Approximation of a nonlinear fractal energy functional on varying Hilbert spaces

Simone Creo; Maria Rosaria Lancia; Alejandro Vélez-Santiago; Paola Vernole

doi:10.3934/cpaa.2018035

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2018, Volume 17, Issue 2: 647-669. Doi: 10.3934/cpaa.2018035

This issue Previous Article A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates Next Article Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature

Approximation of a nonlinear fractal energy functional on varying Hilbert spaces

1.
Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Universitá degli studi di Roma Sapienza, Via A. Scarpa 16,00161 Roma, Italy
2.
Department of Mathematical Sciences, University of Puerto Rico at Mayagüez, Puerto Rico, 00681, USA
3.
Dipartimento di Matematica, Universitá degli Studi di Roma Sapienza, Piazzale Aldo Moro 2,00185 Roma, Italy

* Corresponding author: Maria Rosaria Lancia
* Corresponding author: Maria Rosaria Lancia

Received: October 31, 2016

Revised: June 30, 2017

Published: June 2018

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Abstract

We study a quasi-linear evolution equation with nonlinear dynamical boundary conditions in a two dimensional domain with Koch-type fractal boundary. 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 that the pre-fractal solutions converge in a suitable sense to the limit fractal one via the Mosco convergence of the energy functionals adapted by Tölle to the nonlinear framework in varying Hilbert spaces.

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Mathematics Subject Classification: Primary:35K, 28A80;Secondary:37L05, 31C25.

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Figure 1. The Koch snowflake

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