engleski

Maximally singular weak solutions of Poisson equations

It is known that there exists an explicit function F in L^2(Ω), where Ω is a given bounded open subset of R, such that the corresponding weak solution of the Poisson BVP −∆u = F(x), u ∈ H_0^1(Ω), is maximally singular ; that is, the singular set of u (defined in the Introduction) has the Hausdorff dimension equal to (N − 4)+. This constant is optimal, i.e., the largest possible. Here, we show that much more is true: when N ≥ 5, there exists F ∈ L^2(Ω) such that the corresponding weak solution has the pointwise concentration of singular set of u, in the sense of the Hausdorff dimension, equal to N − 4 at all points of Ω. We also consider the problem of generating weak solutions with the property of contrast ; that is, we construct solutions u that are regular (more specifically, of class C^2, α loc for arbitrary α ∈ (0, 1)) in any prescribed open subset Ωr of Ω, while they are maximally singular in its complement Ω \ Ωr . We indicate several open problems.

Posisson equation ; weak solution ; regularity ; singularity ; maximally singular weak solution ; Hausdorff dimension ; box dimension ; generalized Cantor set ; Stein’s trick

rad je prihvaćen za objavljivanje 08.07.2020.