Elliptic-Parabolic Equations with Hysteresis Boundary Conditions
Ulrich Hornung, Ralph E. Showalter · SIAM Journal on Mathematical Analysis · 1995
A general porous-medium equation is uniquely solved subject to a pair of boundary conditions for the trace of the solution and a second function on the boundary. The use of maximal monotone graphs for the three nonlinearities permits not only the inclusion of the usual boundary conditions of Dirichlet, Neumann, or Robin type, including variational inequality constraints of Signorini type, but also dynamic boundary conditions and those that model hysteresis phenomena. It is shown that the dynamic is determined by a contraction semigroup in a product of $L^1 $ spaces. Several examples and numerical results are described.