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<p>This paper is devoted to establishing the optimal gradient Hölder regularity for weak solutions to nonlinear sub-elliptic systems with drift terms for the super-quadratic growth, under controllable structure conditions and natural structure conditions in the Heisenberg group, respectively. The technique of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {A}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>-harmonic approximation introduced by Simon and developed by Duzaar and Grotowski is adapted to our context, and then several <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Gamma ^{1,\gamma } (0&lt;\gamma &lt;1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="normal">Γ</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>γ</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>&lt;</mo> <mi>γ</mi> <mo>&lt;</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> regularity results are obtained in the sense of the Folland-Stein space. In particular, we establish the optimal Hölder exponent for horizontal gradients of vector-valued weak solutions on its regular set directly. The primary model covered by our analysis is the non-degenerate sub-elliptic <i>p</i>-Laplacian system with the drift term.</p>

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Gradient Hölder regularity for weak solutions of nonlinear sub-elliptic systems with drift terms in the Heisenberg group

  • 期刊论文

  • 发表日期：

    2026年3月23日

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  • Jialin Wang,
  • Guoqiang Duan,
  • Dongni Liao

摘要

This paper is devoted to establishing the optimal gradient Hölder regularity for weak solutions to nonlinear sub-elliptic systems with drift terms for the super-quadratic growth, under controllable structure conditions and natural structure conditions in the Heisenberg group, respectively. The technique of \(\mathcal {A}\) A -harmonic approximation introduced by Simon and developed by Duzaar and Grotowski is adapted to our context, and then several \(\Gamma ^{1,\gamma } (0<\gamma <1)\) Γ 1 , γ ( 0 < γ < 1 ) regularity results are obtained in the sense of the Folland-Stein space. In particular, we establish the optimal Hölder exponent for horizontal gradients of vector-valued weak solutions on its regular set directly. The primary model covered by our analysis is the non-degenerate sub-elliptic p-Laplacian system with the drift term.