Uniqueness of solutions to some quasilinear elliptic equations whose Hamiltonian has natural growth in the gradient

Uniqueness of solutions to some quasilinear elliptic equations whose Hamiltonian has natural... The paper discusses uniqueness of solutions to stationary elliptic problems of the type $$\begin{aligned} A(u)+H(u)=f\in {\mathcal {D}}'(\Omega ), \end{aligned}$$ A ( u ) + H ( u ) = f ∈ D ′ ( Ω ) , where $$\Omega \ \in R^{N},\ $$ Ω ∈ R N , $$u\in W^{1,p}(\Omega )\ (1\le p\le +\infty ),\ A(u)\ $$ u ∈ W 1 , p ( Ω ) ( 1 ≤ p ≤ + ∞ ) , A ( u ) is an elliptic operator, $$H(u)\ $$ H ( u ) is an Hamiltonian that grows with $$\left| {\nabla u}\right| ^{p}$$ ∇ u p and f is given. Methods introduced in Artola (Boll UMI 6(5-B):51–71, 1986), (Proceedings of the International Conference on Generalized Functions, (ICGF 2000). Cambridge Scientific Publishers, Cambridge, 51–92, 2004), (Ricerche di Matematica XLIV, fasc. 2:400–420, 1995) for quasilinear parabolic or elliptic equations, together with properties for some continuity moduli, are used to improve some results from Barles and Murat (Arch Ration Mech Anal 133(1):77–101, 1995) for bounded solutions and from Barles and Porretta (Ann Scuola Norm Sup Pisa Cl Sci 5(1):107–136, 2006), Lions (J Anal Math 45: 234–254, 1985) for unbounded solutions, when 1 $$\le p\le 2.$$ ≤ p ≤ 2 . Unilateral problems are considered and the case where f depends on the solution u is also discussed. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bollettino dell'Unione Matematica Italiana Springer Journals

Uniqueness of solutions to some quasilinear elliptic equations whose Hamiltonian has natural growth in the gradient

Loading next page...
 
/lp/springer_journal/uniqueness-of-solutions-to-some-quasilinear-elliptic-equations-whose-Mk7QFxUv4E
Publisher
Springer Journals
Copyright
Copyright © 2017 by Unione Matematica Italiana
Subject
Mathematics; Mathematics, general
ISSN
1972-6724
eISSN
2198-2759
D.O.I.
10.1007/s40574-017-0130-4
Publisher site
See Article on Publisher Site

Abstract

The paper discusses uniqueness of solutions to stationary elliptic problems of the type $$\begin{aligned} A(u)+H(u)=f\in {\mathcal {D}}'(\Omega ), \end{aligned}$$ A ( u ) + H ( u ) = f ∈ D ′ ( Ω ) , where $$\Omega \ \in R^{N},\ $$ Ω ∈ R N , $$u\in W^{1,p}(\Omega )\ (1\le p\le +\infty ),\ A(u)\ $$ u ∈ W 1 , p ( Ω ) ( 1 ≤ p ≤ + ∞ ) , A ( u ) is an elliptic operator, $$H(u)\ $$ H ( u ) is an Hamiltonian that grows with $$\left| {\nabla u}\right| ^{p}$$ ∇ u p and f is given. Methods introduced in Artola (Boll UMI 6(5-B):51–71, 1986), (Proceedings of the International Conference on Generalized Functions, (ICGF 2000). Cambridge Scientific Publishers, Cambridge, 51–92, 2004), (Ricerche di Matematica XLIV, fasc. 2:400–420, 1995) for quasilinear parabolic or elliptic equations, together with properties for some continuity moduli, are used to improve some results from Barles and Murat (Arch Ration Mech Anal 133(1):77–101, 1995) for bounded solutions and from Barles and Porretta (Ann Scuola Norm Sup Pisa Cl Sci 5(1):107–136, 2006), Lions (J Anal Math 45: 234–254, 1985) for unbounded solutions, when 1 $$\le p\le 2.$$ ≤ p ≤ 2 . Unilateral problems are considered and the case where f depends on the solution u is also discussed.

Journal

Bollettino dell'Unione Matematica ItalianaSpringer Journals

Published: Jun 23, 2017

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 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

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

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off