Variational relaxation and gradient flows for degenerate elliptic conductivity problems
Abstract
We develop a convex-analytic L2-framework for gradient flows of pumping-energy functionals coupled with elliptic state equations and an evolving nonnegative conductivity a≥ 0. The main difficulty is that the conductivity may be merely square-integrable, or degenerate, and formal differentiation need not produce an L2-subgradient. After imposing the strictly positive lower cutoff r+a≥ r>0, we define the pumping-energy functional by a variational formula, prove convexity and lower semicontinuity, and show that its L2-subdifferential is nonempty if and only if the pressure gradient belongs to L4. We also obtain a unique maximal-monotone L2-gradient flow satisfying an energy identity. Moreover, we prove Mosco convergence of the pumping-energy functionals as the cutoff vanishes. An inverse-moment conductivity constraint provides a weak alternative to a uniform positive lower bound. For nonlinear laws a=μ(b), primal and flux representations distinguish two complementary regimes. If μ is concave, the pumping-energy functional remains convex, and we characterize its L2-subdifferential and the induced gradient flow. If 1/μ is concave, the pumping-energy functional is concave, and therefore its negative is convex. We construct its pointwise gradient flow on a uniformly elliptic box. Admissible stationary points for power laws yield p-Laplacian equations with p>2 and 1 < p < 2, respectively.
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BibTeX
@misc{haskovec2026variational,
title = {{Variational relaxation and gradient flows for degenerate elliptic conductivity problems}},
author = {Jan Haskovec and Peter Markowich and Stefano Zampini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33173},
eprint = {2610.33173},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33173},
}
RIS
TY - UNPB
TI - Variational relaxation and gradient flows for degenerate elliptic conductivity problems
AU - Jan Haskovec
AU - Peter Markowich
AU - Stefano Zampini
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.33173
UR - https://arxiv.org/abs/2610.33173
AB - We develop a convex-analytic L^2-framework for gradient flows of pumping-energy functionals coupled with elliptic state equations and an evolving nonnegative conductivity a≥ 0. The main difficulty is that the conductivity may be merely square-integrable, or degenerate, and formal differentiation need not produce an L^2-subgradient. After imposing the strictly positive lower cutoff r+a≥ r>0, we define the pumping-energy functional by a variational formula, prove convexity and lower semicontinuity, and show that its L^2-subdifferential is nonempty if and only if the pressure gradient belongs to L^4. We also obtain a unique maximal-monotone L^2-gradient flow satisfying an energy identity. Moreover, we prove Mosco convergence of the pumping-energy functionals as the cutoff vanishes. An inverse-moment conductivity constraint provides a weak alternative to a uniform positive lower bound. For nonlinear laws a=μ(b), primal and flux representations distinguish two complementary regimes. If μ is concave, the pumping-energy functional remains convex, and we characterize its L^2-subdifferential and the induced gradient flow. If 1/μ is concave, the pumping-energy functional is concave, and therefore its negative is convex. We construct its pointwise gradient flow on a uniformly elliptic box. Admissible stationary points for power laws yield p-Laplacian equations with p>2 and 1 < p < 2, respectively.
ER -