We establish two general nonlinear Liouville theorems for equations of the type $$ -\Delta u=h(x_1)f(u),\; u\geq 0 \; \mbox{ in } R^N,\; \sup_{R^N}u < +\infty. $$ We then show how these Liouville theorems can be used to obtain a priori estimates for positive solutions of indefinite superlinear elliptic equations for several new cases left open in previous research.

We prove the existence of unique regular local in time solutions to the quasi-stationary one-phase Stefan problem with the Gibbs-Thomson correction. The result is optimal with respect to $L_p$ regularity and the obtained phase surface is a submanifold of the $W^{3,1}_p$-class. The proof is based on a Schauder-type estimate for a linearization of the original system.

In this work, we derive a result of exact controllability for a structural acoustic partial differential equation (PDE) model, comprised of a three-dimensional interior acoustic wave equation coupled to a two-dimensional Kirchoff plate equation, with the coupling being accomplished across a boundary interface. For this PDE system, we show that by means of boundary controls, the interior wave and Kirchoff plate initial data can be steered to an arbitrary finite energy state. In this work, key use is made of recent, microlocally-derived, $L^{2}\times H^{-1}$ "recovery" estimates for wave equations with Dirichlet boundary data. Moreover, the coupling of the disparate acoustic wave/Kirchoff plate dynamics is reconciled by means of sharp regularity estimates which are valid for hyperbolic equations of second order.

The study of solutions with fixed energy of certain classes of Lagrangian (or Hamiltonian) systems is reduced, via the classical Maupertuis--Jacobi variational principle, to the study of geodesics in Riemannian manifolds. We are interested in investigating the problem of existence of brake orbits and homoclinics, in which case the Maupertuis--Jacobi principle produces a Riemannian manifold with boundary and with metric degenerating in a nontrivial way on the boundary. In this paper we use the classical Maupertuis--Jacobi principle to show how to remove the degeneration of the metric on the boundary, and we prove in full generality how the brake orbit and the homoclinic multiplicity problem can be reduced to the study of multiplicity of orthogonal geodesic chords in a manifold with regular and strongly concave boundary.