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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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