Let \( D \subset \mathbb{R}^d \) be a domain, and \( \eta \in \mathbb{C}_c^\infty (\mathbb{R}^d) \), with \( 0 \leq \eta(x) \leq 1 \) and \(\eta\equiv 1\) on \(D\). Furthermore, let \(u_\pm\in C^\infty (\mathbb{R}^d) \) denote a solution of the equation given in (0x68c9184d) . Then
\begin{equation}\label{eq:caccioppoli} \int_D |\nabla u_{+}|^2 + \int_D |\nabla u_{-}|^2 \leq \frac{4 M^2}{\lambda^2} \| \nabla \eta \|_{\infty}^2 \left( \int_{\mathrm{supp}\, \eta} |u_{+}|^2 + \int_{\mathrm{supp}\, \eta} |u_{-}|^2 \right). \end{equation}
Remarks
- It is sufficient if \(u_\pm \in C^\infty (\supp \eta)\).
Proof
Since \( u_{\pm} \) are solutions, it follows that
\begin{equation}\label{eq:step1} \begin{aligned} 0 &= \sum_{\pm} \int_{\mathbb{R}^n_{\pm}} \nabla \cdot (A_{\pm} \nabla u_{\pm}) \, u_{\pm} \eta^2 \\[6pt] &= -\sum_{\pm} \left( \int_{\mathbb{R}^n_{\pm}} A_{\pm} \nabla u_{\pm} \cdot \nabla (u_{\pm} \eta^2) \right) +\int_{\mathbb{R}^{n-1}} (A_{+} \nabla u_{+} \cdot \nu - A_{-} \nabla u_{-} \cdot \nu) \, u_{+} \eta^2 \\[6pt] &= -\sum_{\pm} \int_{\mathbb{R}^n_{\pm}} A_{\pm} \nabla u_{\pm} \cdot (\nabla u_{\pm} \, \eta^2 + 2u_{\pm} \eta \nabla \eta), \end{aligned} \end{equation}where we employed the integration by parts formula and the boundary conditions on \( u_{\pm} \).
Then, by ellipticity of \( A_{\pm} \), we deduce
\begin{align*} \sum_{\pm} \int_{\mathbb{R}^n_{\pm}} |\nabla u_{\pm}|^2 \eta^2 &\leq \sum_{\pm} \frac{1}{\lambda} \int_{\mathbb{R}^n_{\pm}} (A_{\pm} \nabla u_{\pm} \cdot \nabla u_{\pm}) \eta^2 \\[6pt] &\leq \sum_{\pm} \frac{2}{\lambda} \int_{\mathbb{R}^n_{\pm}} |A_{\pm} \nabla u_{\pm} \cdot \nabla \eta| \, |u_{\pm}| \eta \\[6pt] &\leq \sum_{\pm} \frac{2M}{\lambda} \int_{\mathbb{R}^n_{\pm}} (|\nabla u_{\pm}|\eta) \, (|\nabla \eta||u_{\pm}|) \\[6pt] &\leq \frac{2M}{\lambda} \left( \sum_{\pm} \int_{\mathbb{R}^n_{\pm}} |\nabla u_{\pm}|^2 \eta^2 \right)^{1/2} \left( \sum_{\pm} \int_{\mathbb{R}^n_{\pm}} |\nabla \eta|^2 |u_{\pm}|^2 \right)^{1/2}, \end{align*}where we used \eqref{eq:step1} and Hölder’s inequality. Multiplying by
\[ \left( \sum_{\pm} \int_{\mathbb{R}^n_{\pm}} |\nabla u_{\pm}|^2 \eta^2 \right)^{-1/2} \]and squaring both sides yields
\begin{align*} \sum_{\pm} \int_D |\nabla u_{\pm}|^2 &\leq \sum_{\pm} \int_{\mathbb{R}^n_{\pm}} |\nabla u_{\pm}|^2 \eta^2 \\[6pt] &\leq \frac{4M^2}{\lambda^2} \| \nabla \eta \|_{\infty}^2 \left( \sum_{\pm} \int_{\mathrm{supp}\, \eta} |u_{\pm}|^2 \right). \end{align*}