Non-linear Gagliardo-Nirenberg inequality involving a second-order elliptic operator in non-divergent form

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F60076658%3A12410%2F25%3A43910132" target="_blank" >RIV/60076658:12410/25:43910132 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/00216208:11320/25:10513201

Výsledek na webu

<a href="https://link.springer.com/article/10.1007/s00030-025-01124-9" target="_blank" >https://link.springer.com/article/10.1007/s00030-025-01124-9</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00030-025-01124-9" target="_blank" >10.1007/s00030-025-01124-9</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Non-linear Gagliardo-Nirenberg inequality involving a second-order elliptic operator in non-divergent form

Popis výsledku v původním jazyce

We obtain the inequalities of the form integral Omega|del u(x)|2h(u(x))dx &lt;= C integral Omega|Pu(x)||TH(u(x))|2h(u(x))dx+Theta,documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$int _{Omega }|nabla u(x)|&lt;^&gt;2h(u(x)),d xle Cint _{Omega } left( sqrt{ |P u(x)||{mathcal {T}}_{H}(u(x))|}right) &lt;^&gt;{2}h(u(x)),d x +Theta ,$$end{document}where Omega subset of Rndocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Omega subset {textbf{R}&lt;^&gt;{n}}$$end{document} is a bounded Lipschitz domain, u is an element of Wloc2,1(Omega)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$uin W&lt;^&gt;{2,1}_{textrm{loc}}(Omega )$$end{document} is non-negative, P is a uniformly elliptic operator in non-divergent form, TH(&lt;middle dot&gt;)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathcal {T}}_{H}(cdot )$$end{document} is certain transformation of the monotone C1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$C&lt;^&gt;1$$end{document} function H(&lt;middle dot&gt;)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$H(cdot )$$end{document}, which is the primitive of the weight h(&lt;middle dot&gt;)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$h(cdot )$$end{document}, and Thetadocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Theta $$end{document} is the boundary term which depends on boundary values of u and del udocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$nabla u$$end{document}, which hold under some additional assumptions. Our results are linked to some results from probability and potential theories, e.g. to some variants of the Douglas formulae.

Název v anglickém jazyce

Non-linear Gagliardo-Nirenberg inequality involving a second-order elliptic operator in non-divergent form

Popis výsledku anglicky

We obtain the inequalities of the form integral Omega|del u(x)|2h(u(x))dx &lt;= C integral Omega|Pu(x)||TH(u(x))|2h(u(x))dx+Theta,documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$int _{Omega }|nabla u(x)|&lt;^&gt;2h(u(x)),d xle Cint _{Omega } left( sqrt{ |P u(x)||{mathcal {T}}_{H}(u(x))|}right) &lt;^&gt;{2}h(u(x)),d x +Theta ,$$end{document}where Omega subset of Rndocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Omega subset {textbf{R}&lt;^&gt;{n}}$$end{document} is a bounded Lipschitz domain, u is an element of Wloc2,1(Omega)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$uin W&lt;^&gt;{2,1}_{textrm{loc}}(Omega )$$end{document} is non-negative, P is a uniformly elliptic operator in non-divergent form, TH(&lt;middle dot&gt;)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathcal {T}}_{H}(cdot )$$end{document} is certain transformation of the monotone C1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$C&lt;^&gt;1$$end{document} function H(&lt;middle dot&gt;)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$H(cdot )$$end{document}, which is the primitive of the weight h(&lt;middle dot&gt;)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$h(cdot )$$end{document}, and Thetadocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Theta $$end{document} is the boundary term which depends on boundary values of u and del udocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$nabla u$$end{document}, which hold under some additional assumptions. Our results are linked to some results from probability and potential theories, e.g. to some variants of the Douglas formulae.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2025

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS

ISSN

1021-9722

e-ISSN

1420-9004

Svazek periodika

32

Číslo periodika v rámci svazku

6

Stát vydavatele periodika

CH - Švýcarská konfederace

Počet stran výsledku

34

Strana od-do

nestránkováno

Kód UT WoS článku

001554505500003

EID výsledku v databázi Scopus

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