Let \(\Omega\subseteq R^d\) be a bounded domain, and let \(\Sigma \subset \Omega\) be a \(C^2\)-hypersurface with constants \(r_0>0\) and \( K>0\) . Assume that \(\Omega\setminus \Sigma\) has two connected components, \(\Omega_+\) and \(\Omega_-\). Furthermore, let \(\mathcal{L}\) be an elliptic operator with constants \(\lambda>0\) and \(M>0\) . If \(u\in H^1(\Omega)\) solves
\[ \mathcal{L}u=0, \]then there exists a positive constant \(h_0\), depending on \(\lambda\), \(M\), \(r_0\), and \(K\), such that the following holds. Assume
- \(h \in (0,h_0)\) and \( r/2 > h\),
- \(D\subset \Omega\) is a connected open subset,
- there is a ball \(B\subseteq D\) with radius \(r\),
- \(\dist(D,\partial\Omega)\ge h\).
Then the estimate
\[ \lVert u\rVert_{L^2(D)}\le C\lVert u\rVert_{L^2(B)}^\delta \lVert u\rVert_{L^2(\Omega)}^{1-\delta} \]holds, where
\[ C = C_1 \biggl(\frac{\lvert \Omega\rvert}{h^d}\biggr)^{1/2}, \qquad \delta\ge \tau^{\frac{C_2\lvert \Omega\rvert}{h^d}}. \]Here \(C_1, C_2>0\) and \(\tau \in (0,1)\) are constants depending only on \(\lambda\), \(M\), \(r_0\), and \(K\) [1, Theorem 1.1].
Links Link to heading
Questions Link to heading
- How the constants \(C_1\), \(C_2\) and \(\tau\) look like for a special case (see (0x68c7ffba) )?
References Link to heading
- C. Cârstea and J. Wang,
Propagation of smallness for an elliptic PDE with piecewise Lipschitz coefficients,
Journal of Differential Equations, vol. 268, no. 12, p. 7609–7628, 2020. doi:10.1016/j.jde.2019.11.088