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Discrete & Continuous Dynamical Systems - S

February 2019 , Volume 12 , Issue 1

Issue on variational convergence and degeneracies in PDES: Fractal domains, composite media, dynamical boundary conditions

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Raffaela Capitanelli, Maria Rosaria Lancia and Maria Agostina Vivaldi
2019, 12(1): ⅰ-ⅰ doi: 10.3934/dcdss.201901i +[Abstract](137) +[HTML](49) +[PDF](71.01KB)
Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets
Kevin Arfi and Anna Rozanova-Pierrat
2019, 12(1): 1-26 doi: 10.3934/dcdss.2019001 +[Abstract](146) +[HTML](34) +[PDF](438.0KB)

In the framework of the Laplacian transport, described by a Robin boundary value problem in an exterior domain in \begin{document} $\mathbb{R}^n$ \end{document}, we generalize the definition of the Poincaré-Steklov operator to \begin{document} $d$ \end{document}-set boundaries, \begin{document} $n-2< d<n$ \end{document}, and give its spectral properties to compare to the spectra of the interior domain and also of a truncated domain, considered as an approximation of the exterior case. The well-posedness of the Robin boundary value problems for the truncated and exterior domains is given in the general framework of \begin{document} $n$ \end{document}-sets. The results are obtained thanks to a generalization of the continuity and compactness properties of the trace and extension operators in Sobolev, Lebesgue and Besov spaces, in particular, by a generalization of the classical Rellich-Kondrachov Theorem of compact embeddings for \begin{document} $n$ \end{document} and \begin{document} $d$ \end{document}-sets.

Dual graphs and modified Barlow-Bass resistance estimates for repeated barycentric subdivisions
Daniel J. Kelleher, Hugo Panzo, Antoni Brzoska and Alexander Teplyaev
2019, 12(1): 27-42 doi: 10.3934/dcdss.2019002 +[Abstract](151) +[HTML](33) +[PDF](812.12KB)

We prove Barlow-Bass type resistance estimates for two random walks associated with repeated barycentric subdivisions of a triangle. If the random walk jumps between the centers of triangles in the subdivision that have common sides, the resistance scales as a power of a constant \begin{document} $ρ$ \end{document} which is theoretically estimated to be in the interval \begin{document} $5/4≤ρ≤3/2$ \end{document}, with a numerical estimate \begin{document} $ρ≈1.306$ \end{document}. This corresponds to the theoretical estimate of spectral dimension \begin{document} $d_S$ \end{document} between 1.63 and 1.77, with a numerical estimate \begin{document} $d_S≈1.74$ \end{document}. On the other hand, if the random walk jumps between the corners of triangles in the subdivision, then the resistance scales as a power of a constant \begin{document} $ρ^T = 1/ρ$ \end{document}, which is theoretically estimated to be in the interval \begin{document} $2/3≤ρ^T≤4/5$ \end{document}. This corresponds to the spectral dimension between 2.28 and 2.38. The difference between \begin{document} $ρ$ \end{document} and \begin{document} $ρ^T$ \end{document} implies that the the limiting behavior of random walks on the repeated barycentric subdivisions is more delicate than on the generalized Sierpinski Carpets, and suggests interesting possibilities for further research, including possible non-uniqueness of self-similar Dirichlet forms.

Asymptotics for quasilinear obstacle problems in bad domains
Raffaela Capitanelli and Salvatore Fragapane
2019, 12(1): 43-56 doi: 10.3934/dcdss.2019003 +[Abstract](159) +[HTML](30) +[PDF](455.7KB)

We study two obstacle problems involving the p-Laplace operator in domains with n-th pre-fractal and fractal boundary. We perform asymptotic analysis for \begin{document} $p \to \infty $ \end{document} and \begin{document} $n \to \infty $ \end{document}.

