A natural approach to the asymptotic mean value property for the p-Laplacian

A natural approach to the asymptotic mean value property for the p-Laplacian Let $$1\le p\le \infty $$ 1 ≤ p ≤ ∞ . We show that a function $$u\in C(\mathbb R^N)$$ u ∈ C ( R N ) is a viscosity solution to the normalized p-Laplace equation $$\Delta _p^n u(x)=0$$ Δ p n u ( x ) = 0 if and only if the asymptotic formula $$\begin{aligned} u(x)=\mu _p(\varepsilon ,u)(x)+o(\varepsilon ^2) \end{aligned}$$ u ( x ) = μ p ( ε , u ) ( x ) + o ( ε 2 ) holds as $$\varepsilon \rightarrow 0$$ ε → 0 in the viscosity sense. Here, $$\mu _p(\varepsilon ,u)(x)$$ μ p ( ε , u ) ( x ) is the p-mean value of u on $$B_\varepsilon (x)$$ B ε ( x ) characterized as a unique minimizer of $$\begin{aligned} \Vert u-\lambda \Vert _{L^p(B_\varepsilon (x))} \end{aligned}$$ ‖ u - λ ‖ L p ( B ε ( x ) ) with respect to $$\lambda \in {\mathbb {R}}$$ λ ∈ R . This kind of asymptotic mean value property (AMVP) extends to the case $$p=1$$ p = 1 previous (AMVP)’s obtained when $$\mu _p(\varepsilon ,u)(x)$$ μ p ( ε , u ) ( x ) is replaced by other kinds of mean values. The natural definition of $$\mu _p(\varepsilon ,u)(x)$$ μ p ( ε , u ) ( x ) makes sure that this is a monotonic and continuous (in the appropriate topology) functional of u. These two properties help to establish a fairly general proof of (AMVP), that can also be extended to the (normalized) parabolic p-Laplace equation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

A natural approach to the asymptotic mean value property for the p-Laplacian

Loading next page...
 
/lp/springer_journal/a-natural-approach-to-the-asymptotic-mean-value-property-for-the-p-Yu6htSXDpf
Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1188-7
Publisher site
See Article on Publisher Site

References

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
discover and read the research
that matters to you.

Enjoy affordable access to
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.

See the journals in your area

Monthly Plan

  • Read unlimited articles
  • Personalized recommendations
  • No expiration
  • Print 20 pages per month
  • 20% off on PDF purchases
  • Organize your research
  • Get updates on your journals and topic searches

$49/month

Start Free Trial

14-day Free Trial

Best Deal — 39% off

Annual Plan

  • All the features of the Professional Plan, but for 39% off!
  • Billed annually
  • No expiration
  • For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.

$588

$360/year

billed annually
Start Free Trial

14-day Free Trial