The Liouville Equation with Singular Data: A Concentration-Compactness Principle via a Local Representation Formula☆
TLDR
For a bounded domain Ω⊂ R 2, the authors established a concentration-compactness result for the following class of singular Liouville equations: −Δu =e u −4π ∑ j=1 m α j δ p j in Ω where pj∈Ω, αj>0 and δpj denotes the Dirac measure with pole at point pj, j= 1, m.About:
This article is published in Journal of Differential Equations.The article was published on 2002-10-10 and is currently open access. It has received 71 citations till now. The article focuses on the topics: Liouville field theory & Dirac measure.read more
Citations
More filters
Journal ArticleDOI
Singular limits in liouville-type equations
TL;DR: In this article, the boundary value problem with homogeneous Dirichlet boundary conditions was considered and conditions under which there exists a solution for any given $m \ge 1$.
Journal ArticleDOI
Analytic aspects of the Toda system: II. Bubbling behavior and existence of solutions
TL;DR: In this article, the authors studied the 2-dimensional Toda lattice for the open case and gave a much more precise bubbling behavior of solutions and studied its existence in some critical cases.
Journal ArticleDOI
Mean Field Equation of Liouville Type with Singular Data: Topological Degree
Chiun-Chuan Chen,Chang-Shou Lin +1 more
TL;DR: In this article, the authors derived the topological degree counting formula for noncritical values of ρ and gave several applications of this formula, including existence of the curvature ǫ + 1 metric with conic singularities, doubly periodic solutions of electroweak theory, and a special case of self-gravitating strings.
Journal ArticleDOI
Compactness of solutions to the Yamabe problem
Yanyan Li,Lei Zhang +1 more
TL;DR: Li et al. as mentioned in this paper established compactness of solutions to the Yamabe problem on any smooth compact connected Riemannian manifold (not conformally diffeomorphic to standard spheres) of dimension n ⩽7 as well as under some additional hypothesis.
Journal ArticleDOI
Compactness of solutions to the Yamabe problem. III
Yanyan Li,Lei Zhang +1 more
TL;DR: For a sequence of blow up solutions of the Yamabe equation on non-locally conformally flat compact Riemannian manifolds of dimension 10 or 11, the authors established sharp estimates on its asymptotic profile near blow up points as well as sharp decay estimates of the Weyl tensor and its covariant derivatives at blow-up points.
References
More filters
Book
Partial Differential Equations
TL;DR: In this paper, the authors present a theory for linear PDEs: Sobolev spaces Second-order elliptic equations Linear evolution equations, Hamilton-Jacobi equations and systems of conservation laws.
Journal ArticleDOI
Uniform estimates and blow–up behavior for solutions of −δ(u)=v(x)eu in two dimensions
Haim Brezis,Frank Merle +1 more
TL;DR: In this article, uniform estimates and blow-up behavior for solutions of −δ(u) = v(x)eu in two dimensions are presented, with a focus on partial differential equations.
Journal ArticleDOI
A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description
TL;DR: In this paper, the canonical Gibbs measure associated to a N-vortex system in a bounded domain Λ, at inverse temperature, was considered and it was shown that, in the limitN→∞, β∈(−8π, + ∞) (here α denotes the vorticity intensity of each vortex), the one particle distribution function ϱN = ϱnx,x∈Λ converges to a superposition of solutions ϱα of the following Mean Field Equation:
Related Papers (5)
Uniform estimates and blow–up behavior for solutions of −δ(u)=v(x)eu in two dimensions
Haim Brezis,Frank Merle +1 more
Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces
Chiun-Chuan Chen,Chang-Shou Lin +1 more