Functional Aspects of the Hardy Inequality: Appearance of a Hidden Energy

Abstract

Starting with a functional difficulty appeared in the paper [14] 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.

Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions

“…coming from inequality (48). Then proceeding as on the Step 1, it can easily be shown that the limit pair (A, y) is admissible to OCP (15)- (18). Hence, the condition (79) 1 is valid.…”

“…The distinguishing feature of optimal control problem (15)- (18) is the fact that the matrix-valued control A ∈ A ad is merely measurable and belongs to the space L 2 Ω; M N (rather than the space of bounded matrices L ∞ Ω; M N ). The unboundedness of the skew-symmetric part of matrix A ∈ A ad can have a reflection in non-uniqueness of weak solutions to the corresponding boundary value problem.…”

“…be a sequence of optimal solutions to regularized problems (38)-(40), where χ Ωε f ε → f strongly in H −1 (Ω). Then there exists an optimal pair (A 0 , y 0 ) ∈ A ad to the original OCP (15)- (18), which is attainable in the following sense Proof. In order to show that this result is a direct consequence of Theorem 1.3, it is enough to establish the compactness property for the sequence of optimal solutions (A 0 ε , v 0 ε , y 0 ε ) ∈ Ξ ε ε>0 in the sense of Definition 3.4.…”

“…Let p 0 ε ∈ H 1 0 (Ω ε ; ∂Ω) ε>0 be a sequence of corresponding adjoint states. Then, the fulfilment of the Hypotheses (H1)-(H2) implies that (A 0 , y 0 ) ∈ A ad × H 1 0 (Ω) is an optimal pair to the original OCP (15)- (18) and there exists an element ψ ∈ H 1 0 (Ω) such that…”

“…Proof. To begin with, we note that due to Theorem 3.6, the sequence of optimal solutions (A 0 ε , v 0 ε , y 0 ε ) ∈ Ξ ε ε>0 to the regularized problems (38)-(40) is compact with respect to w-convergence and each of its w-cluster pairs (A 0 , y 0 ) is an optimal pair to the original problem (15)- (18). Hence, (A 0 , y 0 ) ∈ A ad , and the limit passage in (103)-(104) as ε → 0 leads us to the relation (127) in the sense of distributions.…”