closed, smooth and uniformly convex hypersurface ${\mathcal N}$ such that ${\Omega}_0 = {\rm{Cl}}({\mathcal N})\in {\mathcal K}_o$, we define $\bar{{\mathcal M}}_{\mathcal N}(t)$ as follows:

${\rm{a}}):$ If $\mathcal{J}({\mathcal M}_{\mathcal N}(t)) < A_0$ for all time $t\ge 0$, let $\bar{{\mathcal M}}_{\mathcal N}(t) = {\mathcal M}_{\mathcal N}(t)$ for all $t\ge 0$, where ${\mathcal M}_{\mathcal N}(t)$ is the solution to (1.6).

${\rm{b}}):$ If ${\mathcal J}({\mathcal N}) < A_0$, and $\mathcal{J}({\mathcal M}_{\mathcal N}(t))$ reaches $A_0$ at the first time $t_0 > 0$, we define

$c):$ If ${\mathcal J}({\mathcal N})\geq A_0$, we let $\bar{{\mathcal M}}_{\mathcal N}(t)\equiv {\mathcal N}$ for all $t\geq 0$.

For convenience, we call $\bar{{\mathcal M}}_{\mathcal N}$ a modified flow of (1.6). By the a priori estimates in Section 2, $\bar{{\mathcal M}}_{\mathcal N}(t)$ is smooth for any fixed time $t$, and Lipschitz continuous in time $t$. Moreover, we have the following properties.

${\rm{i}})$ $\bar{{\mathcal M}}_{\mathcal N}(t)$ is defined for all time $t\ge 0$, and by Lemma 3.1, ${\mathcal J}(\bar{{\mathcal M}}_{\mathcal N}(t))$ is non-decreasing. In particular, we have ${\mathcal J}(\bar{{\mathcal M}}_{\mathcal N}(t)) \le \max \{A_0, {\mathcal J}({\mathcal N})\}$ $\forall\ t\ge 0$.

${\rm{ii}})$ If either $\operatorname{dist}(O, {\mathcal N})$ is very small, or ${\rm{Vol}}(\Omega_0)$ is sufficiently large or small, or $e_{{\Omega}_0}$ is sufficiently large, by Proposition 3.2, we have $\bar{{\mathcal M}}_{\mathcal N}(t)\equiv {\mathcal N}$ $\forall \ t\ge 0$.

3.4.   Homology of a class of ellipsoids

Here we recall the homology of a class of ellipsoids $\mathcal A_I$ introduced in ^[13], such that an ellipsoid $E$ with ${\mathcal J}(E) < A_0$ is contained in $\mathcal A_I$. By Proposition 3.2, we have

Corollary 3.6. For the constant $A_0$ given by (3.15), there exist sufficiently small constants $\bar d$ and $\bar v$, and sufficiently large constant $\bar e$, such that for any ${\Omega}\in \mathcal K_o$,

$({\rm{i}})$ if ${\text{dist}}(O, {\partial} \Omega)\leq \bar{d}$, then ${\mathcal J}({\Omega}) > A_0$;

$({\rm{ii}})$ if $e_{\Omega}\ge \bar e$, then ${\mathcal J}({\Omega}) > A_0$;

$({\rm{iii}})$ if ${\rm{Vol}}({\Omega})\le \bar v$ or ${\rm{Vol}}({\Omega})\ge 1/(n+1)^{n+1}\bar v^{-1}$, then ${\mathcal J}({\Omega}) > A_0$.

Let ${\mathcal K}$ be the metric space consisting of non-empty, compact and convex sets in ${\mathbb R}^{n+1}$, equipped with the Hausdorff distance. Denote by $\bar{{\mathcal K}}_o$ the closure of ${\mathcal K}_o$ in ${\mathcal K}$.

Fix the constants $\bar d, \bar v, \bar e$ in Corollary 3.6. Let $\mathcal A_I$ be the set of ellipsoids $E\in \bar{{\mathcal K}}_o$ such that ${{\bar v}}\le {\rm{Vol}}(E)\le {{1/\bar v}}, \ \text{and}\ e_E\le \bar e$. Denote by ${{\mathcal A}}$ the following subset of $\mathcal A_I$

Here, $\omega_n = |B_1(0)|$ is the volume of $B_1(0)$, and $e_E$ is the eccentricity of $E$.

We also denote by $\mathcal{E}_I$ the set of ellipsoids in $\mathcal A_I$ centred at the origin, and by $\mathcal{E}$ the set of ellipsoids in $\mathcal A$ centred at the origin. These sets are all metric spaces by equipping the Hausdorff distance.

