# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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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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2019, 12(1): ⅰ-ⅰ doi: 10.3934/dcdss.201901i +[Abstract](227) +[HTML](53) +[PDF](71.01KB)
Abstract:
2019, 12(1): 1-26 doi: 10.3934/dcdss.2019001 +[Abstract](273) +[HTML](35) +[PDF](438.0KB)
Abstract:

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.

2019, 12(1): 27-42 doi: 10.3934/dcdss.2019002 +[Abstract](337) +[HTML](34) +[PDF](812.12KB)
Abstract:

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.

2019, 12(1): 43-56 doi: 10.3934/dcdss.2019003 +[Abstract](348) +[HTML](30) +[PDF](455.7KB)
Abstract:

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}.

2019, 12(1): 57-64 doi: 10.3934/dcdss.2019004 +[Abstract](160) +[HTML](22) +[PDF](389.58KB)
Abstract:

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

2019, 12(1): 65-90 doi: 10.3934/dcdss.2019005 +[Abstract](404) +[HTML](22) +[PDF](619.1KB)
Abstract:

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.

2019, 12(1): 91-103 doi: 10.3934/dcdss.2019006 +[Abstract](314) +[HTML](24) +[PDF](447.61KB)
Abstract:

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.

2019, 12(1): 105-117 doi: 10.3934/dcdss.2019007 +[Abstract](260) +[HTML](29) +[PDF](461.26KB)
Abstract:

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.

2019, 12(1): 119-128 doi: 10.3934/dcdss.2019008 +[Abstract](178) +[HTML](28) +[PDF](351.03KB)
Abstract:

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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