# Application of the Trace Inequality to the Poisson Equation

Application of the Trace Inequality to the Poisson Equation The purpose of this paper is to show that solutions of the Poisson equation $$-\Delta u=f$$ where f be a complex-valued distribution on $${\mathbb{R}}^{d}$$ , d ≥ 3 and $$f{\in} {\mathcal{M}}\left( \overset{.}{H}^{1}{\rightarrow} \overset{.}{H} ^{-1}\right)$$ satisfy the coercivity property : $$D^{k} u\in \mathcal{M}\left( \overset{.}{H}^{1}\rightarrow \overset{.}{H}^{-1}\right)$$ for all $$k,\left\vert k\right\vert =2$$ . The coercivity of this equation is well studied by Maz’ ya and Verbitsky [14] in the case where f belongs to the class of positive Borel measures. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Application of the Trace Inequality to the Poisson Equation

, Volume 12 (2) – Jan 11, 2008
24 pages

/lp/springer_journal/application-of-the-trace-inequality-to-the-poisson-equation-kNkphLz0FL
Publisher
SP Birkhäuser Verlag Basel
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-007-2125-3
Publisher site
See Article on Publisher Site

### Abstract

The purpose of this paper is to show that solutions of the Poisson equation $$-\Delta u=f$$ where f be a complex-valued distribution on $${\mathbb{R}}^{d}$$ , d ≥ 3 and $$f{\in} {\mathcal{M}}\left( \overset{.}{H}^{1}{\rightarrow} \overset{.}{H} ^{-1}\right)$$ satisfy the coercivity property : $$D^{k} u\in \mathcal{M}\left( \overset{.}{H}^{1}\rightarrow \overset{.}{H}^{-1}\right)$$ for all $$k,\left\vert k\right\vert =2$$ . The coercivity of this equation is well studied by Maz’ ya and Verbitsky [14] in the case where f belongs to the class of positive Borel measures.

### Journal

PositivitySpringer Journals

Published: Jan 11, 2008

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Unlimited reading Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere. ### Stay up to date Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates. ### Organize your research It’s easy to organize your research with our built-in tools. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. ### DeepDyve Freelancer ### DeepDyve Pro Price FREE$49/month

\$360/year
Save searches from