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Convex Analysisの二,三の進展について

Topology and Geometry

We consider the boundary value problem $
\Delta u + \varepsilon ^{2} k{\left( x \right)}e^{u} = 0$
 in a bounded, smooth domain $\Omega$
 in $
\mathbb{R}^{{\text{2}}} 
$

 with homogeneous Dirichlet boundary conditions. Here $$
\varepsilon > 0,k(x)
$$

 is a non-negative, not identically zero function. We find conditions under which there exists a solution $
u_{\varepsilon } 
$
 which blows up at exactly m points as $
\varepsilon \to 0
$
 and satisfies $
\varepsilon ^{2} {\int_\Omega {ke^{{u_{\varepsilon } }} \to 8m\pi } }%
$

 . In particular, we find that if $k\in C^2(\bar\Omega)$
 , $
\inf _{\Omega } k > 0
$
 and $\Omega$
 is not simply connected then such a solution exists for any given $m \ge 1$

/pdf/singular-limits-in-liouville-type-equations-4a2a66zgxn.pdf

Singular limits in liouville-type equations

Motivated by the study of multivortices in the Electroweak Theory of Glashow–Salam–Weinberg [33], we obtain a concentration-compactness principle for the following class of mean field equations: $\left( l \right)_\lambda - \Delta _g v = \lambda K\exp (v)/\int\limits_M {K\exp (v)d\tau _g - W}$ on M, where (M,g) is a compact 2-manifold without boundary, 0 0. We take $W = 4\pi \left( {\sum\limits_{i = 1}^m {\alpha _i \delta _{p_i } - \psi } } \right)$ with αi> 0, δpi the Dirac measure with pole at point pi∈M, i= 1,…,m and ψ∈L∞(M) satisfying the necessary integrability condition for the solvability of (1)λ. We provide an accurate analysis for solution sequences of (1)λ, which admit a “blow up” point at a pole pi of the Dirac measure, in the same spirit of the work of Brezis–Merle [11] and Li–Shafrir [35]. As a consequence, we are able to extend the work of Struwe–Tarantello [49] and Ding–Jost–Li–Wang [21] and derive necessary and sufficient conditions for the existence of periodic N-vortices in the Electroweak Theory. Our result is sharp for N= 1, 2, 3, 4 and was motivated by the work of Spruck–Yang [46], who established an analogous sharp result for N= 1, 2.

Liouville Type Equations with Singular Data¶and Their Applications to Periodic Multivortices¶for the Electroweak Theory

Motivated by the study of selfdual vortices in gauge field theory, we consider a class of Mean Field equations of Liouville-type on compact surfaces involving singular data assigned by Dirac measures supported at finitely many points (the so called vortex points). According to the applications, we need to describe the blow-up behavior of solution-sequences which concentrate exactly at the given vortex points. We provide accurate pointwise estimates for the profile of the bubbling sequences as well as “sup + inf” estimates for solutions. Those results extend previous work of Li [Li, Y. Y. (1999). Harnack type inequality: The method of moving planes. Comm. Math. Phys. 200:421–444] and Brezis et al. [Brezis, H., Li, Y. Shafrir, I. (1993). A sup + inf inequality for some nonlinear elliptic equations involving the exponential nonlinearities. J. Funct. Anal. 115: 344–358] relative to the “regular” case, namely in absence of singular sources.

Profile of Blow-up Solutions to Mean Field Equations with Singular Data

We study the problem of prescribing the Gaussian curvature on surfaces with conical singularities in supercritical regimes. Using a Morse-theoretical approach, we prove a general existence theorem on surfaces with positive genus, with a generic multiplicity result.

Supercritical conformal metrics on surfaces with conical singularities

The Liouville Equation with Singular Data: A Concentration-Compactness Principle via a Local Representation Formula☆

We consider a class of singular Liouville equations on compact surfaces motivated by the study of Electroweak and Self-Dual Chern-Simons theories, the Gaussian curvature prescription with conical singularities and Onsager's description of turbulence. We analyse the problem of existence variationally, and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the Moser-Trudinger inequality, and in turn to lower bounds on the Euler-Lagrange functional. We then discuss existence and non-existence results.

Daniele Bartolucci

Papers

Liouville Type Equations with Singular Data¶and Their Applications to Periodic Multivortices¶for the Electroweak Theory

Profile of Blow-up Solutions to Mean Field Equations with Singular Data

Supercritical conformal metrics on surfaces with conical singularities

The Liouville Equation with Singular Data: A Concentration-Compactness Principle via a Local Representation Formula☆

An Improved Geometric Inequality via Vanishing Moments, with Applications to Singular Liouville Equations