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arXiv · 2607.16767

Existence theory for non-variational systems with free boundaries

Layan El Hajj·Seongmin Jeon·Henrik Shahgholian
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Abstract

We study the existence of solutions for systems of both elliptic and parabolic partial differential equations with potentially singular right-hand sides and free boundaries, posed in general smooth domains. Our results are established within a framework of "meta-theorems." This approach hinges on specific strong properties of the operators and their solutions (in approximate smooth settings) to guarantee the existence of a limit as the approximation parameter tends to zero. The primary challenge lies in applying these meta-theorems to prototype cases, which requires verifying that the necessary strong properties hold. For our analysis, we focus on fully nonlinear and $p$-Laplacian operators and a mixing of these operators, in both elliptic and parabolic contexts. While we focus on these specific cases, the meta-theorems remain valid for any other operators that satisfy the required properties. Beyond the complex proofs of our meta-theorems, and their applications to specific operators, a major challenge is the technical handling of the $p$-parabolic case, which requires proving the regularity of solutions of the $p$-parabolic equation with singular or degenerate right-hand side--addressed in the Appendix--along with several (new) properties, which is missing in the literature.

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Layan El Hajj, Seongmin Jeon, Henrik Shahgholian. 2026-07-18. Existence theory for non-variational systems with free boundaries. https://arxiv.org/abs/2607.16767

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