\[ \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\div}{div} \]

Let \(A_{\pm }\) be Lipschitz symmetric matrix-valued functions satisfying, for given constants \(\lambda\in (0,1]\), \(M>0\),

\begin{align*} \lambda |\xi|^2 \leq A_{\pm }(x)\xi\cdot \xi \leq \lambda^{-1} |\xi|^2, \qquad \forall x, \xi\in \mathbb{R}^d, \end{align*}

\[ \lVert A_{\pm } \rVert_{C^{0,1}(\mathbb{R}^d_{\pm })}\le M. \]

Let \(u_{\pm }\in C^\infty (\mathbb{R}^d)\), and

\[ u=\chi_{\mathbb{R}^d_+}u_+ + \chi_{\mathbb{R}^d_-}u. \]

Also, let

\begin{equation*} \mathcal{L} u = \chi_{\mathbb{R}^d_+}\div (A_+ \nabla u_+) + \chi_{\mathbb{R}^d_-}\div (A_- \nabla u_-) = 0, \end{equation*}

and

\begin{align*} h_0(x)&:=u_+(x,0)-u_-(x,0), \\ h_1(x)&:=A_+(x,0)\nabla u_+(x,0)\cdot \nu - A_-(x,0)\nabla u_-(x,0)\cdot \nu, \end{align*}

where \(\nu=-e_d\).

There exist positive constants \( \alpha_+, \alpha_-, \beta, \delta_0, r_0, \tau_0\), with \(\tau_0\ge 1\), and \(C\) all depending on \( \lambda, M, d\), such that if \( \delta \leq \delta_0 \) and \( \tau \geq \tau_0\), then

\begin{align*} &\sum_{\pm }\Bigg( \tau^{-1}\int_{\mathbb{R}^d_\pm } |D^2 u_\pm |^2 e^{2\tau \phi_\delta} + \tau\int_{\mathbb{R}^d_\pm } |\nabla u_\pm |^2 e^{2\tau \phi_\delta} + \tau^3 \int_{\mathbb{R}^d_\pm } |u_\pm |^2 e^{2\tau \phi_\delta} \Bigg) + \\ &\quad+\sum_{\pm } \Bigg( \tau \int_{\mathbb{R}^{d-1}} |\nabla u_\pm (x,0)|^2 e^{2 \phi_\delta(x,0)} dx +\tau^2 \int_{\mathbb{R}^{d-1}} |u_\pm (x,0)|^2 e^{2 \phi_\delta(x,0)} dx \Bigg) + \\ &\quad+\sum_{\pm } \Biggl( \Big[ \, \nabla ( e^{\tau \phi_\delta} u_{\pm }(\cdot,0)) \, \Big]_{H^{1/2}(\mathbb{R}^{d-1})}^2 +\tau^{2} \Big[ \, e^{\tau \phi_\delta(\cdot,0)} u_\pm (\cdot,0) \, \Big]_{H^{1/2}(\mathbb{R}^{d-1})}^2 \Biggr) \\ &\leq C \Biggl( \sum_{\pm } \int_{\mathbb{R}^d_\pm } | \operatorname{div}(A_\pm \nabla u_\pm )|^2 e^{2\tau \phi_\delta} +\big[ e^{\tau \phi_\delta(\cdot,0)} h_1 \big]^2_{H^{1/2}(\mathbb{R}^{d-1})} +\big[ \nabla ( e^{\tau \phi_\delta} h_0)(\cdot,0) \big]_{H^{1/2}(\mathbb{R}^{d-1})}^2 + \\ &\quad+\tfrac{\tau}{\delta} \int_{\mathbb{R}^{d-1}} |h_1|^2 e^{2\tau \phi_\delta(x,0)} dx +\tfrac{\tau^3}{\delta^3} \int_{\mathbb{R}^{d-1}} |h_0|^2 e^{2\tau \phi_\delta(x,0)} dx \Biggr), \end{align*}

where

\[ \supp u \subset B_{\delta/2}(0) \times [-\delta r_0, \delta r_0], \]

and

\[ \phi_\delta(x,y) = \begin{cases} \tfrac{\alpha_+y}{\delta} + \tfrac{\beta y^2}{2 \delta^2} - \tfrac{|x|^2}{2\delta}, & y \geq 0, \\ \tfrac{\alpha_- y}{\delta} + \tfrac{\beta y^2}{2 \delta^2} - \tfrac{|x|^2}{2\delta}, & y < 0, \end{cases} \]

[1, Theorem 2.1].

See also Link to heading

• fractional Sobolev space

References Link to heading

1. M. Di Cristo, E. Francini, C. Lin, S. Vessella, and J. Wang, Carleman estimate for second order elliptic equations with Lipschitz leading coefficients and jumps at an interface, Journal de Mathématiques Pures et Appliquées, vol. 108, no. 2, p. 163–206, 2017. doi:10.1016/j.matpur.2016.10.015
2. E. Francini, C. Lin, S. Vessella, and J. Wang, Three-region inequalities for the second order elliptic equation with discontinuous coefficients and size estimate, Journal of Differential Equations, vol. 261, no. 10, p. 5306–5323, 2016. doi:10.1016/j.jde.2016.08.002