Convergence and density results for parabolic quasi-linear Venttsel' problems in fractal domains

Simone Creo; Valerio Regis Durante

doi:10.3934/dcdss.2019005

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2019, Volume 12, Issue 1: 65-90. Doi: 10.3934/dcdss.2019005

This issue Previous Article On two-dimensional nonlocal Venttsel' problems in piecewise smooth domains Next Article Monotone wave fronts for $(p, q)$-Laplacian driven reaction-diffusion equations

Convergence and density results for parabolic quasi-linear Venttsel' problems in fractal domains

Simone Creo^, and
Valerio Regis Durante

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Università degli studi di Roma Sapienza, Via A. Scarpa 16, 00161 Roma, Italy

* Corresponding author: Simone Creo
* Corresponding author: Simone Creo

Received: March 31, 2017

Revised: October 31, 2017

Published: July 2018

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Abstract

In this paper we study a quasi-linear evolution equation with nonlinear dynamical boundary conditions in a three dimensional fractal cylindrical domain $Q$ , whose lateral boundary is a fractal surface $S$ . 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 $Q$ and $S$ . 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.

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Mathematics Subject Classification: Primary: 35K, 28A80; Secondary: 35K90, 46E35, 31C25.

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Figure 1. The pre-fractal curve $F_h$ for $h = 3$.

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Figure 2. The fractal domain $Q$.

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