We prove the existence of smooth solutions of two-dimensional nonuniformly elliptic curvature equations subject to a nonlinear oblique boundary condition. These are equations whose principal part is given by a suitable symmetric function of the principal curvatures of the graph of the solution $u$. The types of boundary conditions we are able to treat are the same as those we considered in earlier work on Hessian equations.

We consider the fourth-order nonlinear degenerate parabolic equation $$ u_t+(|u|^n u_{xxx})_x=0 $$ which arises in lubrication models for thin viscous films and spreading droplets as well as in the flow of a thin neck of fluid in a Hele-Shaw cell. We prove that if $0<n<2$ this equation has finite speed of propagation for nonnegative ``strong" solutions and hence there exists an interface or free boundary separating the regions where $u>0$ and $u=0$. Then we prove that the interface is Hölder continuous if $1/2<n<2$ and right-continuous if $0<n\leq 1/2$. Finally we study the Cauchy problem and obtain optimal asymptotic rates as $t\to\infty$ for the solution and for the interface when $0<n<2$; these rates exactly match those of the source-type solutions. If $0<n<1$ the property of finite speed of propagation is also proved for changing sign solutions.

We prove that the linear system of thermoelastic plates, with finite speed of propagation of the elastic components, is controllable in the following sense: If the control time is large enough and we act in the equation of displacement by means of a control supported in a neighborhood of the boundary of the plate, then we may control exactly the displacement and simultaneously the temperature in an approximate way. The method of proof is an adaptation of the techniques developed by the second author for the proof of the exact-approximate controllability for the three-dimensional system of thermolelasticity and combines: (i) a decoupling result based on an idea due to Henry, Lopes and Perissinotto for three-dimensional thermoelasticity, (ii) the variational approach to controllability developed by Fabre, Puel and Zuazua and (iii) some observability inequalities for the system of thermoelastic plates.

In this paper we characterize the existence of coexistence states for the classical Lotka-Volterra predator-prey model with periodic coefficients and analyze the dynamics of positive solutions of such models. Among other results we show that if some trivial or semi-trivial positive state is linearly stable, then it is globally asymptotically stable with respect to the positive solutions. In fact, the model possesses a coexistence state if, and only if, any of the semi-trivial states is unstable. Some permanence and uniqueness results are also found. An example exhibiting a unique coexistence state that is unstable is given.

Infinite-dimensional elliptic equations, with Hölder-continuous coefficients are here studied by purely analytic methods. In particular, Schauder estimates for solutions of such equations are derived.

The paper establishes several minimization theorems for noncoercive functionals defined on a Hilbert (or reflexive Banach) space which are subject to constraints. Applications to critical point theory and variational inequalities are given. The results are also applied to obtain the existence of solutions of several nonlinear boundary and unilateral problems.