<p>We prove local Hölder continuity for nonnegative, locally bounded, local weak solutions to the class of doubly nonlinear parabolic equations <Equation ID="Equ57"> <EquationSource Format="TEX">$\begin{aligned} \partial _t (u^q)- \text {div}(\left| {Du}\right| ^{p-2} Du) = 0, \qquad p>2, \quad 0<q<p-1 . \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>u</mi> <mi>q</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mtext>div</mtext> <mrow> <mo stretchy="false">(</mo> <msup> <mfenced close="|" open="|"> <mrow> <mi mathvariant="italic">Du</mi> </mrow> </mfenced> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>D</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="2em" /> <mi>p</mi> <mo>></mo> <mn>2</mn> <mo>,</mo> <mspace width="1em" /> <mn>0</mn> <mo><</mo> <mi>q</mi> <mo><</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>The proof relies on expansion of positivity results combined with the study of an alternative (related to DeGiorgi-type lemmas) and an exponential shift which allows us to deal with the intrinsic geometry associated with the problem.</p>

Hölder regularity for a class of doubly nonlinear PDEs

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2026年1月19日

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Filippo Maria Cassanello,
Eurica Henriques

摘要

We prove local Hölder continuity for nonnegative, locally bounded, local weak solutions to the class of doubly nonlinear parabolic equations $\begin{aligned} \partial _t (u^q)- \text {div}(\left| {Du}\right| ^{p-2} Du) = 0, \qquad p>2, \quad 0<q<p-1 . \end{aligned}$ $\begin{matrix} \partial_{t} (u^{q}) - div ({|Du|}^{p - 2} D u) = 0, p > 2, 0 < q < p - 1 . \end{matrix}$ The proof relies on expansion of positivity results combined with the study of an alternative (related to DeGiorgi-type lemmas) and an exponential shift which allows us to deal with the intrinsic geometry associated with the problem.