The authors study the abstract Cauchy problem associated to the operator \(Au=\nabla(\alpha\nabla u)\) with Wentzell boundary conditions in an open bounded set \(\Omega\subset\mathbb{R}^n\) with smooth boundary. The function \(\alpha\) is continuous, \(\alpha>0\) in \(\Omega\) and \(\alpha|_{\partial\Omega}=0\). The problem is reduced to a suitable Dirichlet problem, and it is proven that \(A\) generates an analytic semigroup in \(L^2(\Omega)\) with weight \({1/\alpha}\). Under additional assumptions on \(\alpha\), \(D(A)\) is completely characterized. Analogous generation results are proven in \(L^2(0,1)\). Also results on the analyticity of the semigroup generated by \(A\) are proven in \(L^p(\Omega)\) with weight \(\alpha^{{1/p}}\) with \(10\), \(00\); where \(\alpha\in C([0,1])\), \(\alpha(0)=\alpha(1)=0\), \(\alpha(x)>0\), \(0