<p>We study interior <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C^{2,\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> regularity estimates for solutions of fully nonlinear uniformly elliptic equations of the general form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$F(D^2u)=0$</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 stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> in two independent variables and without any geometric condition on <i>F</i>. By means of the theory of divergence form equations we prove that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> solutions of the previous equation are <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C^{2,\bar{\alpha }(\lambda /\Lambda )}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>2</mn> <mo>,</mo> <mover accent="true"> <mrow> <mi>α</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">/</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> in the interior of the domain, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$0<\lambda \le \Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>λ</mi> <mo>≤</mo> <mi mathvariant="normal">Λ</mi> </mrow> </math></EquationSource> </InlineEquation> are the ellipticity constants. We finally exploit the theory of nondivergence equations in the plane to obtain <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$C^{2,\tilde{\alpha }}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>2</mn> <mo>,</mo> <mover accent="true"> <mi>α</mi> <mo stretchy="false">~</mo> </mover> </mrow> </msup> </math></EquationSource> </InlineEquation> regularity for an explicit exponent <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\tilde{\alpha }=\tilde{\alpha }(\lambda /\Lambda )>\lambda /\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>α</mi> <mo stretchy="false">~</mo> </mover> <mo>=</mo> <mover accent="true"> <mi>α</mi> <mo stretchy="false">~</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">/</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">)</mo> </mrow> <mo>></mo> <mi>λ</mi> <mo stretchy="false">/</mo> <mi mathvariant="normal">Λ</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the smoothness of solutions of fully nonlinear second order equations in the plane

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2026年2月18日

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Alessandro Goffi

摘要

We study interior $C^{2,\alpha }$ $C^{2, α}$ regularity estimates for solutions of fully nonlinear uniformly elliptic equations of the general form $F(D^2u)=0$ $F (D^{2} u) = 0$ in two independent variables and without any geometric condition on F. By means of the theory of divergence form equations we prove that $C^2$ $C^{2}$ solutions of the previous equation are $C^{2,\bar{\alpha }(\lambda /\Lambda )}$ $C^{2, \bar{α} (λ / Λ)}$ in the interior of the domain, where $0<\lambda \le \Lambda $ $0 < λ \leq Λ$ are the ellipticity constants. We finally exploit the theory of nondivergence equations in the plane to obtain $C^{2,\tilde{\alpha }}$ $C^{2, \tilde{α}}$ regularity for an explicit exponent $\tilde{\alpha }=\tilde{\alpha }(\lambda /\Lambda )>\lambda /\Lambda $ $\tilde{α} = \tilde{α} (λ / Λ) > λ / Λ$ .