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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.