On hyperbolic variational inequalities of first order and some applications
The author studies strong solutions and their coincidence sets to the obstacle problem for linear transport operators. He starts with the elliptic variational inequality \[ \quad u_\delta\in \mathbb K^1_-:\delta\int_Q \nabla u_\delta\cdot \nabla(v-u_\delta)+(Hu_\delta-f)(v-u_\delta)\geq 0,\;\forall v\in \mathbb K^1_-, \] where \(\delta>0,\;f\in L^2(Q),\;Q\subset \mathbb R^N\;\) is an open domain with a Lipschitz boundary and the first order operator \(H\) is given by \(\;Hv=\mathbf b\cdot\nabla v+cv,\;\mathbf b=(b_i(x)).\;\) The convex set \(\mathbb K^1_-\) has the form \(\;\mathbb K^1_-=\mathbb K\cap H^1_{\Sigma_-}(Q),\) \( H^1_{\Sigma_-}(Q)=\{v\in H^1(Q):v=0\;\text{in}\;\Sigma_-\}, \;\Sigma_-=\{x\in \partial Q:\mathbf b\cdot\mathbf n<0\}. \) The existence and uniqueness of a solution is an immediate consequence of the general theory for the second order obstacle problem. The strong convergence of the solutions and the characteristic functions of their coincidence sets is obtained in for \(\;\delta\to 0\;\) in the passage to the first order problem \(\quad u\in \mathbb K_-:\int_Q (Hu_\delta-f)(v-u_\delta)\geq 0,\;\forall v\in \mathbb K.\) An application to a steady state chemotaxis problem is given. The extension to the two obstacles problem of first order is also presented.
scientific article; zbMATH DE number 2111479