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<p>In this paper, we study the interior <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(C^2\)</EquationSource> </InlineEquation> regularity problem for the Hessian quotient equation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\left(\frac{\sigma_n}{\sigma_k}\right)(D^2u)=f\)</EquationSource> </InlineEquation>. We give a complete answer to this longstanding problem: for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(k=n-1,n-2\)</EquationSource> </InlineEquation>, we establish an interior <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(C^2\)</EquationSource> </InlineEquation> estimate; for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(k\leq n-3\)</EquationSource> </InlineEquation>, we show that interior <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(C^2\)</EquationSource> </InlineEquation> estimate fails by finding a singular solution.</p>

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Interior \(C^2\) Estimate for Hessian Quotient Equation in General Dimension

  • 期刊论文

  • 发表日期：

    2025年6月30日

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  • Siyuan Lu

摘要

In this paper, we study the interior \(C^2\) regularity problem for the Hessian quotient equation \(\left(\frac{\sigma_n}{\sigma_k}\right)(D^2u)=f\) . We give a complete answer to this longstanding problem: for \(k=n-1,n-2\) , we establish an interior \(C^2\) estimate; for \(k\leq n-3\) , we show that interior \(C^2\) estimate fails by finding a singular solution.