Dirichlet boundary conditions on linear combination of functions

variational formulation
1

Hi.
I am solving a linear system of PDEs in the functions u_1(r) and u_2(r).
I need to impose a Dirichlet boundary condition of the type

a u_1(r) + b u_2(r) = c

when r is in a given subdomain of the boundary.
Here a, b, and c are known constants.
I know how to use DirichletBC to implement things like

u_1(r) = c_1

but I don’t know how to handle the linear combination.
Thanks in advance for your help!

2

Linear combinations is a multi-point constraint, and can be done with Lagrange multipliers, Nitsches method or matrix reduction techniques. See for instance: No-penetration_bounadry_condition