In this paper we obtain the strong maximum principle for the following degenerate elliptic quasilinear equation <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1977_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="195" /> </InlineMediaObject> <EquationSource Format="TEX">$G_h \bigl ( Du, D^2u \bigr ) = f(u,|Du|)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mi>h</mi> </msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>D</mi> <mi>u</mi> <mo>,</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> <mi>u</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mo stretchy="false">|</mo> <mi>D</mi> <mi>u</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in an n-dimensional domain, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1977_Article_IEq2.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="323" /> </InlineMediaObject> <EquationSource Format="TEX">$G_h \bigl ( Du, D^2u \bigr ) = |Du|^{h-3} \sum _{i,j=1}^n D_i u D_j u D_{ij} u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mi>h</mi> </msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>D</mi> <mi>u</mi> <mo>,</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> <mi>u</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>D</mi> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>h</mi> <mo>-</mo> <mn>3</mn> </mrow> </msup> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msub> <mi>D</mi> <mi>i</mi> </msub> <mi>u</mi> <msub> <mi>D</mi> <mi>j</mi> </msub> <mi>u</mi> <msub> <mi>D</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msub> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation>, which equals the h-homogeneous infinity-Laplacian <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1977_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta _{\infty }^h u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="normal">Δ</mi> <mrow> <mi>∞</mi> </mrow> <mi>h</mi> </msubsup> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation> of u when <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1977_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$Du \ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mi>u</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1977_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="204" /> </InlineMediaObject> <EquationSource Format="TEX">$f:[0,\infty ) \times [0,\infty ) \rightarrow [0,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> satisfies some conditions.

Strong maximum principle for some degenerate elliptic quasilinear equation involving infinity-Laplacian

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2025年9月23日

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Makoto Kodama,
Kazuhiro Takimoto

摘要

In this paper we obtain the strong maximum principle for the following degenerate elliptic quasilinear equation $G_h \bigl ( Du, D^2u \bigr ) = f(u,|Du|)$ $G_{h} (D u, D^{2} u) = f (u, | D u |)$ in an n-dimensional domain, where $G_h \bigl ( Du, D^2u \bigr ) = |Du|^{h-3} \sum _{i,j=1}^n D_i u D_j u D_{ij} u$ $G_{h} (D u, D^{2} u) = {| D u |}^{h - 3} \sum_{i, j = 1}^{n} D_{i} u D_{j} u D_{ij} u$ , which equals the h-homogeneous infinity-Laplacian $\Delta _{\infty }^h u$ $Δ_{\infty}^{h} u$ of u when $Du \ne 0$ $D u \neq 0$ , and $f:[0,\infty ) \times [0,\infty ) \rightarrow [0,\infty )$ $f : [0, \infty) \times [0, \infty) \to [0, \infty)$ satisfies some conditions.