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<p>In this note, we generalize Savin’s small perturbation theorem [<CitationRef CitationID="CR31">31</CitationRef>] to nonhomogeneous fully nonlinear equations <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(F(D^2u, Du, u,x)=f\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> <mi>u</mi> <mo>,</mo> <mi>D</mi> <mi>u</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> provided the coefficients and the right-hand side terms are Hölder small perturbations. As an application, we establish a partial regularity result for the sigma-<i>k</i> Hessian equation <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\sigma _{k}(D^2u)=f\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

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A generalization of Savin’s small perturbation theorem for fully nonlinear elliptic equations and applications

  • 期刊论文

  • 发表日期：

    2026年4月30日

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  • Zhenyu Fan

摘要

In this note, we generalize Savin’s small perturbation theorem [31] to nonhomogeneous fully nonlinear equations \(F(D^2u, Du, u,x)=f\) F ( D 2 u , D u , u , x ) = f provided the coefficients and the right-hand side terms are Hölder small perturbations. As an application, we establish a partial regularity result for the sigma-k Hessian equation \(\sigma _{k}(D^2u)=f\) σ k ( D 2 u ) = f .