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<p>We consider a wide class of fully nonlinear integro-differential equations that degenerate when the gradient of the solution vanishes. By using compactness and perturbation arguments, we give a complete characterization of the regularity of viscosity solutions according to different diffusion orders. More precisely, when the order of the fractional diffusion is sufficiently close to 2, we obtain Hölder continuity for the gradient of any viscosity solutions and further derive an improved gradient regularity estimate at the origin. For the order of the fractional diffusion in the interval (1,&#xa0;2), we prove that there is at least one solution of class <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3023_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">\(C^{1, \alpha }_\textrm{loc}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>C</mi> <mtext>loc</mtext> <mrow> <mn>1</mn> <mo>,</mo> <mi>α</mi> </mrow> </msubsup> </math></EquationSource> </InlineEquation>. Additionally, if the order of the fractional diffusion is in the interval (0,&#xa0;1], the local Hölder continuity of solutions is inferred.</p>

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Regularity for a class of degenerate fully nonlinear nonlocal elliptic equations

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月27日

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  • Yuzhou Fang,
  • Vicenţiu D. Rădulescu,
  • Chao Zhang

摘要

We consider a wide class of fully nonlinear integro-differential equations that degenerate when the gradient of the solution vanishes. By using compactness and perturbation arguments, we give a complete characterization of the regularity of viscosity solutions according to different diffusion orders. More precisely, when the order of the fractional diffusion is sufficiently close to 2, we obtain Hölder continuity for the gradient of any viscosity solutions and further derive an improved gradient regularity estimate at the origin. For the order of the fractional diffusion in the interval (1, 2), we prove that there is at least one solution of class \(C^{1, \alpha }_\textrm{loc}\) C loc 1 , α . Additionally, if the order of the fractional diffusion is in the interval (0, 1], the local Hölder continuity of solutions is inferred.