{
  "abstract": "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 $a\\geq 0$. 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 $r+a\\geq r>0$, 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33173",
  "author_total": 3,
  "authors": [
    {
      "name": "Jan Haskovec"
    },
    {
      "name": "Peter Markowich"
    },
    {
      "name": "Stefano Zampini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e97jd4f5c8kan.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33173",
  "fulltext_url": null,
  "id": "e97jd4f5c8kan",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33173",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7223800252",
  "pdf_url": "https://arxiv.org/pdf/2610.33173",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e97jd4f5c8kan.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33173",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33173"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Variational relaxation and gradient flows for degenerate elliptic conductivity problems",
  "title_text": "Variational relaxation and gradient flows for degenerate elliptic conductivity problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/e97jd4f5c8kan",
  "venue": "arXiv",
  "versions": []
}