It was proved in ^[13] that $\mathcal E_I$ is contractible and so the homology $H_k(\mathcal E_I) = 0$ for all $k\ge1$. Moreover, $\mathcal A_I$ is homeomorphic to $\mathcal E_I\times B_1(0)$. Hence, $\mathcal A_I$ is contractible and the homology

Denote

It is the boundary of $\mathcal A_I$ if we regard $\mathcal A_I$ as a set in the topological space of all ellipsoids. Moreover, there is a retraction $\Psi$ from $\mathcal A_I\setminus \{B_1\}$ to $\mathcal P$. Namely, $\Psi: \mathcal A_I\setminus \{B_1\}\to \mathcal P$ is continuous and $\Psi|_{ \mathcal P} = id$. The following two theorems were also proved in ^[13].

Proposition 3.7. We have the following results.

$({\rm{i}})$ $H_{k+1}(\mathcal P) = H_k({{\mathcal A}})$ for all $k\ge1$.

$({\rm{ii}})$ There is a long exact sequence

Proposition 3.8. Let $n_* = \frac{n(n+1)}{2}$. The homology group $H_{n^*+n-1}(\mathcal E) = \mathbb Z$.

3.5.   Selection of a good initial condition

In this subsection, we use Propositions 3.7 and 3.8 to select a special initial condition in $\mathcal A_I$ such that the solution to the Gauss curvature flow (1.6) satisfies the uniform estimate. The idea is similar to that in ^[13].

For any ellipsoid ${\mathcal N}$ such that ${\rm{Cl}}({\mathcal N})\in \mathcal A_I$, let $\bar{{\mathcal M}}_{\mathcal N}(t)$ be the solution to the modified flow. We have the following properties:

$1)$ If ${\rm{Cl}}({\mathcal N})$ is close to $\mathcal P$ in Hausdorff distance or in $\mathcal P$, we have $\mathcal{J}({\mathcal N})\ge A_0$ and so $\bar{{\mathcal M}}_{\mathcal N}(t)\equiv {\mathcal N}$ for all $t$ (see Corollary 3.6).

$2)$ If ${\rm{Cl}}({\mathcal N})$ is close to $B_1(0)$ in Hausdorff distance, then ${\mathcal J}({\mathcal N}) < A_0$.

$3)$ By our definition of the modified flow, ${\mathcal J} (\bar{{\mathcal M}}_{\mathcal N}(t)) < \max\{A_0, {\mathcal J}({\mathcal N})\}$ for all $t$. Hence by Remark 3.5, if $\bar{{\mathcal M}}_{\mathcal N}(t)$ is not identical to $\bar{{\mathcal M}}_{\mathcal N}(0) = {\mathcal N}$, then

With these properties, we can prove the following key lemma.

Lemma 3.9. For every $t > 0$, there exists ${\mathcal N} = {\mathcal N}_t$ with ${\mathit{\rm{Cl}}}({\mathcal N})\in \mathcal{A}_I$, such that the minimum ellipsoid of $\bar{{\mathcal M}}_{{\mathcal N}}(t)$ is the unit ball $B_1(0)$.

Proof. Suppose by contradiction that there exists $t' > 0$ such that, for any ${\Omega}\in\mathcal A_I$, $E_{\mathcal N}(t')\ne B_1(0)$, where ${\mathcal N} = {\partial}{\Omega}$ and $E_{\mathcal N}(t')$ is the minimum ellipsoid of ${\Omega}_{{\mathcal N}}(t'): = {\rm{Cl}}(\bar{{\mathcal M}}_{\mathcal N}(t'))$.

By Corollary 3.6, $E_{\mathcal N}(t')\in\mathcal A_I$. Hence we can define a continuous map $T: \mathcal A_I\to \mathcal P$ by

where $\Psi$ is the retraction after (3.18), and $B_1 = B_1(0)$ for short. Note that when $\Omega\in \mathcal P$, we have ${\mathcal J}(\Omega)\ge A_0$ and thus $E_{\mathcal N}(t') = E_{\mathcal N}(0) = \Omega$. This implies that $T|_{\mathcal P} = id_{\mathcal P}$. Hence, $T$ is a retraction from $\mathcal A_I$ to $\mathcal P$, and so there is an injection from $H_*(\mathcal P)$ to $H_*(\mathcal A_I)$. By (3.17), we then have

It follows from Proposition 3.7 (ⅱ) that

Computing the left-hand side by the Künneth formula and using the fact $H_k({\mathbb S}^n) = \mathbb Z$ if $k = 0$ or $k = n$, and $H_k({\mathbb S}^n) = 0$ otherwise, we further obtain

However, this contradicts Proposition 3.8 by taking $k = n^*+2n-1$ in the above.

