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- W2083307655 abstract "We study the extension problem on the Sierpinski Gasket (SG). In the first part we consider minimizing the functional $mathcal {E}_{lambda }(f) = mathcal {E}(f,f) + lambda int f^{2} d mu $ with prescribed values at a finite set of points where $mathcal {E}$ denotes the energy (the analog of $int |nabla f|^{2}$ in Euclidean space) and μ denotes the standard self-similiar measure on SG. We explicitly construct the minimizer $f(x) = sum _{i} c_{i} G_{lambda }(x_{i}, x)$ for some constants c i , where G λ is the resolvent for λ≥0. We minimize the energy over sets in SG by calculating the explicit quadratic form $mathcal {E}(f)$ of the minimizer f. We consider properties of this quadratic form for arbitrary sets and then analyze some specific sets. One such set we consider is the bottom row of a graph approximation of SG. We describe both the quadratic form and a discretized form in terms of Haar functions which corresponds to the continuous result established in a previous paper. In the second part, we study a similar problem this time minimizing $int _{SG} |Delta f(x)|^{2} d mu (x)$ for general measures. In both cases, by using standard methods we show the existence and uniqueness to the minimization problem. We then study properties of the unique minimizers." @default.
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- W2083307655 date "2014-05-31" @default.
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- W2083307655 title "Extensions and their Minimizations on the Sierpinski Gasket" @default.
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- W2083307655 doi "https://doi.org/10.1007/s11118-014-9415-8" @default.
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