Nonlinear ${\cal N}=2$ Supersymmetry and 3D Supersymmetric Born-Infeld Theory

D$p$-branes acquire effective nonlinear descriptions whose bosonic part is related to the Born-Infeld action. This nonlinearity has been proven to be a consequence of the partial ${\cal N}=2\to{\cal N}=1$ supersymmetry breaking, originating from the solitonic nature of the branes. In this work, we focus on the effective descriptions of D2-branes. Using the Goldstone multiplet interpretation of the action and the method of nilpotent ${\cal N}=2$ superfields, we construct the 3D, ${\cal N}=1$ superspace effective action which makes the first supersymmetry manifest and realizes the second, spontaneously broken, supersymmetry nonlinearly. We show that there are two such supersymmetric extensions of the 3D Born-Infeld action which correspond to the dynamics of the 3D Maxwell-Goldstone multiplet and the 3D projection of the Tensor-Goldstone multiplet respectively. Moreover, we demonstrate that these results are derived by applying the constrained superfield approach on the ${\cal N}=2, D=3$ vector and chiral multiplets after expanding them around a nontrivial vacuum. We find that these two descriptions are related by a duality transformation which results in the inversion of a dimensionless parameter. For both descriptions we derive the explicit bosonic and fermionic parts of the 3D super Born-Infeld action. Finally, consider the deformation of the Maxwell-Goldstone superspace action by the characteristic Chern-Simons-like, gauge invariant, mass term.