AI 中文总结
该研究针对散度型微分算子的Lions极大正则性问题，构造出1/2-Hölder端点处的反例，证明仅该连续性不蕴含极大L²正则性，且可推广到高维有界区域。
AI 中文摘要
本研究针对散度型微分算子的Lions问题，给出了极大L²正则性的一个反例。在有界区间上，我们构造了一个有界、一致椭圆型的实标量扩散系数，其在时间上具有1/2-Hölder连续性，取值属于空间L^∞，且可任意逼近热方程的常系数。对于零初始数据及在时间上连续、空间值为平方可积的强迫项，唯一的Lions变分解的时间导数在时空上并非平方可积。因此，即使对于热方程的任意小标量扰动，仅1/2-Hölder连续性也不蕴含极大L²正则性。该构造基于一族缺失的振荡三角模式，这些模式定位于不断收缩的时间区间上，其空间剖面及其一阶空间导数在两个端点处均为零，这使得反例可零延拓至实直线。通过张量积及抛物尺度变换的局部化，最终得到在R^d上以及R^d的任意有界区域Ω上的实对称各向同性反例，其中d≥1。
英文摘要
In this work we give a counterexample to maximal $\mathrm{L}^2$-regularity in Lions' problem for divergence-form differential operators. On a bounded interval, we construct a bounded, uniformly elliptic, real scalar diffusion coefficient that is $\frac{1}{2}$-Hölder continuous in time with values in spatial $\mathrm{L}^\infty$. It can be chosen arbitrarily close to the constant coefficient of the heat equation. For zero initial data and a forcing term that is continuous in time with square-integrable spatial values, the unique Lions variational solution has a time derivative that is not square integrable in space-time. Thus $\frac{1}{2}$-Hölder continuity alone does not imply maximal $\mathrm{L}^2$-regularity, even for arbitrarily small scalar perturbations of the heat equation. The construction is based on a lacunary family of oscillatory trigonometric modes localised on shrinking time intervals. The spatial profile and the oscillatory modes, together with their first spatial derivatives, vanish at both endpoints. This permits zero extension of the counterexample to the real line. Tensorisation and localisation by parabolic rescaling then yield real symmetric isotropic counterexamples on $\mathbb{R}^d$ and on every bounded domain $Ω\subset\mathbb{R}^d$, for all $d\ge1$.
CommentsAdded one reference and two remarks. No changes to the proofs
