Gradient estimates for the solutions of higher order curvature equations with prescribed contact angle

1.   Introduction

Let $\Omega$ be a bounded $C^3$ domain in $\mathbb{R}^n$ and $u\in C^{3}(\overline{\Omega})$. In this paper, we will establish a priori gradient estimates for solutions of the prescribed $k$-curvature equation with the prescribed contact angle boundary value

where $\kappa = (\kappa_1, \cdots, \kappa_n)$ are the principal curvatures of the graph $\mathcal{M} = \{(x, u(x))\in\mathbb{R}^{ n+1}| x\in\Omega\}$, $n \geq2$, $f$ is a smooth, positive function in $\Omega$ and $\phi$ is a smooth function on $\overline{\Omega}$ such that $-1 < \phi < 1$. And for any $k = 1, 2, \cdots, n$,

the $k$-order fundamental symmetric function of $\lambda\in \mathbb{R}^n$. For $k = 1$, $n$, the (1.1) is the mean curvature and Gaussian curvature equation respectively.

The gradient estimate for the prescribed mean curvature equation has been extensively studied. The interior gradient estimate, for the minimal surface equation, was obtained in the case of two variables by Finn ^[2]. Bombieri-De Giorgi-Miranda ^[1] obtained the estimate for high dimensional cases. For the general mean curvature equation, such an estimate had also been obtained by Ladyzhenskaya and Ural'tseva ^[10], Trudinger ^[17] and Simon ^[13]. All their methods were used by test function argument and a resulting Sobolev inequality. A more detailed history could be found in Gilbarg and Trudinger ^[3]. In 1983, Korevaar ^[5] introduced the normal variation technique and got the maximum principle proof for the interior gradient estimate on the minimal surface equation, then in 1987 Korevaar ^[6] got the interior gradient estimates for the higher order curvature equations. Trudinger ^[18] also studied the curvature equations and got the interior gradient estimates for a class curvature equation. In 1998, Wang ^[19] gave new proof for the interior gradient estimates on the general $k$-curvature equation via the standard Bernstein technique. In 2012, Sheng-Trudinger-Wang ^[16] also gave a new proof for the general Weingarten curvatures equation by the moving frame on the hypersurface.

For the mean curvature equation with the Neumann boundary value problem, Ma and Xu ^[11] used the technique developed by Spruck ^[14], Lieberman ^[7], Wang ^[19] and Jin-Li-Li ^[4] to get the global gradient estimates. As a consequence, they obtained an existence theorem for a class of mean curvature equations with the Neumann boundary value. For a fully nonlinear elliptic equation with Neumann boundary value or oblique derivative problem, we recommend Lieberman ^[8] to readers. Recently, Ma and Wang ^[12] used the technique developed by Sheng-Trudinger-Wang ^[16] to give a simpler new proof of the gradient estimates for the mean curvature equation with Neumann boundary value and prescribed contact angle boundary value. In this paper, we use the same technique to get the global gradient estimates for the $k$-curvature equation with the prescribed contact angle boundary value. Precisely, we have the following theorem.

Theorem 1.1. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with a $C^3$ boundary, $n\geq2$, $\nu$ is the unit inner normal to $\partial\Omega$. Suppose $f\in C^1(\overline{\Omega}\times[-M_0, M_0])$ satisfies that $f_z\geq0$ with $\inf_{\Omega}f\geq f_0 > 0$, and $\phi\in C^3(\overline{\Omega})$ and $-1 < \phi < 1$. If $u\in C^2(\overline{\Omega})\cap C^3(\Omega)$ is a bounded $k$-admissible solution of the $k$-curvature equation (1.1), then we have

where $C$ is a positive constant depending on $n$, $k, f_0$, $\Omega$, $|f|_{C^1(\overline{\Omega}\times[-M_0, M_0])}$, $|\phi|_{C^3(\overline{\Omega})}$, and $M_0 = |u|_{C^0(\overline{\Omega})}$.

Remark 1.2. We define the Garding's cone as $\Gamma_k = \{\lambda\in\mathbb{R}^n|\sigma_i(\lambda) > 0, \ 1\leq i\leq k\}$. Then we say a function $u$ is $k$-admissible if $\lambda(D^2u)\in\Gamma_k$, where $\lambda(D^2u) = (\lambda_1, \cdots, \lambda_n)$ are eigenvalues of the Hessian matrix $D^2u$.

Remark 1.3. In order to prove the existence theorem for the $k$-curvature equations with the prescribed contact angle boundary value problem, we still need global estimates for second-order derivatives. In another paper, we had gotten the global gradient estimates for the $k$-curvature equation with the Neumann boundary value problem.

