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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In the framework of the Laplacian transport, described by a Robin boundary value problem in an exterior domain in
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
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
We establish the regularity results for solutions of nonlocal Venttsel' problems in polygonal and piecewise smooth two-dimensional domains.
In this paper we study a quasi-linear evolution equation with nonlinear dynamical boundary conditions in a three dimensional fractal cylindrical domain
We study the existence of monotone heteroclinic traveling waves for the
where the non-homogeneous operator appearing on the right-hand side is of
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.
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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