Regularizing effect of the interplay between coefficients in some noncoercive integral functionals

Zhang, Aiping, Feng, Zesheng, and Gao, Hongya. "Regularizing effect of the interplay between coefficients in some noncoercive integral functionals." Czechoslovak Mathematical Journal 74.3 (2024): 915-925. <http://eudml.org/doc/299304>.

@article{Zhang2024,
abstract = {We are interested in regularizing effect of the interplay between the coefficient of zero order term and the datum in some noncoercive integral functionals of the type \[ \mathcal \{J\} (v)= \int \_\Omega j(x,v,\nabla v) \{\rm d\}x +\int \_\Omega a(x) |v|^\{2\} \{\rm d\} x -\int \_\Omega fv \{\rm d\}x, \quad v\in W^\{1,2\}\_\{0\}(\Omega ), \] where $\Omega \subset \mathbb \{R\}^N$, $j$ is a Carathéodory function such that $\xi \mapsto j(x,s,\xi )$ is convex, and there exist constants $ 0\le \tau <1$ and $M>0$ such that \[ \frac\{ |\xi |^\{2\}\}\{(1+|s|)^\{\tau \}\}\le j(x,s,\xi )\le M|\xi |^2 \] for almost all $x\in \Omega $, all $s\in \mathbb \{R\}$ and all $\xi \in \mathbb \{R\}^N$. We show that, even if $0<a(x)$ and $f(x)$ only belong to $L^\{1\}(\Omega )$, the interplay \[|f(x)|\le 2 Qa(x) \] implies the existence of a minimizer $u \in W_0^\{1,2\} (\Omega )$ which belongs to $L^\{\infty \}(\Omega )$.},
author = {Zhang, Aiping, Feng, Zesheng, Gao, Hongya},
journal = {Czechoslovak Mathematical Journal},
keywords = {regularizing effect; interplay; minimizer; noncoercive integral functional},
language = {eng},
number = {3},
pages = {915-925},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Regularizing effect of the interplay between coefficients in some noncoercive integral functionals},
url = {http://eudml.org/doc/299304},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Zhang, Aiping
AU - Feng, Zesheng
AU - Gao, Hongya
TI - Regularizing effect of the interplay between coefficients in some noncoercive integral functionals
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 3
SP - 915
EP - 925
AB - We are interested in regularizing effect of the interplay between the coefficient of zero order term and the datum in some noncoercive integral functionals of the type \[ \mathcal {J} (v)= \int _\Omega j(x,v,\nabla v) {\rm d}x +\int _\Omega a(x) |v|^{2} {\rm d} x -\int _\Omega fv {\rm d}x, \quad v\in W^{1,2}_{0}(\Omega ), \] where $\Omega \subset \mathbb {R}^N$, $j$ is a Carathéodory function such that $\xi \mapsto j(x,s,\xi )$ is convex, and there exist constants $ 0\le \tau <1$ and $M>0$ such that \[ \frac{ |\xi |^{2}}{(1+|s|)^{\tau }}\le j(x,s,\xi )\le M|\xi |^2 \] for almost all $x\in \Omega $, all $s\in \mathbb {R}$ and all $\xi \in \mathbb {R}^N$. We show that, even if $0<a(x)$ and $f(x)$ only belong to $L^{1}(\Omega )$, the interplay \[|f(x)|\le 2 Qa(x) \] implies the existence of a minimizer $u \in W_0^{1,2} (\Omega )$ which belongs to $L^{\infty }(\Omega )$.
LA - eng
KW - regularizing effect; interplay; minimizer; noncoercive integral functional
UR - http://eudml.org/doc/299304
ER -