Abstract.
Let Ω be a bounded subset of R N, \( a \in C^1(\overline\Omega) \) with \( a>0 \) in Ω and A be the operator defined by \( Au := \nabla\cdot (a\nabla u) \) with the generalized Wentzell boundary condition.¶¶\( Au + \beta\frac{\partial u}{\partial n} + \gamma u=0\qquad \hbox{on} \quad\partial \Omega. \)¶¶If \( \partial\Omega \) is in C 2, β and γ are nonnegative functions in \( C^1(\partial\Omega), \) with β > O, and \( \Gamma:=\{x\in\partial\Omega: a(x)>0\}\neq\emptyset \), then we prove the existence of a (C 0 ) contraction semigroup generated by \( \overline{A} \), the closure of A, on a suitable L p space, \( 1\le p $<$\infty \) and on \( C(\overline{\Omega}).\) Moreover, this semigroup is analytic if \( 1 $<$ p $<$\infty. \)
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Favini, A., Goldstein, G., Goldstein, J. et al. The heat equation with generalized Wentzell boundary condition. J.evol.equ. 2, 1–19 (2002). https://doi.org/10.1007/s00028-002-8077-y
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DOI: https://doi.org/10.1007/s00028-002-8077-y