The resolvent kernel for PCF self-similar fractals

Abstract

For the Laplacian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:annotation encoding="application/x-tex">\Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defined on a p.c.f. self-similar fractal, we give an explicit formula for the resolvent kernel of the Laplacian with Dirichlet boundary conditions and also with Neumann boundary conditions. That is, we construct a symmetric function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Superscript left-parenthesis lamda right-parenthesis"> <mml:semantics> <mml:msup> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi> λ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">G^{(\lambda )}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which solves <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis lamda double-struck upper I minus normal upper Delta right-parenthesis Superscript negative 1 Baseline f left-parenthesis x right-parenthesis equals integral upper G Superscript left-parenthesis lamda right-parenthesis Baseline left-parenthesis x comma y right-parenthesis f left-parenthesis y right-parenthesis d mu left-parenthesis y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> λ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">I</mml:mi> </mml:mrow> <mml:mo> − </mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo> ∫ </mml:mo> <mml:msup> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi> λ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>d</mml:mi> <mml:mi> μ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\lambda \mathbb {I} - \Delta )^{-1} f(x) = \int G^{(\lambda )}(x,y) f(y) \, d\mu (y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The method is similar to Kigami’s construction of the Green kernel and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Superscript left-parenthesis lamda right-parenthesis"> <mml:semantics> <mml:msup> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi> λ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">G^{(\lambda )}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is expressed as a sum of scaled and “translated” copies of a certain function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi Superscript left-parenthesis lamda right-parenthesis"> <mml:semantics> <mml:msup> <mml:mi> ψ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi> λ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\psi ^{(\lambda )}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which may be considered as a fundamental solution of the resolvent equation. Examples of the explicit resolvent kernel formula are given for the unit interval, standard Sierpinski gasket, and the level-3 Sierpinski gasket <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper G 3"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">SG_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .