OUHABAZ, El Maati

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OUHABAZ, El Maati
Institut de Mathématiques de Bordeaux [IMB]

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Ce document a été publié dans

Journal of Spectral Theory. 2018-12-31

European Mathematical Society

Résumé en anglais

Given a general symmetric elliptic operator $$ L_{a} := \sum_{k,,j=1}^d \p_k (a_{kj} \p_j ) + \sum_{k=1}^d a_k \p_k - \p_k( \overline{a_k} . ) + a_0$$we define the associated Dirichlet-to-Neumann (D-t-N) operator with partial data, i.e., data supported in a part of the boundary. We prove positivity, $L^p$-estimates and domination properties for the semigroup associated with this D-t-N operator. Given $L_a $ and $L_b$ of the previous type with bounded measurable coefficients $a = \{ a_{kj}, \ a_k, a_0 \}$ and $b = \{ b_{kj}, \ b_k, b_0 \}$, we prove that if their partial D-t-N operators (with $a_0$ and $b_0$ replaced by $a_0 -\la$ and $b_0 -\la$) coincide for all $\la$, then the operators $L_a$ and $L_b$, endowed with Dirichlet, mixed or Robin boundary conditions are unitary equivalent. In the case of the Dirichlet boundary conditions, this result was proved recently by Behrndt and Rohleder \cite{BR12} for Lipschitz continuous coefficients. We provide a different proof which works for bounded measurable coefficients and other boundary conditions.< Réduire

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https://oskar-bordeaux.fr/handle/20.500.12278/194108

Project ANR

Aux frontières de l'analyse Harmonique - ANR-12-BS01-0013

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Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251