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<p>In this paper, we are concerned with non-divergence form parabolic equations in a bounded cylindrical domain with a finite number of subdomains. Under the assumption that the interfacial boundaries are <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C^{1,\text {Dini}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mtext>Dini</mtext> </mrow> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(C^{\gamma _0}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <msub> <mi>γ</mi> <mn>0</mn> </msub> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\gamma _0&gt;\frac{1}{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mn>0</mn> </msub> <mo>&gt;</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation> in the spatial variables and the time variable, respectively, we derive the Hessian estimates and piecewise <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(C^{1,2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>-regularity provided the coefficients and the inhomogeneous terms are of piecewise Dini mean oscillation. The corresponding results for the adjoint operator are also established.</p>

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Partial estimates for non-divergence form parabolic equations with piecewise DMO coefficients

  • 期刊论文

  • 发表日期：

    2025年12月5日

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  • Huaijun Teng,
  • Longjuan Xu

摘要

In this paper, we are concerned with non-divergence form parabolic equations in a bounded cylindrical domain with a finite number of subdomains. Under the assumption that the interfacial boundaries are \(C^{1,\text {Dini}}\) C 1 , Dini and \(C^{\gamma _0}\) C γ 0 with \(\gamma _0>\frac{1}{2}\) γ 0 > 1 2 in the spatial variables and the time variable, respectively, we derive the Hessian estimates and piecewise \(C^{1,2}\) C 1 , 2 -regularity provided the coefficients and the inhomogeneous terms are of piecewise Dini mean oscillation. The corresponding results for the adjoint operator are also established.