Abstract
Diffusion problems on topological networks (one-dimensional networks) have been introduced by G. Lumer [Lu. 1–4] and are also considered by F. Ali Mehmeti [AM] and the author [N.1–3]. According to the ideas of G. Lumer [Lu. 5], we develop here a local approach to diffusion problems on higher dimensional ramified spaces. We consider the variational formulation of such problems (see [L-U, G-T, Li, Sh, Lu. 5]). The transmission operator is the sum of weak Ventcel'-Visik boundary operators [B-C-P] (it is either a first order operator or a second order operator). Finally, like Gilbarg-Trudinger [G-T], we establish a continuity result which will be used in [N. 5] to show that one of the assumptions of the Lumer-Phillips theorem [P] (density of the range) is fulfilled.
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Nicaise, S. Elliptic operators on elementary ramified spaces. Integr equ oper theory 11, 230–257 (1988). https://doi.org/10.1007/BF01272120
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DOI: https://doi.org/10.1007/BF01272120