In the following we prove the convergence of the flow (1.6) with a specially chosen initial condition. Take a sequence $t_k\to\infty$ and let ${\mathcal N}_k = {\mathcal N}_{t_k}$ be the initial data from Lemma 3.9. By our choice of $A_0$ (see (3.15) and (3.16)), Lemma 3.9 implies that

Hence, by the monotonicity of the functional ${\mathcal J}$, we have

Since ${\rm{Cl}}({\mathcal N}_k)\in\mathcal A_I$ and $B_{\bar d}(0) \subset {\rm{Cl}}({\mathcal N}_k)$, by Blaschke's selection theorem, there is a subsequence of ${\mathcal N}_k$ which converges in Hausdorff distance to a limit ${\mathcal N}_*$ such that ${\rm{Cl}}({\mathcal N}_*)\in\mathcal A_I$ and $B_{\bar d}\subset {\rm{Cl}}({\mathcal N}_*)$.

Next, we show that the flow (1.6) starting from ${\mathcal N}_*$ satisfying ${\mathcal J}({\mathcal M}_{{\mathcal N}_*}(t)) < A_0$ for all $t$.

Lemma 3.10. For any $t\geq 0$, we have

Hence

Proof. For any given $t > 0$, since ${\mathcal N}_k\to {\mathcal N}_*$ and $t_k\to\infty$, when $k$ is sufficiently large such that $t_k > t$, we have

By the monotonicity of the functional ${\mathcal J}$,

Combining above two inequalities with (3.20), we obtain that

This completes the proof.

3.6.   Convergence of the flow and existence of solutions to (1.4)

Let ${\Omega}_{{\mathcal N}_*}(t) = {\rm{Cl}}({\mathcal M}_{{\mathcal N}_*}(t))$ and $u(\cdot, t)$ be its support function. By Lemma 3.10, ${{\mathcal M}}_{{\mathcal N}_*}(t)$ satisfies (3.19). Hence,

where $C = (n+1)/(\bar v\omega_{n-1}\bar d^n)$. Hence, condition (2.4) holds, and we obtain the existence of solutions to (1.4) as follows.

Proof of Theorem 1.2. Denote ${\mathcal M}(t) = {\mathcal M}_{{\mathcal N}_*}(t)$ and $\mathcal{J}(t) = \mathcal{J}({\mathcal M}(t))$. By Lemma 3.1 and Lemma 3.10,

Therefore,

This implies that there exists a sequence $t_i\to \infty$ such that

Passing to a subsequence, we obtain by the a priori estimates (2.9) that $u(\cdot, t_i)\to u_\infty$ in $C^{3, \alpha}({\mathbb S}^{n})$-topology and $u_\infty$ satisfies

where $\lambda = \lim_{t_i\to\infty}\eta(t_i)$. As $p\ne q$, it can be seen that $u_\infty^\lambda = \lambda^{\frac{1}{p-q}} u_\infty$ satisfies (1.4).

4.   Further remarks

This section is devoted to some variants of Theorem 1.2. We show that when $g$ is a positive function defined on ${\mathbb R}^{n+1}$ instead of ${\mathbb S}^n$, our argument still works. We first show Eq (4.1) below admits a solution $u$ up to some multiplier $\lambda > 0$ when $p$ and $q$ are in the same range as in Theorem 1.2. Since now $g$ is a function on ${\mathbb R}^{n+1}$, it does not imply that the equation with $\lambda = 1$ has a solution by a scaling argument as mentioned in Section 3.6.

Theorem 4.1. Let $f\in C^{1, 1}({\mathbb S}^n)$ and $g\in C^{1, 1}({\mathbb R}^{n+1})$ be two positive functions satisfying $c_0^{-1}\le f, g\le c_0$ for some $c_0 > 1$. Suppose $q > n$ and $p$ satisfies (1.5). Then there is a constant $\lambda > 0$, and a uniformly convex and $C^{3, \alpha}$ function $u$, $\alpha\in (0, 1)$, such that

Proof. Similarly to (1.6), we study the flow

where

The same computation as in Lemma 3.1 implies that the functional

is non-decreasing along (4.2).

