We study the structure of generalized Verma modules over a semi-simple complex finite-dimensional Lie algebra, which are induced from simple modules over a parabolic subalgebra. We consider the case when the annihilator of the starting simple module is a minimal primitive ideal if we restrict this module to the Levi factor of the parabolic subalgebra. We show that these modules correspond to pr...

Using suitable polychromatic superpositions of helical Mathieu and parabolic nondiffracting beams, we study for the first time, to the best of our knowledge, the higher-order helical Mathieu X waves and the traveling and stationary parabolic X waves. The mathematical and physical properties of these new kinds of localized pulses are discussed.

In this paper we obtain quantitative estimates of strong unique continuation for solutions to parabolic equations. We apply these results to prove stability estimates of logarithmic type for an inverse problem consisting in the determination of unknown portions of the boundary of a domain Ω in Rn, from the knowledge of overdetermined boundary data for parabolic boundary value problems.

In this paper we study optimal control problems governed by semilinear parabolic equations. We obtain necessary optimality conditions in the form of an exact Pontryagin’s minimum principle for distributed and boundary controls (which can be unbounded) and bounded initial controls. These optimality conditions are obtained thanks to new regularity results for linear and nonlinear parabolic equati...