Basics for P.D.E. Systems

The main object in this chapter is the stress-energy tensor, which is an algebraic fact implying several useful identities like the (weak) monotonicity formula, Gui’s Hamiltonian identities, and Pohozaev’ identities, for all solutions and all potentials W ≥ 0. Modica’s inequality holds in the scalar case and implies a strong monotonicity formula, but is not generally valid in the vector case. The triple junction on the plane is also introduced.