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MSC: 35B65 35J25 35K20Keywords: Second order linear elliptic equations Lipschitz domains Robin boundary conditions Hölder regularity L ∞ -coefficients Parabolic equations Strongly continuous semigroups on C(Ω) Wentzell-Robin boundary conditions For a linear, strictly elliptic second order differential operator in divergence form with bounded, measurable coefficients on a Lipschitz domain Ω we show that solutions of the corresponding elliptic problem with Robin and thus in particular with Neumann boundary conditions are Hölder continuous up to the boundary for sufficiently L p -regular right-hand sides. From this we deduce that the parabolic problem with Robin or Wentzell-Robin boundary conditions is well-posed on C(Ω).

We consider an one-dimensional lattice system of unbounded and continuous spins. The Hamiltonian consists of a perturbed strictly-convex single-site potential and with longe-range interaction. We show that if the interactions decay algebraically of order 2 + α, α > 0 then the correlations also decay algebraically of order 2 +α for someα > 0. For the argument we generalize a method from Zegarlinski from finite-range to infinite-range interaction to get a preliminary decay of correlations, which is improved to the correct order by a recursive scheme based on Lebowitz inequalities. Because the decay of correlations yields the uniqueness of the Gibbs measure, the main result of this article yields that the on-phase region of a continuous spin system is at least as large as for the Ising model.

Abstract. Convergence of operators acting on a given Hilbert space is an old and well studied topic in operator theory. The idea of introducing a related notion for operators acting on varying spaces is natural. Many previous contributions to this subject consider either concrete examples of perturbations, or an abstract setting where weak or strong convergence of the resolvents is used. However, it seems that the first results on norm resolvent convergence in this direction have been obtained only recently, to the best of our knowledge. Here we consider sectorial operators on Hilbert spaces that depend on a parameter. We define a notion of convergence that generalises convergence of the resolvents in operator norm to the case when the operators act on different spaces. In addition, we show that this kind of convergence is compatible with the functional calculus of the operator and moreover implies convergence of the spectrum. Finally, we present examples for which this convergence can be checked, including convergence of coefficients of parabolic problems. Convergence of a manifold (roughly speaking consisting of thin tubes) towards the manifold's skeleton graph plays a prominent role, being our main application.Mathematics subject classification (2010): 34B45, 35P05, 47D06.

Based on a simple geometric description of orthogonal projections onto closed, convex sets we find an implicit formula for the L 2 -projection onto the L p -unit ball. This allows us to to prove the L p -quasi-contractivity of semigroups generated by linear elliptic operators and quasi-linear operators of q-Laplace-type in a simple way.Mathematics Subject Classification. Primary 35K15; Secondary 35B45.

We consider a class of evolution equations taking place on the edges of a finite network and allow for feedback effects between different, possibly non-adjacent edges. This generalizes the setting that is common in the literature, where the only considered interactions take place at the boundary, i. e., in the nodes of the network. We discuss well-posedness of the associated initial value problem as well as contractivity and positivity properties of its solutions. Finally, we discuss qualitative properties that can be formulated in terms of invariance of linear subspaces of the state space, i. e., of symmetries of the associated physical system. Applications to a neurobiological model as well as to a system of linear Schrödinger equations on a quantum graph are discussed.2000 Mathematics Subject Classification. 34B45, 70S10, 47D06. Key words and phrases. parabolic diffusion equations on networks; symmetries of dynamical systems.

In the first part of the article we characterize automatic continuity of positive operators. As a corollary we consider complete norms for which a given cone E+ in an infinite dimensional Banach space E is closed and we obtain the following result: every two such norms are equivalent if and only if E+ ∩ (−E+) = {0} and E+ − E+ has finite codimension. Without preservation of an order structure, on an infinite dimensional Banach space one can always construct infinitely many mutually non-equivalent complete norms. We use different techniques to prove this. The most striking is a set theoretic approach which allows us to construct infinitely many complete norms such that the resulting Banach spaces are mutually non-isomorphic. Mathematics Subject Classification (2000). 46B03, 46B40, 03E75, 46B26.

Abstract. We prove Hölder continuity up to the boundary for solutions of quasi-linear degenerate elliptic problems in divergence form, not necessarily of variational type, on Lipschitz domains with Neumann and Robin boundary conditions. This includes the p-Laplace operator for all p ∈ (1, ∞), but also operators with unbounded coefficients. Based on the elliptic result we show that the corresponding parabolic problem is wellposed in the space C(Ω) provided that the coefficients satisfy a mild monotonicity condition. More precisely, we show that the realization of the elliptic operator in C(Ω) is m-accretive and densely defined. Thus it generates a non-linear strongly continuous contraction semigroup on C(Ω). Mathematics Subject Classification (1991