Deformable medical image registration is a fundamental task in medical image analysis. While deep learning-based methods have demonstrated superior accuracy and computational efficiency compared to traditional techniques, they often overlook the critical role of regularization in ensuring robustness and anatomical plausibility. We propose DARE (Deformable Adaptive Regularization Estimator), a novel registration framework that dynamically adjusts elastic regularization based on the gradient norm of the deformation field. Our approach integrates strain and shear energy terms, which are adaptively modulated to balance stability and flexibility. To ensure physically realistic transformations, DARE includes a folding-prevention mechanism that penalizes regions with negative deformation Jacobian. This strategy mitigates non-physical artifacts such as folding, avoids over-smoothing, and improves both registration accuracy and anatomical plausibility