On the Laplace equation with dynamical boundary conditions of reactive-diffusive type
type: Item
instance of: scholarly article
title: On the Laplace equation with dynamical boundary conditions of reactive-diffusive type
zbMATH Open document ID: 1167.35019
author: Enzo Vitillaro
author: Juan Luis Vázquez
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-05-19T00:00:00Z
review text: This paper deals with the Laplace equation in a bounded regular domain \(\Omega\) of \(\mathbb R^N\) \((N\geq 2)\) coupled with a dynamical boundary condition of reactive-diffusive type. In particular we study the problem \[ \begin{alignedat}{2} &\Delta u=0 &&\text{in }(0,\infty)\times\Omega,\\ &u_t=ku_\nu+l\Delta_\Gamma u\quad &&\text{on }(0,\infty)\times\Gamma,\\ &u(0,x)= u_0(x) &&\text{on }\Gamma, \end{alignedat} \] where \(u=u(t,x)\), \(t\geq 0\), \(x\in\Omega\), \(\Gamma=\partial\Omega\), \(\Delta=\Delta_x\) denotes the Laplacian operator with respect to the space variable, while \(\Delta_\Gamma\) denotes the Laplace-Beltrami operator on \(\Gamma\), \(\nu\) is the outward normal to \(\Omega\), and \(k\) and \(l\) are given real constants. Well-posedness is proved for any given initial distribution \(u_0\) on \(\Gamma\), together with the regularity of the solution. Moreover the Fourier method is applied to represent it in term of the eigenfunctions of a related eigenvalue problem.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35A22
Mathematics Subject Classification ID: 35C05
zbMATH DE Number: 5555225
zbMATH Keywords: Laplace equation
zbMATH Keywords: dynamical boundary conditions
zbMATH Keywords: reactive terms
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.01.023
OpenAlex ID: W2038059005
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DOI: 10.1016/J.JMAA.2009.01.023
description: scientific article; zbMATH DE number 5555225