Semilinear Neumann boundary value problems for measure-valued maps

@misc{kroshnin2026semilinear,
  title = {{Semilinear Neumann boundary value problems for measure-valued maps}},
  author = {Aleksei Kroshnin and Hugo Lavenant and Dmitry Vorotnikov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31353},
  eprint = {2609.31353},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31353},
}

TY  - UNPB
TI  - Semilinear Neumann boundary value problems for measure-valued maps
AU  - Aleksei Kroshnin
AU  - Hugo Lavenant
AU  - Dmitry Vorotnikov
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31353
UR  - https://arxiv.org/abs/2609.31353
AB  - We study the Wasserstein lift of nonlinear Poisson systems with inhomogeneous Neumann boundary conditions. We establish the existence of minimizers using new estimates involving the p-moments of measure-valued traces. We also derive the optimality conditions, which turn out to be matrix-valued generalizations of pressureless Euler equations. While deterministic minimizers exist under suitable convexity assumptions and for one-dimensional source domains, we show that genuinely measure-valued minimizers arise in higher dimensions. In particular, we construct examples in which the lifted and classical variational problems have different minimum values, thereby precluding the solutions to the lifted problem from being deterministic, despite the absence of any a priori mechanism enforcing measure-valued behavior. To the best of our knowledge, this phenomenon is new in the nonlinear elliptic theory.
ER  -