Ill-posedness of the Dirichlet problem for 2D Lagrangian mean curvature equation

Wentao Cao; Zhehui Wang

Ill-posedness of the Dirichlet problem for 2D Lagrangian mean curvature equation

Analysis of PDEs 2025-12-11 v2

Authors: Wentao Cao , Zhehui Wang

Abstract

We investigate the Dirichlet problem of the two dimensional Lagrangian mean curvature equation in a bounded domain. Infinitely many $C^{1, \alpha} (\alpha\in (0,\frac{1}{5}))$ very weak solutions are built through Nash-Kuiper construction. Moreover, we note there are infinitely many $C^{1, \alpha}$ very weak solutions that can not be improved to be $C^{2, \alpha}$ .

Keywords

Dirichlet problem Lagrangian mean curvature equation ill-posedness Nash-Kuiper construction very weak solutions

Cite

@article{arxiv.2409.04816,
  title  = {Ill-posedness of the Dirichlet problem for 2D Lagrangian mean curvature equation},
  author = {Wentao Cao and Zhehui Wang},
  journal= {arXiv preprint arXiv:2409.04816},
  year   = {2025}
}

Comments

to be published in SCIENCE CHINA Mathematics

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