Regularity of the stress field for degenerate and/or singular elliptic problems

Lledos, Benjamin

Annales mathématiques Blaise Pascal, Tome 31 (2024) no. 1, pp. 83-135

We investigate the regularity of the solutions to degenerate and/or singular elliptic equations. We prove the continuity of $G\left(\nabla u\right)$ where $u$ is a locally Lipschitz solution of $\mathrm{div}G\left(\nabla u\right)=\lambda \in ℝ$ in dimension two under some growth assumptions on $G$. Additionally, we establish a result that holds in any dimension, indicating that the separation between $\nabla u$ and the degeneracy set of $G$ is continuous.

Nous étudions la régularité des solutions d’équations elliptiques dégénérées et/ou singulières. Nous prouvons la continuité de $G\left(\nabla u\right)$ où $u$ est une solution localement Lipschitz de $\mathrm{div}G\left(\nabla u\right)=\lambda \in ℝ$ en dimension deux sous certaines hypothèses de croissance sur $G$. De plus, nous établissons un résultat valable en toute dimension, indiquant que la séparation entre $\nabla u$ et l’ensemble de dégénérescence de $G$ est continue.

Classification : 35B65, 35J62, 35J70, 35J75, 49N99
Keywords: Regularity, elliptic PDE, dimension two

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