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

Aiping Zhang; Zesheng Feng; Hongya Gao

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 3, page 915-925
  • ISSN: 0011-4642

Abstract

top
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
where , is a Carathéodory function such that is convex, and there exist constants and such that
for almost all , all and all . We show that, even if and only belong to , the interplay
implies the existence of a minimizer which belongs to .

How to cite

top

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 -

References

top
  1. Arcoya, D., Boccardo, L., 10.1016/j.jfa.2014.11.011, J. Func. Anal. 268 (2015), 1153-1166. (2015) Zbl1317.35082MR3304596DOI10.1016/j.jfa.2014.11.011
  2. Arcoya, D., Boccardo, L., 10.1016/j.matpur.2017.08.001, J. Math. Pures Appl. (9) 111 (2018), 106-125. (2018) Zbl1390.35067MR3760750DOI10.1016/j.matpur.2017.08.001
  3. Boccardo, L., Croce, G., 10.1515/9783110315424, De Gruyter Studies in Mathematics 55. Walter De Gruyter, Berlin (2013). (2013) Zbl1293.35001MR3154599DOI10.1515/9783110315424
  4. Boccardo, L., Orsina, L., Existence and regularity of minima for integral functionals noncoercive in the energy space, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 25 (1997), 95-130. (1997) Zbl1015.49014MR1655511
  5. Capone, C., Napoli, A. Passarelli di, 10.57262/ade029-0910-757, Adv. Differ. Equ. 29 (2024), 757-782. (2024) Zbl07854804MR4740327DOI10.57262/ade029-0910-757
  6. Capone, C., Radice, T., 10.1007/s41808-020-00082-w, J. Elliptic Parabol. Equ. 6 (2020), 751-771. (2020) Zbl1451.35042MR4169451DOI10.1007/s41808-020-00082-w
  7. Degiovanni, M., Marzocchi, M., 10.1016/j.na.2019.03.008, Nonlinear Anal., Theory Methods Appl., Ser. A 185 (2019), 206-215. (2019) Zbl1421.35130MR3928322DOI10.1016/j.na.2019.03.008
  8. Giusti, E., 10.1142/5002, World Scientific, Singapore (2003). (2003) Zbl1028.49001MR1962933DOI10.1142/5002
  9. Li, Z., 10.1007/s10440-018-0217-7, Acta Appl. Math. 163 (2019), 145-156. (2019) Zbl1423.35206MR4008700DOI10.1007/s10440-018-0217-7
  10. Moreno-Mérida, L., Porzio, M. M., 10.3233/ASY-191558, Asymptotic Anal. 118 (2020), 143-159. (2020) Zbl1454.35206MR4113595DOI10.3233/ASY-191558
  11. Radice, T., 10.4064/sm230104-18-3, Stud. Math. 276 (2024), 1-17. (2024) Zbl7887996MR4756832DOI10.4064/sm230104-18-3
  12. Zhang, C., Zhou, S., 10.5186/aasfm.2017.4205, Ann. Acad. Sci. Fenn., Math. 42 (2017), 95-103. (2017) Zbl1367.35072MR3558517DOI10.5186/aasfm.2017.4205

NotesEmbed ?

top

You must be logged in to post comments.