<p>In the framework of Potential Theory we prove existence for the Perron-Weiner-Brelot-Bauer solution to the Dirichlet problem related to a family of totally degenerate, in the sense of Bony, differential operators. We also state and prove a Wiener-type criterium and an exterior cone condition for the regularity of a boundary point. Our results apply to a wide family of strongly degenerate operators that includes the following example <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathcal {L}}= t^2\Delta _x + \langle x, \nabla _y \rangle -\partial _t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo>=</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> <msub> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </msub> <mo>+</mo> <mrow> <mo stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <msub> <mi mathvariant="normal">∇</mi> <mi>y</mi> </msub> <mo stretchy="false">⟩</mo> </mrow> <mo>-</mo> <msub> <mi>∂</mi> <mi>t</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(x,y,t) \in {{\mathbb {R}}}^N \times {{\mathbb {R}}}^{N} \times {{\mathbb {R}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>×</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

The Dirichlet problem for a family of totally degenerate differential operators

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发表日期：

2025年8月20日

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Maria Manfredini,
Mirco Piccinini,
Sergio Polidoro

摘要

In the framework of Potential Theory we prove existence for the Perron-Weiner-Brelot-Bauer solution to the Dirichlet problem related to a family of totally degenerate, in the sense of Bony, differential operators. We also state and prove a Wiener-type criterium and an exterior cone condition for the regularity of a boundary point. Our results apply to a wide family of strongly degenerate operators that includes the following example ${\mathcal {L}}= t^2\Delta _x + \langle x, \nabla _y \rangle -\partial _t$ $L = t^{2} Δ_{x} + ⟨ x, \nabla_{y} ⟩ - \partial_{t}$ , for $(x,y,t) \in {{\mathbb {R}}}^N \times {{\mathbb {R}}}^{N} \times {{\mathbb {R}}}$ $(x, y, t) \in R^{N} \times R^{N} \times R$ .