Symmetry in nonlinear PDEs: Some open problems

In this note, we discuss symmetry properties of solutions for simple scalar and vector-valued systems of nonlinear elliptic partial differential equations (PDEs). The systems of interest are of variational nature, they appear, for instance, in some first-order phase transition models in mathematical physics (e.g., phase separation, superconductivity, liquid crystals) and are naturally related to some PDEs in geometry (minimal surfaces and harmonic maps). We review some known results and present few open problems about symmetry of minimizers for all these models.