**5. Conclusions**

The well-posedness of the Cauchy problem, as well as the well-formulation of the junction conditions, are crucial aspects of any theory of gravity. In fact, a well-posed initial value problem ensures uniqueness, continuity, and causality of solutions from initial data; at the same time, well-formulated junction conditions allow us to understand if and how two different space-times can be soldered at a given hypersurface, with obvious applications and consequences, for example, on an astrophysical level.

In this paper, we have discussed the Cauchy problem and the junction conditions within the framework of *f*(*R*)-gravity with torsion.

For what concerns the Cauchy problem, we have seen that the problem is always well-posed in vacuo and, in the absence of spin, every time the trace of the matter stress–energy tensor is constant; indeed, in such a circumstance, the theory amounts to an Einstein-like theory for which the well-posedness of the initial value problem is well-established: for instance, this is what happens in the case of coupling to an electromagnetic field or a Yang–Mills field. On the contrary, when the stress–energy tensor trace is not constant, the problem needs to be discussed case-by-case.

Here, we have faced the coupling to a perfect fluid and a Klein–Gordon scalar field. In both cases, we have derived sufficient conditions ensuring the well-posedness of the initial value problem. We have also proved that there exist *f*(*R*) models with torsion, which actually satisfy the stated conditions: the model *f*(*R*) = *R* + *αR* <sup>2</sup> does it. The key idea to achieve these results has been implementing a conformal transformation from the Jordan to the Einstein frame, proving that the conservation laws are formally preserved under such a transformation; this has allowed us to apply well-known Bruhat's results, holding in GR.

On the junction conditions, we have deduced the general requirements needed to solder at a given hypersurface two different solutions of *f*(*R*)-gravity with torsion. Despite a formal resemblance, junction conditions for *f*(*R*)-gravity with torsion differ from those holding in the ECSK theory because they involve the trace of the matter stress–energy tensor and its first derivatives; this is due to the contributions that the non-linearity of the gravitational function *f*(*R*) gives to the contorsion tensor and, in general, it results in specific conditions that the matter fields have to satisfy at the separation hypersurface. In order to better clarify this aspect, we have given two illustrative examples, considering the model *f*(*R*) = *R* + *αR* 2 coupled to a spin fluid and a Dirac field, respectively.

Finally, we have shown that the study of the initial value problem, as well as the junction conditions in the context of *f*(*R*)-gravity with torsion, singles out suitable conditions on the gravitational Lagrangian function *f*(*R*) itself, which may be used as selection criteria for viable *f*(*R*) models.

**Conflicts of Interest:** The author declares no conflict of interest.
