Two exponential stabilization results are proved for a vibrating structure subject to an interfacial slip. More precisely, the structure consists of two identical beams of Timoshenko type and clamped together but allowing for a longitudinal movement between the layers. We will stabilize the system through a transverse friction and also through a viscoelastic damping.

In beginning of the 90s, Bernard Helffer and Jöhannes Sjostrand introduced operators serving to develop methods for the study of integrals in high dimensions of the type that appear in statistical mechanics and Euclidean field theory. In these papers, the authors studied a certain class of unbounded spin models by means of the spectrum of the Witten Laplacians. The decay of correlations, the analyticity of the free energy, the Poincaré and log-Sobolev inequalities turned out to be the relevant tools for investigating phase transitions in certain classical continuous models. The present paper proposes a direct and more general method for investigating phase transitions in classical continuous models of Kac type. We discuss hypotheses on the source term that will result in a direct proof of the analyticity of the free energy without using the truncated correlations. We also use the Witten Laplacians to derive mixing properties and a decay of correlations that lead to the logarithmic Sobolev inequality. The novelty, as compared to previous work, is that our method is more direct and does not use the one-dimensional Witten Laplacians.