On the Laplace equation with dynamical boundary conditions of reactive-diffusive type
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.
scientific article; zbMATH DE number 5555225
scientific article