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Register a new userModulus and Poincare inequalities on non-self-similar Sierpinski carpets
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A carpet is a metric space homeomorphic to the Sierpinski carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincare inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poincare inequalities: these examples have no manifold points, yet embed isometrically as subsets of Euclidean space.
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We construct symmetric self-similar Dirichlet forms on unconstrained Sierpinski carpets, which are natural extension of planner Sierpinski carpets by allowing the small cells to live off the $1/k$ grids. The intersection of two cells can be a line segment of irrational length, and we also drop the non-diagonal assumption in this recurrent setting. The proof of the existence is purely analytic. A uniqueness theorem is also provided. Moreover, the additional freedom of unconstrained Sierpinski carpets allows us to slide the cells around. In this situation, we view unconstrained Sierpinski carpets as moving fractals, and we prove that the self-similar Dirichlet forms will vary continuously in a $Gamma$-convergence sense.
Let $E$ be a Bedford-McMullen carpet associated with a set of affine mappings ${f_{ij}}_{(i,j)in G}$ and let $mu$ be the self-affine measure associated with ${f_{ij}}_{(i,j)in G}$ and a probability vector $(p_{ij})_{(i,j)in G}$. We study the asymptotics of the geometric mean error in the quantization for $mu$. Let $s_0$ be the Hausdorff dimension for $mu$. Assuming a separation condition for ${f_{ij}}_{(i,j)in G}$, we prove that the $n$th geometric error for $mu$ is of the same order as $n^{-1/s_0}$.
A well-known family of determinantal inequalities for mixed volumes of convex bodies were derived by Shephard from the Alexandrov-Fenchel inequality. The classic monograph Geometric Inequalities by Burago and Zalgaller states a conjecture on the validity of higher-order analogues of Shephards inequalities, which is attributed to Fedotov. In this note we disprove Fedotovs conjecture by showing that it contradicts the Hodge-Riemann relations for simple convex polytopes. Along the way, we make some expository remarks on the linear algebraic and geometric aspects of these inequalities.
We prove that on an essentially non-branching $mathrm{MCP}(K,N)$ space, if a geodesic ball has a volume lower bound and satisfies some additional geometric conditions, then in a smaller geodesic ball (in a quantified sense) we have an estimate on the isoperimetric constants.
We show that self-similar solutions for the mean curvature flow, surface diffusion and Willmore flow of entire graphs are stable upon perturbations of initial data with small Lipschitz norm. Roughly speaking, the perturbed solutions are asymptotically self-similar as time tends to infinity. Our results are built upon the global analytic solutions constructed by Koch and Lamm cite{KochLamm}, the compactness arguments adapted by Asai and Giga cite{Giga2014}, and the spatial equi-decay properties on certain weighted function spaces. The proof for all of the above flows are achieved in a unified framework by utilizing the estimates of the linearized operator.
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