The rest of the paper is organized as follows. In Section 2, we first give the definitions and some notations. We also give some basic properties of the fundamental symmetric functions. In Section 3, we prove the main Theorem 1.1 by the moving frame on the hypersurface.

2.   Preliminary

$A$, $B$, $\cdots$ will be from $1$ to $n + 1$ and $i$, $j$, $\alpha$, $\cdots$ from $1$ to $n$, the repeated indices denote summation over the indices.

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ and $u\in C^{\infty}(\overline{\Omega})$. Then the graph of $u$ is a hypersurface in $\mathbb{R}^{n+1}$, denoted by $\mathcal{M}$, given by the smooth embedding $X : \Omega \rightarrow \mathbb{R}^{ n+1}$,

Denote $u_i = u_{x_i}$, $u_{ij} = u_{x_ix_j}$, and $Du = (u_1, \cdots, u_n)$. Then the downward unit normal of $\mathcal{M}$ is

Let $\{\varepsilon_1, \varepsilon_2, \cdots, \varepsilon_{n+1}\}$ be the standard orthonormal basis in $\mathbb{R}^{n+1}$. We choose an orthonormal frame in $\mathbb R^{n+1}$ such that $\{e_1, e_2, ..., e_n\}$ are tangent to $\mathcal{M}$ and $e_{n+1} = N$ is the downward unit normal. Let the corresponding coframe be denoted by $\{\omega_A\}$ and the connection forms by $\{\omega_{A, B}\}$. The pullback of them through the embedding are still denoted by $\{\omega_A \}$, $\{\omega_{A, B}\}$ in the abuse of notation. Therefore on $\mathcal{M}$

The second fundamental form is defined by the symmetry matrix $\{h_{ij}\}$ with

The principal curvatures $\kappa = (\kappa_1, \kappa_2, \cdots, \kappa_n)$ are the eigenvalues of the second fundamental form $(h_{ij})$.

The first and second-order covariant derivatives will be denoted by $\nabla_i$, $\nabla_i\nabla_j$ respectively. We recall the following fundamental formulas of a hypersurface in ${\mathbb R}^{n+1}$.

We denote

It is well known that there exists a small positive universal constant $\mu_0$ such that $d(x)\in C^3(\overline{\Omega_{\mu}}), \ \forall0 < \mu\leq\mu_0$, provided $\partial\Omega\in C^3$. As in Simon-Spruck ^[15] or Lieberman ^[7] (on page 331), we can extend $\nu$ by $\nu = D d$ in $\Omega_{\mu}$ and note that $\nu$ is a $C^3(\overline{\Omega_{\mu}})$ vector field. As mentioned in the book ^[7], we also have the following formulas

Lemma 2.1. Denote $v = \sqrt{1+|Du|^2}$ and $e_A^B = \langle e_A, \varepsilon_B\rangle$ for $A, B = 1, \cdots, n+1$. We have

Proof. Note that $u = \langle X, \varepsilon_{n+1}\rangle$. Using the Gauss formula and Weingarten equation above, we obtain

Similarly, we have

It follows that, recall $u_l = v\langle e_{n+1}, \varepsilon_l\rangle = ve_{n+1}^l$,

Furthermore, we have

noting that

then, since $\nabla_jh_{ir} = \nabla_rh_{ij}$ (Codazzi equation),

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Now we give some basic properties of elementary symmetric functions, which could be found in ^[9].

First, we denote by $\sigma_{k}(\lambda|i)$ the symmetric function with $\lambda_{i} = 0$ and $\sigma_{k}(\lambda|ij)$ the symmetric function with $\lambda_{i} = \lambda_{j} = 0$.

Proposition 2.2. Let $\lambda = (\lambda_{1}, \cdots, \lambda_{n})\in \mathbb{R}^{n}$ and $k = 1, \cdots, n$, then

Recall that Garding's cone is defined as

Proposition 2.3. Let $\lambda \in\Gamma_{k}$ and $k\in\{1, 2, \cdots, n\}$. Suppose that

then we have

Then the $k$-curvature equation (1.1) is elliptic if the principal curvatures $\kappa\in\Gamma_k$.

3.   Gradient estimate for prescribed contact angle

We consider the following $k$-curvature equation with the prescribed angle condition and obtain a gradient estimate of $k$-admissible solution. We state it again in the following theorem.

