A new approach to solving a quasilinear boundary value problem with p -Laplacian using optimization

Michaela Bailová; Jiří Bouchala

Applications of Mathematics (2023)

  • Volume: 68, Issue: 4, page 425-439
  • ISSN: 0862-7940

Abstract

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We present a novel approach to solving a specific type of quasilinear boundary value problem with p -Laplacian that can be considered an alternative to the classic approach based on the mountain pass theorem. We introduce a new way of proving the existence of nontrivial weak solutions. We show that the nontrivial solutions of the problem are related to critical points of a certain functional different from the energy functional, and some solutions correspond to its minimum. This idea is new even for p = 2 . We present an algorithm based on the introduced theory and apply it to the given problem. The algorithm is illustrated by numerical experiments and compared with the classic approach.

How to cite

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Bailová, Michaela, and Bouchala, Jiří. "A new approach to solving a quasilinear boundary value problem with $p$-Laplacian using optimization." Applications of Mathematics 68.4 (2023): 425-439. <http://eudml.org/doc/299559>.

@article{Bailová2023,
abstract = {We present a novel approach to solving a specific type of quasilinear boundary value problem with $p$-Laplacian that can be considered an alternative to the classic approach based on the mountain pass theorem. We introduce a new way of proving the existence of nontrivial weak solutions. We show that the nontrivial solutions of the problem are related to critical points of a certain functional different from the energy functional, and some solutions correspond to its minimum. This idea is new even for $p=2$. We present an algorithm based on the introduced theory and apply it to the given problem. The algorithm is illustrated by numerical experiments and compared with the classic approach.},
author = {Bailová, Michaela, Bouchala, Jiří},
journal = {Applications of Mathematics},
keywords = {$p$-Laplacian operator; quasilinear elliptic PDE; critical point and value; optimization algorithm; gradient method},
language = {eng},
number = {4},
pages = {425-439},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A new approach to solving a quasilinear boundary value problem with $p$-Laplacian using optimization},
url = {http://eudml.org/doc/299559},
volume = {68},
year = {2023},
}

TY - JOUR
AU - Bailová, Michaela
AU - Bouchala, Jiří
TI - A new approach to solving a quasilinear boundary value problem with $p$-Laplacian using optimization
JO - Applications of Mathematics
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 4
SP - 425
EP - 439
AB - We present a novel approach to solving a specific type of quasilinear boundary value problem with $p$-Laplacian that can be considered an alternative to the classic approach based on the mountain pass theorem. We introduce a new way of proving the existence of nontrivial weak solutions. We show that the nontrivial solutions of the problem are related to critical points of a certain functional different from the energy functional, and some solutions correspond to its minimum. This idea is new even for $p=2$. We present an algorithm based on the introduced theory and apply it to the given problem. The algorithm is illustrated by numerical experiments and compared with the classic approach.
LA - eng
KW - $p$-Laplacian operator; quasilinear elliptic PDE; critical point and value; optimization algorithm; gradient method
UR - http://eudml.org/doc/299559
ER -

References

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