Since $g$ is bounded and positive, we have

Therefore, Proposition 3.2 is also valid for the functional ${\mathcal J}^{(1)}$.

As a result, we can define the modified flow for (4.2) as in Section 3.3 with $A_0 = 10c_0|{\mathbb S}^n|+ 10(n+1)\|f\|_{L^{1}({\mathbb S}^n)}$. By using the topological argument as in Section 3.5, we can show the existence of an initial hypersurface ${\mathcal M}_0$ with ${\rm{Cl}}({\mathcal M}_0)\in{\mathcal K}_o$ such that the evolving hypersurface ${\mathcal M}_t = \partial \Omega_t$ of the flow (4.2) satisfies

for some constants $R > r > 0$ which are independent of time $t$. Equivalently, the support function of ${\mathcal M}_t$, which satisfies the parabolic equation

enjoys the $C^0$-estimates:

Since (4.3) is of the form (2.8), Theorem 2.1 is valid for (4.3). Therefore, $u(\cdot, t)$ exists for all time $t\ge0$ and is of $C^{3, \alpha}$-smooth.

The same argument as in Section 3.6 implies that $u(\cdot, t_i)$ converges to a solution of (4.1) with $\lambda = \lim_{t_i\to\infty}\eta(t_i)$.

When $q\ge n+1$ and $p < -q$, we can show the existence of solutions to (4.1) with $\lambda = 1$.

Theorem 4.2. Let $f\in C^{1, 1}({\mathbb S}^n)$ and $g\in C^{1, 1}({\mathbb R}^{n+1})$ be two positive functions satisfying $c_0^{-1}\le f, g\le c_0$ for some $c_0 > 1$. Suppose $q\ge n+1$ and $p < -q$. Then there is a uniformly convex and $C^{3, \alpha}$ function to (4.1) with $\lambda = 1$, where $\alpha\in (0, 1)$.

Proof. Consider the functional

and the flow

Similar calculations as in Lemma 3.1 show that ${\mathcal J}^{(2)}$ is non-decreasing under (4.4).

One can verify that Proposition 3.2 holds for the functional ${\mathcal J}^{(2)}$.

Step 1: if $d = \operatorname{dist}(O, {\partial}{\Omega})$ is sufficiently small, then ${\mathcal J}^{(2)}({\Omega})$ is sufficiently large. We adopt the same notations as in Proposition 3.2. By virtue of (3.3) and (3.4), we obtain

where $C > 0$ depends only on $n, p, q$ and $c_0$. As $q\ge n+1$ and $a_1\geq d$, we have

This shows that if $d$ is tiny, then ${\mathcal J}^{(2)}({\Omega})$ is huge.

Step 2: if ${\rm{Vol}}({\Omega})$ or $[{\rm{Vol}}({\Omega})]^{-1}$ is sufficiently small, then ${\mathcal J}^{(2)}({\Omega})$ is sufficiently large. Since ${\rm{Vol}}({\Omega})$ being small implies that $d$ is tiny, we are done by Step 1 in this case. Suppose that ${\rm{Vol}}({\Omega})$ is huge. The conclusion then follows from

Step 3: if $e_{\Omega}$ is sufficiently large, then so is ${\mathcal J}^{(2)}({\Omega})$. By Step 1, we assume without loss of generality that $d$ is bounded from below by a constant $C > 0$ depending on $n, p, q$ and $c_0$. Using (4.5),

This proves Step 3.

To complete the proof, we introduce a modified flow of (4.4) and use the topological argument as in Section 3 to find the needed initial hypersurface ${\mathcal N}$, such that the evolving hypersurfaces ${\mathcal M}_t$ with ${\mathcal M}_0 = {\mathcal N}$ satisfy $B_r(0)\subset{\Omega}_t\subset B_{R}(0)$ for some constants $R > r > 0$. Since the support function $u(\cdot, t)$ of (4.4) satisfies a parabolic equation of the form (2.8), the higher order derivative estimates follow by Theorem 2.1. The remaining proof follows exactly as that of Theorem 1.2. Since (4.4) does not contain an integral term like $\eta(t)$ in (4.2), along a sequence of times $\{t_i\}_{i = 1}^\infty$, $u(\cdot, t_i)$ converges to a solution of (4.1) with $\lambda = 1$.

Acknowledgments

The first author was supported by ARC DE210100535. The second author was supported by NSFC12031017 and the Fundamental Research Funds for the Central Universities (No. 226-2022- 00157). The third author was supported by ARC DP200101084.

Conflict of interest

The authors declare no conflict of interest.