Let $\Omega$ be a smooth bounded domain in $\R^N$ and let \begin{equation*} Lu=\sum_{j,k=1}^N \p_{x_j}\left(a_{jk}(x)\p_{x_k} u ight), \end{equation*} in $\Omega$ and \begin{equation*} Lu+\beta(x)\sum\limits_{j,k=1}^N a_{jk}(x)\partial_{x_j} u \, n_k+\gamma (x)u-q\beta(x)\sum_{j,k=1}^{N-1}\p_{ au_k}\left(b_{jk}(x)\p_{ au_j}u ight)=0, \end{equation*} on $\p\Omega$ define a generalized Laplacian on $\Omega$ with a Wentzell boundary condition involving a generalized Laplace-Beltrami operator on the boundary. Under some smoothness and positivity conditions on the coefficients, this defines a nonpositive selfadjoint operator, $-S^2$, on a suitable Hilbert space. If we have a sequence of such operators $S_0,\,S_1,\,S_2,...$ with corresponding coefficients \begin{equation*} \Phi_n=\left(a_{jk}^{(n)},\, b_{jk}^{(n)},\, \beta_n,\gamma_n,\,q_n ight) \end{equation*} satisfying $\Phi_n o\Phi_o$ uniformly as $n o\infty$, then $u_n(t) o u_o(t)$ where $u_n$ satisfies \begin{equation*} i\frac{du_n}{dt}=S_n^m u_n, \end{equation*} or \begin{equation*} \frac{d^2u_n}{dt^2}+S_n^{2m} u_n=0, \end{equation*} or \begin{equation*} \frac{d^2u_n}{dt^2}+F(S_n)\frac{du_n}{dt}+S_n^{2m} u_n=0, \end{equation*} for $m=1,\,2,$ initial conditions independent of $n$, and for certain nonnegative functions $F$. This includes Schr\"odinger equations, damped and undamped wave equations, and telegraph equations.

We prove a very general form of the Angle Concavity Theorem, which says that if ((T(t)) defines a one parameter semigroup acting over various L^p spaces (over a fixed measure space), which is analytic in a sector of opening angle heta_p, then the maximal choice for heta_p is aconcave function of 1-1/p. This and related results are applied to get improved estimates on the optimal L^p angle of ellipticity for a parabolic equation of the form {\partial u}{\partial t}=Au, where A is a uniformly elliptic second order partial differential operator with Wentzell or dynamic boundary condition. Similar results are obtained for the higher order equation {\partial u}{\partial t}=(-1)^{m+1}A^m u for all positive integers m.

We consider operators in divergence and in nondivergence form with degeneracy at the interior of the space domain. Characterizing the domain of the operators, we prove that they generate positive analytic semigroupson spaces of L^2 type.Finally, some applications to linear and semilinear parabolic evolution problems and to linear hyperbolic ones are presented.

We present a survey of recent results concerning heat and telegraph equations, equipped with Goldstein-Wentzell boundary conditions (already known as general Wentzell boundary conditions). We focus on the generation of analytic semigroups and continuous dependence of the solutions of the associated Cauchy problems from the boundary conditions.

Let us consider the operator A_nu:=(-1)^{n+1}\alpha (x)u^(2n) on H^n_0(0,1) with domain D(A_n):={u\in H^n_0(0,1)\cap H^{2n}_{loc}(0,1): A_n u\in H^n_0(0,1)}, where n\in\bold N, \alpha\in H^n_0(0,1), \alpha (x)>0 in (0,1).Under additional boundedness and integrability conditions on \alpha with respect to x^{2n} (1-x)^{2n}, we prove that (A_n, D(A_n)) is nonpositive and selfadjoint, thus it generates a cosine function, hence an analytic semigroup in the right half plane on H^n_0(0,1). Analyticity results are also proved in H^n(0,1). In particular, all results work well when \alpha (x)=x^j (1-x)^j, for |j-n|<1/2. Hardy type inequalities are also obtained.

Si prova che le soluzioni dell'equazione del telegrafo in molte classi di domini non limitati corrispondono asintoticamente a soluzioni dell'equazione del calore, con diverse condizioni al bordo, tra cui quelle di Wentzell generali. Si provano anche risultati di dipendenza continua dalle condizioni al bordo.