Under the action of a thermal dipole, Green’s functions are constructed for the stationary heat conduction and thermoelasticity problems for a half-space with a free, rigidly, smoothly or flexibly fixed boundary, on which a zero temperature or thermal insulation is given. The harmonic double-layer potentials are used in the solution of the heat conduction problem. Also, the thermoelastic displacement potentials are applied in the infinite space with two thermal dipoles mirrored with respect to the half-space boundary. To satisfy the boundary conditions, biharmonic Love functions are constructed. Expressions for temperature, displacements and stresses are presented. These expressions can be used to determine the half-space thermoelastic state caused by the given heat flux perturbation by a thin heat-proof inclusion parallel to the half-space boundary.