We discuss the self-adjointness in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-setting of the operators acting as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$-\nabla \cdot h\nabla $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mi>h</mi> <mi mathvariant="normal">∇</mi> </mrow> </math></EquationSource> </InlineEquation>, with piecewise constant functions h having a jump along a Lipschitz hypersurface <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation>, without explicit assumptions on the sign of h. We establish a number of sufficient conditions for the self-adjointness of the operator with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H^s$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>s</mi> </msup> </math></EquationSource> </InlineEquation>-regularity for suitable <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$s\in [1,\frac{3}{2}]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, in terms of the jump value and the regularity and geometric properties of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation>. An important intermediate step is a link with Fredholm properties of the Neumann-Poincaré operator on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation>, which is new for the Lipschitz setting.

On Neumann-Poincaré operators and self-adjoint transmission problems

期刊论文
发表日期：

2025年9月29日

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Badreddine Benhellal,
Konstantin Pankrashkin

摘要

We discuss the self-adjointness in $L^2$ $L^{2}$ -setting of the operators acting as $-\nabla \cdot h\nabla $ $- \nabla \cdot h \nabla$ , with piecewise constant functions h having a jump along a Lipschitz hypersurface $\Sigma $ $Σ$ , without explicit assumptions on the sign of h. We establish a number of sufficient conditions for the self-adjointness of the operator with $H^s$ $H^{s}$ -regularity for suitable $s\in [1,\frac{3}{2}]$ $s \in [1, \frac{3}{2}]$ , in terms of the jump value and the regularity and geometric properties of $\Sigma $ $Σ$ . An important intermediate step is a link with Fredholm properties of the Neumann-Poincaré operator on $\Sigma $ $Σ$ , which is new for the Lipschitz setting.