تقرير
On a Hardy-Morrey inequality
العنوان: | On a Hardy-Morrey inequality |
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المؤلفون: | Hynd, Ryan, Larson, Simon, Lindgren, Erik |
سنة النشر: | 2024 |
المجموعة: | Mathematics |
مصطلحات موضوعية: | Mathematics - Analysis of PDEs, Mathematics - Functional Analysis, 26D10, 46E35, 35P30 |
الوصف: | Morrey's classical inequality implies the H\"older continuity of a function whose gradient is sufficiently integrable. Another consequence is the Hardy-type inequality $$ \lambda\biggl\|\frac{u}{d_\Omega^{1-n/p}}\biggr\|_{\infty}^p\le \int_\Omega |Du|^p \,dx $$ for any open set $\Omega\subsetneq \mathbb{R}^n$. This inequality is valid for functions supported in $\Omega$ and with $\lambda$ a positive constant independent of $u$. The crucial hypothesis is that the exponent $p$ exceeds the dimension $n$. This paper aims to develop a basic theory for this inequality and the associated variational problem. In particular, we study the relationship between the geometry of $\Omega$, sharp constants, and the existence of a nontrivial $u$ which saturates the inequality. |
نوع الوثيقة: | Working Paper |
URL الوصول: | http://arxiv.org/abs/2401.05781 |
رقم الأكسشن: | edsarx.2401.05781 |
قاعدة البيانات: | arXiv |
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