2011/02/28 by J. L. Vázquez, Vázquez, J. L., Nikolaos B. Zographopoulos +1 · 2 citations
Computer Science · Mathematics · Engineering · #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1102.5661
Starting with a functional difficulty appeared in the paper \citevz00 by Vázquez and Zuazua, we obtain new insights into the Hardy Inequality and the evolution problem associated to it by means of a reformulation of the problem. Surprisingly, the connection of the energy of the new formulation with the standard Hardy functional is nontrivial, due to the presence of a Hardy singularity energy. This corresponds to a loss for the total energy. The problem arises when the equation is posed in a bounded domain, and also when posed in the whole space. We also consider an equivalent problem with inverse square potential on an exterior domain. The extra energy term is then present as an effect that comes from infinity, a kind of hidden energy. In this case, in an unexpected way, this term is additive to the total energy, and it may even constitute the main part of it.