Let \(\Omega\subseteq R^d\) be a bounded domain. Suppose \(A\in C^{0,1} (\Omega)\) with
\begin{align*} \lambda |\xi|^2 \leq A\xi\cdot \xi \leq \lambda^{-1} |\xi|^2, \qquad \forall x\in \Omega, \forall \xi\in \mathbb{R}^d, \end{align*}\[ \lVert A \rVert_{C^{0,1}(\Omega)}\le M, \]
and \( u \in H^1(\Omega) \) is a weak solution of
\[ \operatorname{div}(A \nabla u) = 0. \]Let \( h \in (0, r/2), \, D \subset U \) connected, open with \( B_{r/2}(x_0) \subset D \), and \( \operatorname{dist}(D, \partial \Omega) \geq h \). Then
\[ \| u \|_{L^\infty(D)} \leq C \, \| u \|_{L^2(B_r(x_0))}^{\delta} \, \| u \|_{L^2(\Omega)}^{1-\delta}, \]where
\[ C = C_1 \left( \frac{|\Omega|}{h^d} \right)^{1/2}, \quad \delta \geq \tau^{\frac{C_2 |\Omega|}{h^d}}, \]with \( C_1, C_2 > 0, \, \tau \in (0,1) \) depending on \( \lambda, M \) [1, Theorem 5.1]
Questions
- What if \( \operatorname{dist}(x_0, D) = r/2 + \varepsilon \) and \( h = \varepsilon'\) with \( \varepsilon, \varepsilon' < r/2 \). Then \( B_r(x_0) \nsubseteq U \).
- What weak solution means in this context?
Links Link to heading
References Link to heading
- G. Alessandrini, L. Rondi, E. Rosset, and S. Vessella,
The stability for the Cauchy problem for elliptic equations,
Inverse Problems, vol. 25, no. 12, p. 123004, 2009. doi:10.1088/0266-5611/25/12/123004