通过抛物容量得到的 Morrey 测度对偶条件

中文摘要 AI 辅助

我们建立了 Morrey 型条件，以确保一个有限符号 Radon 测度属于 $p$-Laplacian 型非线性抛物方程解的能量空间的对偶空间。更精确地说，对于抛物柱体 $Q_{r,r^p}(z):=B_r(x)\times(t-r^p,t+r^p)$，我们证明 $|\mu|\bigl(Q_{r,r^p}(z)\cap(\Omega\times (0,T))\bigr) \leq Mr^{n+p-\vartheta}$（其中 $\vartheta<p$）蕴含对偶性，并且这个阈值在一般情况下是尖锐的。更一般地，对于时间长度与 $r^q$（$q>1$）成比例的柱体，充分阈值是 $\vartheta<\min\{p,q\}$。最后，我们讨论了由此产生的带测度数据的非线性抛物方程的变分、能量和重整化解理论，并将我们的结论与现有文献联系起来。

英文摘要

We establish Morrey-type conditions ensuring that a finite signed Radon measure belongs to the dual of the energy space of solutions to nonlinear parabolic equations of $p$-Laplacian type. More precisely, for the parabolic cylinders $Q_{r,r^p}(z):=B_r(x)\times(t-r^p,t+r^p)$, we prove that \[ |μ|\bigl(Q_{r,r^p}(z)\cap(Ω\times (0,T))\bigr) \leq Mr^{n+p-\vartheta}, \qquad \vartheta<p, \] implies duality, and this threshold is sharp in general. More generally, for cylinders with time length proportional to $r^q$, $q>1$, the sufficient threshold is $\vartheta<\min\{p,q\}$. Finally, we discuss the resulting variational, energy, and renormalized solution theories for nonlinear parabolic equations with measure data, relating our conclusions to the existing literature.

发表机构

• Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara（费拉拉大学数学与计算机科学系）

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