Variational formulation of the semiconductor electrostatic problem
Applicable Nonlinear Analysis, Volume 3, Issue 1, April 2026, Pages: 66–75
MYKOLA YAREMENKO
National Technical University of Ukraine, “Igor Sikorsky Kyiv Polytechnic Institute”, 37, Prospect Beresteiskyi (former Peremohy), Kyiv, Ukraine, 03056
Abstract
We develop a complete mathematical theory for the nonlinear Poisson-Boltzmann equation governing electrostatics in double-gate semiconductor devices. The cornerstone of our approach is the identification of an electrostatic free energy functional \[J[\phi] = \int_\Omega \left[ \frac{1}{2}|\nabla\phi|^2 + \lambda^2\eta e^{(\phi-w)/\eta} \right] d\Omega\] whose Euler-Lagrange equation yields the governing PDE. We prove that \(J[\phi]\) is strictly convex and coercive, ensuring the existence of a unique global minimizer via the direct method. Regularity theory establishes that weak solutions belong to \(C^{2,\alpha}(\Omega)\), justifying formal asymptotic expansions. Stability analysis reveals a positive-definite spectrum for the linearized operator and global convergence under gradient flow dynamics. The singular perturbation limit \(\eta \to 0\) is examined, with rigorous analysis of the resulting free boundary problem and boundary layer structure. Our work provides the mathematical foundation for the asymptotic methods employed in nanoscale transistor modeling while developing techniques applicable to a broad class of semilinear elliptic equations with exponential nonlinearities.
Cite this Article as
Mykola Yaremenko, Variational formulation of the semiconductor electrostatic problem, Applicable Nonlinear Analysis, 3(1), 66–75, 2026