Theorem 3.1. Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with $C^{3}$ boundary. $f\in C^1(\Omega\times[-M_0, M_0])$ satisfies that $f_z\geq0$. Assume $u$ is a $k$-admissible solution of the equation

where $\nu$ be the unit inner normal vector on $\partial\Omega$ and $\phi\in C^3\left(\bar \Omega, (-1, 1)\right)$. We have

Proof. Denote $v = \sqrt{1+|Du|^{2}}$ and $M_{0} = \sup_{\overline{\Omega}}|u|$. Let

The constant $\alpha_{1}$ will be determined later. Fix a small $0 < \mu\leq\mu_0$ and consider the auxiliary function

There are three cases to be considered.

Case 1. $G(x)$ attains maximum at $x_{0}\in\partial\Omega_{\mu}\cap\Omega$.

By the interior gradient estimates of Korevaar ^[6] and Wang ^[19], we have

Case 2. $G(x)$ attains maximum at $x_{0}\in\partial \Omega$.

Assume $U\subset\mathbb{R}^{n}$ be a neighborhood of $x_{0}$. We choose a geodesic coordinate $\{x_{i}\}_{i}^{n-1}$ on $U\cap\partial\Omega$ centered at $x_0$. We let $\partial_{x_n} = \nu$ at $x_0$. In the following, we take all calculations at $x_0$.

Denote $(b_{ij})$ the second fundamental form of $\partial\Omega$ with respect to $\nu$. We have

and

Denote $a = w\log w$ for simplicity. Note that

Choose $l = n$ in (3.10), then plug into (3.11) to get

By (3.8) and (3.10),

From the boundary data $u_n = \phi v$ and (3.13), we have

It follows that

Plugging (3.15) into (3.12), we get

Here we use the fact

Putting (3.17) into (3.7), we have

Thus we have $v\leq C$.

Case 3. $G(x)$ attains its maximum at $x_0\in\Omega_{\mu}$.

Direct computation shows that

and

From (2.7)–(2.10) and (3.3), we have

and

By selecting a suitable moving frame, we assume $(h_{ij})$ is diagonal at $x_0$. At the maximum point $x_0\in\Omega_{\mu}$, from (3.19) and $\nabla G = 0$, we see that

Together with (3.21), we also have

We divide the indexes $i\in I = \{1, 2, \cdots, n\}$ into two subsets as follows.

For $i\in J$, recall $|\nabla d|\leq 1$,

For $i\in K$, we have

Now we assume $v$, $w$ large enough, i.e., $v\geq\max\{\frac{\alpha_1}{\epsilon}, \exp{\frac{2}{\epsilon}}\}$, then

and

for a small positive number $\epsilon$. Then, using (3.24), we get

and

Thus, plugging (3.28) into (3.32), we have

provided we set $\epsilon = \frac{1}{100}$.

By the maximum principle, by (3.20), we have

We use the fact $\nabla_i\nabla_ju = \frac{h_{ij}}{v}$ in the last inequality and assume $v$ large.

From (3.22) we obtain

By (3.24), $f_z\geq0$, $|\langle\nabla f, \nabla u\rangle |\leq \frac{|Df|}{v}$, and the Cauchy-Schwartz inequality, we have

Multiplying $\nabla_id$ with both sides of (3.24), we have

It follows that, by the Cauchy-Schwartz inequality,

Similarly, multiplying $\nabla_iu$ with both sides of (3.24), we have

By the Cauchy-Schwartz inequality, recall $|\alpha_1\nabla_iu|\leq9, \ \text{for}\ i\in J$,

Here we point out that $C$ is independent of $\alpha_1$.

Plugging (3.37) and (3.38) into (3.36), we have

Set $\epsilon_1 = \frac{1}{3}$, $\epsilon_2 = \frac{1}{16}$, and assume $v$ is sufficiently large, then

Putting (3.40) into (3.34), we get

In view of (3.28), we see that

On the other hand, by (3.31) and (3.33),

Particularly, there is $i_0\in K$, say $i_0 = 1$, such that $|\nabla_1u|\geq\frac{1}{2\sqrt{n}}$ and $F^{11}\geq\frac{1}{n}\mathcal{F}$.

Plugging (3.42) and (3.43) into (3.41) and choosing $\alpha_1 = \max\{4\sqrt{n}, 16nC\}$, we have

Thus we have $(1-|\phi|)v\leq w\leq\exp{\frac{128n^2C}{\alpha_1^2}}$ and finish the proof.

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Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The research of the first author was supported by China Postdoctoral Science Foundation 2022M722474. The research of the second author was supported by NSFC 12141105, NSFC 11871255 and National Key Research and Development Project SQ2020YFA070080.

Conflict of interest

The authors declare no conflict of interest.