多孔介质方程的$\alpha$-凹性不保持性
Non-preservation of $α$-concavity for the porous medium equation
AI总结：
本研究针对多孔介质方程压力的α-凹性保持问题，通过构造显式例子证明0≤α<1/2或1/2<α≤1时该凹性通常不保持，解决了Vázquez的公开问题，强化了Ishige-Salani的早期结果，且因1/2-凹性已被证明保持而具有紧性，凹性破坏可发生在初始支集的内点或边界点。
摘要：
我们证明，对于$0\le\alpha<1/2$或$1/2<\alpha\le 1$，多孔介质方程通常不保持压力的$\alpha$-凹性。这一结果解决了Vázquez提出的关于多孔介质方程是否保持压力凹性的公开问题。我们的结果强化了Ishige-Salani的早期工作，后者仅研究了$\alpha>0$较小的情形。由于Daskalopoulos-Hamilton-Lee已证明$1/2$-凹性是保持的，我们的结果是紧的。我们的显式例子表明，凹性可在初始数据支集的内点处瞬时被破坏。对于$0\le\alpha<1/2$的情形，我们还给出了另一组例子，证明凹性可在边界点处被破坏。
英文摘要：
We show that the porous medium equation does not in general preserve $α$-concavity of the pressure for $0\leα<1/2$ or $1/2<α\le 1$. In particular, this resolves an open problem of Vázquez on whether concavity of pressure is preserved by the porous medium equation. Our results strengthen an earlier work of Ishige-Salani, who considered the case of small $α>0$. Since Daskalopoulos-Hamilton-Lee showed that $1/2$-concavity is preserved, our result is sharp. Our explicit examples show that concavity can be instantaneously broken at an interior point of the support of the initial data. For $0\leα<1/2$, we give another set of examples to show that concavity can be broken at a boundary point.
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