The non-uniqueness of the limit solutions of the scalar Chern-Simons equations with signed measures

Adilson Eduardo Presoto

Mathematica Bohemica (2021)

  • Volume: 146, Issue: 3, page 235-249
  • ISSN: 0862-7959

Abstract

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We investigate the effect of admitting signed measures as a datum at the scalar Chern-Simons equation - Δ u + e u ( e u - 1 ) = μ in Ω with the Dirichlet boundary condition. Approximating μ by a sequence ( μ n ) n ∈ ℕ of L 1 functions or finite signed measures such that this equation has a solution u n for each n ∈ ℕ , we are interested in establishing the convergence of the sequence ( u n ) n ∈ ℕ to a function u # and describing the form of the measure which appears on the right-hand side of the scalar Chern-Simons equation solved by u # .

How to cite

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Presoto, Adilson Eduardo. "The non-uniqueness of the limit solutions of the scalar Chern-Simons equations with signed measures." Mathematica Bohemica 146.3 (2021): 235-249. <http://eudml.org/doc/297571>.

@article{Presoto2021,
abstract = {We investigate the effect of admitting signed measures as a datum at the scalar Chern-Simons equation \[ -\Delta u + \{\rm e\}^u(\{\rm e\}^u-1) =\mu \quad \mbox\{in\}\ \Omega \] with the Dirichlet boundary condition. Approximating $\mu $ by a sequence $(\mu _n)_\{n \in \mathbb \{N\}\}$ of $L^1$ functions or finite signed measures such that this equation has a solution $u_n$ for each $n\in \mathbb \{N\}$, we are interested in establishing the convergence of the sequence $(u_n)_\{n\in \mathbb \{N\}\}$ to a function $u^\{\#\}$ and describing the form of the measure which appears on the right-hand side of the scalar Chern-Simons equation solved by $u^\{\#\}$.},
author = {Presoto, Adilson Eduardo},
journal = {Mathematica Bohemica},
keywords = {elliptic equation; exponential nonlinearity; scalar Chern-Simons equation; signed measure},
language = {eng},
number = {3},
pages = {235-249},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The non-uniqueness of the limit solutions of the scalar Chern-Simons equations with signed measures},
url = {http://eudml.org/doc/297571},
volume = {146},
year = {2021},
}

TY - JOUR
AU - Presoto, Adilson Eduardo
TI - The non-uniqueness of the limit solutions of the scalar Chern-Simons equations with signed measures
JO - Mathematica Bohemica
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 146
IS - 3
SP - 235
EP - 249
AB - We investigate the effect of admitting signed measures as a datum at the scalar Chern-Simons equation \[ -\Delta u + {\rm e}^u({\rm e}^u-1) =\mu \quad \mbox{in}\ \Omega \] with the Dirichlet boundary condition. Approximating $\mu $ by a sequence $(\mu _n)_{n \in \mathbb {N}}$ of $L^1$ functions or finite signed measures such that this equation has a solution $u_n$ for each $n\in \mathbb {N}$, we are interested in establishing the convergence of the sequence $(u_n)_{n\in \mathbb {N}}$ to a function $u^{\#}$ and describing the form of the measure which appears on the right-hand side of the scalar Chern-Simons equation solved by $u^{\#}$.
LA - eng
KW - elliptic equation; exponential nonlinearity; scalar Chern-Simons equation; signed measure
UR - http://eudml.org/doc/297571
ER -

References

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