\phi\>\phi_\nu\Big)dA. \nonumber      
    \end{aligned}$$ Integrating by parts for the last line of and noting $-[\tau+2(\g-\a)]+(\g+(n-1)\a)=-2\g$, we get the assertion.

Heintz-Karcher type and Minkowski type inequalities {#sec inequalities}
===================================================

In this section, we will give proofs to the Heintz-Karcher type and Minkowski type inequalities stated in Theorem \[thm ineq\].\
[*Proof of Theorem \[thm ineq\] (i).*]{}Recall that $\bd^D \phi=V^{\g-\a} \left[\bd \phi+(2\g+ n\a) \bar g\left(\bn u, \bn \phi\right)\right].$ We know from the standard elliptic PDE theory that the following Dirichlet boundary value problem $$\label{Dirichlet3}
\left\{
    \begin{array}[]{rlll}
        \di \bd^D \phi&=&1&\mathrm{ in }\ \O,\\
        \phi&=& 0 &\mathrm{ on }\ \S,\\
    \end{array}
    \right.$$ admits a unique smooth solution $\phi\in C^{\infty}(\overline{\O})$. We will use the solution $\phi$ of the Dirichlet problem (\[Dirichlet3\]) in . For $\tau=(n+1)\a+\g$, we have $$\begin{aligned}
\label{case2-1}
    \frac{n-1}{n}\int_\O V^\tau=&\frac{n-1}{n}\int_\O V^{\tau}(\bd^D \phi)^2\nonumber\\ 
    \geq&  \int_\O V^{\tau}\Big[|\bd^D \phi|^2-|\bn^{2,D}\phi|^2\Big]\nonumber\\
   \ge &\int_{\S} V^{\tau}H^D \<\bn^D \phi,\nu\>^2,
 \end{aligned}$$ where in the first inequality, we have used the Cauchy-Schwarcz inequality, in the second inequality we have used integral formula (\[formula RF 1\]), the nonnegativity of $Ric^{D}$, and the Dirichlet boundary condition.

On the other hand, using equation , divergent structure , and integration by parts,we have $$\begin{aligned}
\label{case2-2}
    \int_\O V^\tau &=&\int_{\O}V^{\tau} \bd^D \phi\\
    &=&\int_{\S} V^{\tau}\<\bn^D \phi,\nu\>. \nonumber\end{aligned}$$ Combining , and using Hölder’s inequality, 