A singular perturbation in a linear parabolic equation with terms concentrating on the boundary (Q365169)

Let \(\Omega \subset \mathbb R^N\) be open and bounded with \(C^2\)-boundary \(\partial \Omega\). Further let \(\Gamma \subset \partial \Omega\) be a subset of the boundary which is isolated from the rest of the boundary. For sufficiently small \(\varepsilon\) one can then consider the strip of width \(\varepsilon\) with base \(\Gamma\) \[ \omega_\varepsilon =\{ x-\sigma n(x): x \in \Gamma, \sigma \in [0,\varepsilon) \}, \] where \(n(x)\) denotes the outer normal at \(x\). To illustrate the findings of the author, we present a special case of the results obtained. For \(\varepsilon\) as above consider the parabolic problem \[ \begin{cases} u_t^\varepsilon - \mathrm{div}(a(x) \nabla u^\varepsilon ) + c(x)u^\varepsilon = \frac{1}{\varepsilon} \mathcal{X}_{\omega_\varepsilon} V_\varepsilon(x) u^\varepsilon\qquad &\text{in } \Omega\\ a(x)\frac{\partial u^\varepsilon}{\partial n} + b(x) u^\varepsilon = 0 \qquad &\text{on } \Gamma\\ u^\varepsilon = 0 \qquad &\text{on } \partial \Omega \setminus \Gamma\\ u^\varepsilon(0) = u_0 \qquad &\text{in } \Omega.\end{cases} \] The author is interested in the convergence of the solutions of the above problem for \(\varepsilon \to 0\) if the potentials \(V_\varepsilon\) concentrate on the boundary for \(\varepsilon \to 0\) in the following sense: there exists a \(V_0 \in L^r(\Gamma)\) for some \(r > N - 1\) such that for any \(\varphi \in C^\infty(\overline{\Omega})\) one has \[ \lim_{\varepsilon \to 0} \frac{1}{\varepsilon} \int_{\omega_\varepsilon} V_\varepsilon \varphi = \int_\Gamma V_0 \varphi. \] It is shown that for \(\varepsilon\) sufficiently small the above problem is well-defined and its solutions \(u^\varepsilon\) converge uniformly on bounded time intervals away from \(t = 0\) to the solution \(u\) of the problem \[ \begin{cases} u_t - \mathrm{div}(a(x) \nabla u) + c(x)u = 0 \qquad &\text{in } \Omega\\ a(x)\frac{\partial u}{\partial n} + b(x) u = V_0(x)u \qquad &\text{on } \Gamma\\ u = 0 \qquad &\text{on } \partial \Omega \setminus \Gamma\\ u(0) = u_0 \qquad &\text{in } \Omega.\end{cases} \] Hence, under suitable assumptions potentials at the boundary can be approximated by potentials living on the interior of the domain. In fact, results of the above type are obtained for more general potentials and a quantitative estimate on the convergence rate of the solutions towards \(u\) is given. Moreover, the convergence of the first eigenvalues of the problems is studied as well.