On two-dimensional nonlocal Venttsel' problems in piecewise smooth domains
Simone Creo, Maria Rosaria Lancia, Alexander Nazarov and Paola Vernole
2019, 12(1): 57-64 doi: 10.3934/dcdss.2019004 +[Abstract](72) +[HTML](21) +[PDF](389.58KB)

We establish the regularity results for solutions of nonlocal Venttsel' problems in polygonal and piecewise smooth two-dimensional domains.

Convergence and density results for parabolic quasi-linear Venttsel' problems in fractal domains
Simone Creo and Valerio Regis Durante
2019, 12(1): 65-90 doi: 10.3934/dcdss.2019005 +[Abstract](262) +[HTML](21) +[PDF](619.1KB)

In this paper we study a quasi-linear evolution equation with nonlinear dynamical boundary conditions in a three dimensional fractal cylindrical domain \begin{document} $Q$ \end{document}, whose lateral boundary is a fractal surface \begin{document} $S$ \end{document}. We consider suitable approximating pre-fractal problems in the corresponding pre-fractal varying domains. After proving existence and uniqueness results via standard semigroup approach, we prove density results for the domains of energy functionals defined on \begin{document} $Q$ \end{document} and \begin{document} $S$ \end{document}. Then we prove that the pre-fractal solutions converge in a suitable sense to the limit fractal one via the Mosco convergence of the energy functionals.

Monotone wave fronts for $(p, q)$-Laplacian driven reaction-diffusion equations
Maurizio Garrione and Marta Strani
2019, 12(1): 91-103 doi: 10.3934/dcdss.2019006 +[Abstract](118) +[HTML](22) +[PDF](447.61KB)

We study the existence of monotone heteroclinic traveling waves for the \begin{document}$\end{document}-dimensional reaction-diffusion equation

where the non-homogeneous operator appearing on the right-hand side is of \begin{document}$(p, q)$\end{document}-Laplacian type. Here we assume that \begin{document}$2 ≤ q < p$\end{document} and \begin{document}$f$\end{document} is a nonlinearity of Fisher type on \begin{document}$[0, 1]$\end{document}, namely \begin{document}$f(0) = 0 = f(1)$\end{document} and \begin{document}$f > 0$\end{document} on \begin{document}$]0, 1[\, $\end{document}. We give an estimate of the critical speed and we comment on the roles of \begin{document}$p$\end{document} and \begin{document}$q$\end{document} in the dynamics, providing some numerical simulations.

Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces
Michel L. Lapidus, Goran Radunović and Darko Žubrinić
2019, 12(1): 105-117 doi: 10.3934/dcdss.2019007 +[Abstract](105) +[HTML](25) +[PDF](461.26KB)

We establish pointwise and distributional fractal tube formulas for a large class of compact subsets of Euclidean spaces of arbitrary dimensions. These formulas are expressed as sums of residues of suitable meromorphic functions over the complex dimensions of the compact set under consideration (i.e., over the poles of its fractal zeta function). Our results generalize to higher dimensions (and in a significant way) the corresponding ones previously obtained for fractal strings by the first author and van Frankenhuijsen. They are illustrated by several examples and applied to yield a new Minkowski measurability criterion.

Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations
Paola Mannucci, Claudio Marchi and Nicoletta Tchou
2019, 12(1): 119-128 doi: 10.3934/dcdss.2019008 +[Abstract](82) +[HTML](27) +[PDF](351.03KB)

This paper concerns singular perturbation problems where the dynamics of the fast variable evolve in the whole space according to an operator whose infinitesimal generator is formed by a Grushin type second order part and a Ornstein-Uhlenbeck first order part.

We prove that the dynamics of the fast variables admits an invariant measure and that the associated ergodic problem has a viscosity solution which is also regular and with logarithmic growth at infinity. These properties play a crucial role in the main theorem which establishes that the value functions of the starting perturbation problems converge to the solution of an effective problem whose operator and initial datum are given in terms of the associated invariant measure.

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