The Robin heat kernel and its expansion via Robin eigenfunctions

We prove the existence and uniqueness of the Robin heat kernel on compact Riemannian manifolds with smooth boundary for Robin parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha element-of double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mi> α </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha \in \mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , expressed as a spectral expansion in terms of Robin eigenvalues and eigenfunctions. For the non-negative parameter regime ( <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mi> α </mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha \ge 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ), we present a direct proof based on trace Sobolev inequalities and eigenfunction estimates. The case of negative parameters ( <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi> α </mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha &gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) requires novel analytical techniques to handle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> estimates of Robin eigenfunctions, addressing challenges not present in the non-negative case. Our result extends the classical Dirichlet and Neumann cases to the less-studied negative parameter regime.