The Lin-Ni's problem for mean convex domains / Olivier Druet, Frédéric Robert, Juncheng Wei.

Format:

Book

Author/Creator:

Druet, Olivier, 1976- author.

Robert, Frédéric, 1974- author.

Wei, Juncheng, 1968- author.

Series:

Memoirs of the American Mathematical Society ; Volume 218, Number 1027.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 218, Number 1027

Language:

English

Subjects (All):

Neumann problem.

Differential equations, Elliptic.

Blowing up (Algebraic geometry).

Convex domains.

Physical Description:

1 online resource (105 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2011.

Language Note:

English

Summary:

The authors prove some refined asymptotic estimates for positive blow-up solutions to $\Delta u+\epsilon u=n(n-2)u^{\frac{n+2}{n-2}}$ on $\Omega$, $\partial_\nu u=0$ on $\partial\Omega$, $\Omega$ being a smooth bounded domain of $\mathbb{R}^n$, $n\geq 3$. In particular, they show that concentration can occur only on boundary points with nonpositive mean curvature when $n=3$ or $n\geq 7$. As a direct consequence, they prove the validity of the Lin-Ni's conjecture in dimension $n=3$ and $n\geq 7$ for mean convex domains and with bounded energy. Recent examples by Wang-Wei-Yan show that the bound on the energy is a necessary condition.

Contents:

""Contents""; ""Abstract""; ""Introduction""; ""Chapter 1. L-bounded solutions""; ""Chapter 2. Smooth domains and extensions of solutions to elliptic equations""; ""Chapter 3. Exhaustion of the concentration points""; ""Chapter 4. A first upper-estimate""; ""Chapter 5. A sharp upper-estimate""; ""Chapter 6. Asymptotic estimates in C1()""; ""Chapter 7. Convergence to singular harmonic functions""; ""1. Convergence at general scale""; ""2. Convergence at appropriate scale""; ""Chapter 8. Estimates of the interior blow-up rates""; ""Chapter 9. Estimates of the boundary blow-up rates""

""Chapter 10. Proof of Theorems 1 and 2""""Appendix A. Construction and estimates on the Green's function""; ""Appendix B. Projection of the test functions""; ""Bibliography""

Notes:

"July 2012, Volume 218, Number 1027 (end of volume)."

Includes bibliographical references.

Description based on print version record.

ISBN:

0-8218-9016-6