The heat equation with generalized Wentzell boundary condition

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.$