<p>We modify Pogorelov’s classic construction to demonstrate the absence of a priori <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> estimates for the equations <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\det (D^2 u \pm Du \otimes Du) = f(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">det</mo> <mrow> <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>D</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in dimension <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n \ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We construct a sequence of solutions <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$z_\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>z</mi> <mi>ε</mi> </msub> </math></EquationSource> </InlineEquation> with second derivatives blowing up at the origin as <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\varepsilon \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, while the corresponding right-hand sides <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$f_\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>f</mi> <mi>ε</mi> </msub> </math></EquationSource> </InlineEquation> admit uniform <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$C^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> estimates. Specifically, the counterexamples are given by <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$z_\varepsilon (x_1, \dots , x_n) = (1+x_1^2)(1+x_2^2)(\varepsilon ^2 + \eta ^2)^{\alpha /2},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>z</mi> <mi>ε</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msubsup> <mi>x</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msubsup> <mi>x</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>ε</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>η</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>α</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\eta = \sqrt{x_3^2 + \dots + x_n^2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>η</mi> <mo>=</mo> <msqrt> <mrow> <msubsup> <mi>x</mi> <mn>3</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>x</mi> <mi>n</mi> <mn>2</mn> </msubsup> </mrow> </msqrt> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\alpha = 2 - \frac{2}{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mn>2</mn> <mo>-</mo> <mfrac> <mn>2</mn> <mi>n</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>.</p>

On counterexamples to interior $C^2$ estimates for Monge-Ampère type equations

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2026年6月25日

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Cheuk Yan Fung

摘要

We modify Pogorelov’s classic construction to demonstrate the absence of a priori $C^2$ $C^{2}$ estimates for the equations $\det (D^2 u \pm Du \otimes Du) = f(x)$ $det (D^{2} u \pm D u \otimes D u) = f (x)$ in dimension $n \ge 3$ $n \geq 3$ . We construct a sequence of solutions $z_\varepsilon $ $z_{ε}$ with second derivatives blowing up at the origin as $\varepsilon \rightarrow 0$ $ε \to 0$ , while the corresponding right-hand sides $f_\varepsilon $ $f_{ε}$ admit uniform $C^2$ $C^{2}$ estimates. Specifically, the counterexamples are given by $z_\varepsilon (x_1, \dots , x_n) = (1+x_1^2)(1+x_2^2)(\varepsilon ^2 + \eta ^2)^{\alpha /2},$ $z_{ε} (x_{1}, \dots, x_{n}) = (1 + x_{1}^{2}) (1 + x_{2}^{2}) {(ε^{2} + η^{2})}^{α / 2},$ where $\eta = \sqrt{x_3^2 + \dots + x_n^2}$ $η = \sqrt{x_{3}^{2} + \dots + x_{n}^{2}}$ and $\alpha = 2 - \frac{2}{n}$ $α = 2 - \frac{2}{n}$ .

On counterexamples to interior \(C^2\) estimates for Monge-Ampère type equations

摘要