A Dual-Parameter Quadratic Regularization Framework in Sobolev Spaces

We introduce a two-parameter variational energy framework based on weighted Sobolev norms, termed the K–R energy functional, defined by where and independently regulate amplitude energy and smoothness energy, respectively. This formulation establishes a controllable and physically interpretable trade-off between data fidelity and regularity within a unified Hilbert-space geometry. We develop a complete well-posedness theory for the associated variational problem, proving coercivity, strict convexity, existence, and uniqueness of minimizers in . Explicit stability estimates with respect to data perturbations are derived, revealing precise quantitative dependence on the parameters and . The corresponding Euler–Lagrange equation is shown to be a screened Poisson (Helmholtz-type) partial differential equation, thereby linking the K–R framework to classical reaction–diffusion dynamics. A spectral analysis of the induced elliptic operator characterizes the frequency-selective behavior governed by the K–R balance, and a structure-preserving discrete formulation is presented for numerical implementation. The proposed framework unifies and extends classical Tikhonov regularization by enabling independent and physically meaningful control of zeroth- and first-order energies. Applications to inverse problems, signal and image smoothing, and VLSI thermal and power optimization are discussed, demonstrating the broad applicability and robustness of the K–R energy paradigm.