In this paper we prove that the Dirichlet-to-Neumann semigroup $S(t)$ is an analytic compact Markov irreducible semigroup in $C(\partial \Omega)$ in any bounded smooth domain $\Omega$. By a generalization of the Lax semigroup, we construct an approximating family for $S(t)$. We prove some regularizing characters and compactness of this family. By using the ergodic properties of $S(t)$, we deduce its asymptotic behavior. At the end we conjecture some open problems.

In a previous paper [7], we investigated the asymptotic behaviour of subsonic travelling waves of finite energy for the Gross-Pitaevskii equation in every dimension $N \geq 2$. In particular, we gave their first-order asymptotics in case they were axisymmetric. In the present paper, we compute their first-order asymptotics at infinity in the general case.

This paper is concerned with the mean field equation for equilibrium turbulence with arbitrarily signed vortices. We develop blow-up analysis and establish a functional inequality of the Trudinger-Moser type.

We study smoothness of generalized solutions of nonlocal elliptic problems in plane bounded domains with piecewise smooth boundary. The case where the support of nonlocal terms can intersect the boundary is considered. We find conditions that are necessary and sufficient for any generalized solution to possess an appropriate smoothness (in terms of Sobolev spaces). Both homogeneous and nonhomogeneous nonlocal boundary-value conditions are studied.