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December 2021 Maximally singular weak solutions of Laplace equations
Josipa-Pina Milišić, Darko Žubrinić
Rocky Mountain J. Math. 51(6): 2147-2157 (December 2021). DOI: 10.1216/rmj.2021.51.2147


It is known that there exists an explicit function F in L2(Ω), where Ω is a given bounded open subset of N, such that the corresponding weak solution of the Poisson boundary value problem Δu=F(x), uH01(Ω), is maximally singular; that is, the singular set of u has Hausdorff dimension equal to (N4)+. This constant is optimal, i.e., the largest possible. Here, we show that much more is true: when N5, there exists FL2(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 N4 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 Cloc2,α for arbitrary α(0,1)) in any prescribed open subset Ωr of Ω, while they are maximally singular in its complement ΩΩr. We indicate several open problems.


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Josipa-Pina Milišić. Darko Žubrinić. "Maximally singular weak solutions of Laplace equations." Rocky Mountain J. Math. 51 (6) 2147 - 2157, December 2021.


Received: 15 June 2021; Published: December 2021
First available in Project Euclid: 22 March 2022

Digital Object Identifier: 10.1216/rmj.2021.51.2147

Primary: 26A30 , 28A75 , 28A78

Keywords: box dimension , generalized Cantor set , Hausdorff dimension , maximally singular weak solution , Poisson equation , regularity , singularity , Stein’s trick , Weak solution

Rights: Copyright © 2021 Rocky Mountain Mathematics Consortium


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Vol.51 • No. 6 • December 2021
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