<p>We prove the weak Harnack inequality for the functions <i>u</i> which belong to the corresponding De Giorgi classes <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10231_2025_1553_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$DG^{-}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <msup> <mi>G</mi> <mo>-</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> under the additional assumption that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10231_2025_1553_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$u\in L^{s}_{loc}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>∈</mo> <msubsup> <mi>L</mi> <mrow> <mi mathvariant="italic">loc</mi> </mrow> <mi>s</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with some <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10231_2025_1553_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$s> 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In particular, our result covers new cases of functionals with a variable exponent or double-phase functionals under the non-logarithmic condition.</p>

A note on the weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth

期刊论文
发表日期：

2025年2月11日

查看全文

Mariia Savchenko,
Igor Skrypnik,
Yevgeniia Yevgenieva

摘要

We prove the weak Harnack inequality for the functions u which belong to the corresponding De Giorgi classes $DG^{-}(\Omega )$ $D G^{-} (Ω)$ under the additional assumption that $u\in L^{s}_{loc}(\Omega )$ $u \in L_{loc}^{s} (Ω)$ with some $s> 0$ $s > 0$ . In particular, our result covers new cases of functionals with a variable exponent or double-phase functionals under the non-logarithmic condition.