具有自由边界的非变分系统的存在性理论
Existence theory for non-variational systems with free boundaries
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中文总结 AI 辅助
研究具有潜在奇异右端项和自由边界的椭圆与抛物偏微分方程组解的存在性，通过“元定理”框架，聚焦特定算子，克服应用挑战及p -抛物型情况技术难题，证明了解的正则性等性质。
中文摘要 AI 辅助
我们研究了在一般光滑区域中具有潜在奇异右端项和自由边界的椭圆型和抛物型偏微分方程组解的存在性。我们的结果是在“元定理”框架内建立的。这种方法依赖于算子及其解（在近似光滑设置下）的特定强性质，以保证当近似参数趋于零时极限的存在性。主要挑战在于将这些元定理应用于原型案例，这需要验证必要的强性质成立。我们的分析集中在完全非线性和p -拉普拉斯算子以及这些算子的混合，包括椭圆型和抛物型情况。虽然我们专注于这些特定情况，但元定理对任何满足所需性质的其他算子仍然有效。除了元定理的复杂证明及其对特定算子的应用外，一个主要挑战是p -抛物型情况的技术处理，这需要证明具有奇异或退化右端项的p -抛物型方程解的正则性——在附录中解决——以及文献中未有的几个（新）性质。
英文摘要
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.
