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Carleman estimates for anisotropic elliptic operators with jumps at an interface. (English) Zbl 1319.47038
Let \(\Omega\) be an open set of \( \mathbb R^{n}\) and let \(\Sigma\) be a \(\mathcal{C}^{\infty}\) oriented hypersurface of \(\Omega\) such that \[ \Omega = \Omega_{+} \cup \Sigma \cup \;\Omega_{-}, \;\overline{\Omega}_{\pm} = \Omega_{\pm} \cup \Sigma, \] \(\Omega_{\pm}\) being open subsets of \( \mathbb R^n\). Consider the elliptic second-order operator defined by \[ \mathcal{L} = -\mathrm{div} (A (x) \nabla), \] where \(A (x)\) is a symmetric positive definite \(n \times n\) matrix admitting the representation \[ A = H_{-} A_{-} + H_{+} A_{+}, \;A_{\pm} \in \mathcal{C}^{\infty} (\Omega) \] (\(H_{\pm}\) denote the Heaviside functions). Define the vector space \[ W : = \{ H_{-} w_{-} +H_{+} w_{+} \}, \] where the functions \(w_{\pm}\) are in \(\mathcal{C}^{\infty} (\Omega)\) and satisfy the conditions \[ w_{+} = w_{-} \text{ at } \Sigma, \]
\[ \langle d w_{+}, A_{+} \nu \rangle = \langle d w_{-}, A_{-} \nu \rangle, \] \(\nu\) is the unit conormal vector field to \(\Sigma\) pointing into \(\Omega_{\pm}\).
The authors prove that under certain conditions made for any compact subset \(K\) of \(\Omega\), there exist a weight function \(\varphi\) and positive constants \(C, \tau_{1}\) such that for all \(\tau \geq \tau_{1}\) and all \(w \in W\) with \(\operatorname{supp} w \subset K\), the following inequality holds: \[ C \| e^{\tau \varphi} \mathcal{L} w \|_{L^{2} (\mathbb{R}^{n})} \geq \]
\[ \geq \tau^{3/2} \| e^{\tau \varphi} w \|_{L^{2} (\mathbb{R}^{n})} + \tau^{1/2} \| H_{+} e^{\tau \varphi} \nabla w_{+} \|_{L^{2} (\mathbb{R}^{n})} + \tau^{1/2} \| H_{-} e^{\tau \varphi} \nabla w_{-} \|_{L^{2} (\mathbb{R}^{n})} + \]
\[ + \tau^{3/2} \| (e^{\tau \varphi} w)/_{\Sigma} \|_{L^{2} (\Sigma)} + \tau^{1/2} \| (e^{\tau \varphi} \nabla w_{+})/_{\Sigma} \|_{L^{2} (\Sigma)} + \tau^{1/2} \| (e^{\tau \varphi} \nabla w_{-})/_{\Sigma} \|_{L^{2} (\Sigma)}. \] This result is generalized to the nonhomogeneous case when \(w_{+} - w_{-} = \theta\) and \(\langle A_{+} d w_{+} - A_{-} d w_{-}, \nu \rangle = \Theta \) at \(\Sigma\). Some examples regarded as applications are also presented.

MSC:
47F05 General theory of partial differential operators
35J15 Second-order elliptic equations
35J57 Boundary value problems for second-order elliptic systems
35J75 Singular elliptic equations
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