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VIDEO LIBRARY |
Conference on Complex Analysis and Mathematical Physics, dedicated to the 70th birthday of A. G. Sergeev
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Boundary value problems on hypersurfaces and Roland Duduchavaab a The University of Georgia b A. Razmadze Mathematical Institute |
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Abstract: We consider two examples of boundary value problems (BVPs) on hypersurfaces: heat conduction by an "isotropic" media, governed by the Laplace equation and bending of elastic "isotropic" media governed by Láme equations. The boundary conditions are classical Dirichlet-Neumann mixed type. The domain The object of the investigation is what happens with the above mentioned mixed boundary value problems when the thickness of the layer converges to zero The suggested approach is based on the fact that the Laplace and Láme operators are represented in terms of Günter's tangential and normal (to the surface) derivatives. Namely, if $$ \mathcal{D}_j:=\partial_j-\nu_j\mathcal{D}_4, \qquad \mathcal{D}_4 =\partial_\nu=\displaystyle\sum\limits_{k=1}^3\nu_k\partial_k,\qquad j=1,\ldots,n $$ and the Laplace-Beltrami operator on the surface $$ \Delta_\mathcal{C}=\mathcal{D}_1^2+\mathcal{D}_2^2+\mathcal{D}_3^2. $$ Moreover, the Laplace and the Láme operators in the domain $$ \Delta_{\Omega^h}=\partial _1^2+\partial_2^2+\partial _3^2,\qquad \mathcal{L}_{\Omega^h}=-2\mu\,\Delta-(\lambda+2\mu)\,\nabla{\rm div} $$ are represented as follows: $$ \Delta _{\Omega ^h}= \displaystyle\sum\limits_{j=1}^{4} \mathcal{D}_{j}^{2}+2\mathcal{H}_\mathcal{C}\mathcal{D}_4,\qquad \mathcal{L}_{\Omega^h}=-2\mu\,\Delta_{\Omega^h} -(\lambda+2\mu)\,\nabla_{\Omega^h}{\rm div} _{\Omega^h}. $$ Here $$ \nabla_{\Omega^h}\varphi:=\Bigl\{\mathcal{D}_1\varphi,...,\mathcal{D}_4 \varphi\Bigr\}^\top,\qquad {\rm div}_{\Omega^h}{\mathbf U}:=\sum_{j=1}^4\mathcal{D}_jU^0_j+\mathcal{H}_\mathcal{C} U^0_4,\\ {\mathbf U}=(U_1,U_2,U_3)^\top,\qquad U^0_j:=U_j-U^0_4,\quad U_4^0:=\langle\nu,{\mathbf U}\rangle,\quad j=1,2,3 $$ are the gradient and divergence. The work is carried out in collaboration with T. Buchukuri and G. Tephnadze (Tbilisi). Language: English |