We prove that the class of regular saddle surfaces in the hyperbolic or spherical three-space coincides with the class of regular surfaces with curvature not greater than the curvature of the surrounding space. We also show that a similar result for nonregular surfaces is incorrect.

We define Orlicz-Sobolev spaces on an arbitrary metric space with a Borel regular outer measure, and we develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. We prove that each Orlicz-Sobolev function has a quasi-continuous representative. We give estimates for the capacity of balls when the measure is doubling. Under additional regularity assumption on the measure, we establish some relations between capacity and Hausdorff measures.

We study the internal exact null controllability of a nonlinear heat equation with homogeneous Dirichlet boundary condition. The method used combines the Kakutani fixed-point theorem and the Carleman estimates for the backward adjoint linearized system. The result extends to the case of boundary control.

We study the dependence on the control $q$ of the interval of definition of the solution $u$ of the Cauchy problem $\iota u^{\prime }+\Delta u=-\lambda {\left|u\right|}^{2}u-\iota qu$ in ${ℝ}^2\times \left(0,T\right),u\left(x,0\right)=\omega$ in ${ℝ}^2$, and we prove a version of Fibich′s conjecture. Feedback laws for an inverse problem of the above equation with experimental data, measured on a portion of the boundary of an open, bounded subset of ${ℝ}^2$ are established.