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### Open Access Journals

DCDS

For $\Omega$ a bounded open set in $\R^N$ we consider the space
$H^1_0(\bar{\Omega})=${$u_{|_{\Omega}}: u \in H^1(\R^N):$
$u(x)=0$ a.e. outside $\bar{\Omega}$}. The set $\Omega$ is
called

*stable*if $H^1_0(\Omega)=H^1_0(\bar{\Omega})$. Stability of $\Omega$ can be characterised by the convergence of the solutions of the Poisson equation$ -\Delta u_n = f$ in $D(\Omega_n)^$´, $ u_n \in H^1_0(\Omega_n)$

and also the Dirichlet Problem with respect to $\Omega_n$ if
$\Omega_n$ converges to $\Omega$ in a sense to be made precise. We
give diverse results in this direction, all with purely analytical
tools not referring to abstract potential theory as in Hedberg's
survey article [Expo. Math. **11** (1993), 193--259]. The most
complete picture is obtained when $\Omega$ is supposed to be Dirichlet
regular. However, stability does not imply Dirichlet regularity as
Lebesgue's cusp shows.

CPAA

Given a family $A(t)$ of closed unbounded operators on a UMD Banach space $X$
with common domain $W,$ we investigate various properties of the operator
$D_{A}:=\frac{d}{dt}-A(\cdot)$ acting from $\mathcal{W}_{per}^{p}:=\{u\in
W^{1,p}(0,2\pi ;X)\cap L^{p}(0,2\pi ;W):u(0)=u(2\pi)\}$ into $\mathcal{X}
^{p}:=L^{p}(0,2\pi ;X)$ when $p\in (1,\infty).$ The primary focus is on the
Fredholmness and index of $D_{A},$ but a number of related issues are also
discussed, such as the independence of the index and spectrum of $D_{A}$
upon $p$ or upon the pair $(X,W)$ as well as sufficient conditions ensuring
that $D_{A}$ is an isomorphism. Motivated by applications when $D_{A}$
arises as the linearization of a nonlinear operator, we also address similar
questions in higher order spaces, which amounts to proving (nontrivial)
regularity properties. Since we do not assume that $\pm A(t)$ generates any
semigroup, approaches based on evolution systems are ruled out. In
particular, we do not make use of any analog or generalization of Floquet's
theory. Instead, some arguments, which rely on the autonomous case (for
which results have only recently been made available) and a partition of
unity, are more reminiscent of the methods used in elliptic PDE theory with
variable coefficients.

CPAA

If $\Omega$ is any compact Lipschitz domain, possibly in a Riemannian manifold,
with boundary $\Gamma = \partial \Omega$, the Dirichlet-to-Neumann operator $\mathcal{D}_\lambda$
is defined on $L^2(\Gamma)$ for any real $\lambda$.
We prove a close relationship
between the eigenvalues of $\mathcal{D}_\lambda$ and those of the Robin Laplacian $\Delta_\mu$,
i.e. the Laplacian with Robin boundary conditions $\partial_\nu u =\mu u$.
This is used to give another proof of the Friedlander inequalities between Neumann and
Dirichlet eigenvalues, $\lambda^N_{k+1} \leq \lambda^D_k$, $k \in N$, and to sharpen
the inequality to be strict, whenever $\Omega$ is a Lipschitz domain in $R^d$. We give
new counterexamples to these inequalities in
the general Riemannian setting. Finally, we prove that the semigroup generated by
$-\mathcal{D}_\lambda$, for $\lambda$ sufficiently small or negative, is irreducible.

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