65 research outputs found
Asymptotics and quantization for a mean-field equation of higher order
Given a regular bounded domain , we describe the
limiting behavior of sequences of solutions to the mean field equation of order
, , under the Dirichlet boundary condition and
the bound . We emphasize the connection with the problem of
prescribing the -curvature.Comment: 21 page
A theory of -dissipative solvers for scalar conservation laws with discontinuous flux
We propose a general framework for the study of contractive semigroups
of solutions to conservation laws with discontinuous flux. Developing the ideas
of a number of preceding works we claim that the whole admissibility issue is
reduced to the selection of a family of "elementary solutions", which are
certain piecewise constant stationary weak solutions. We refer to such a family
as a "germ". It is well known that (CL) admits many different contractive
semigroups, some of which reflects different physical applications. We revisit
a number of the existing admissibility (or entropy) conditions and identify the
germs that underly these conditions. We devote specific attention to the
anishing viscosity" germ, which is a way to express the "-condition" of
Diehl. For any given germ, we formulate "germ-based" admissibility conditions
in the form of a trace condition on the flux discontinuity line (in the
spirit of Vol'pert) and in the form of a family of global entropy inequalities
(following Kruzhkov and Carrillo). We characterize those germs that lead to the
-contraction property for the associated admissible solutions. Our
approach offers a streamlined and unifying perspective on many of the known
entropy conditions, making it possible to recover earlier uniqueness results
under weaker conditions than before, and to provide new results for other less
studied problems. Several strategies for proving the existence of admissible
solutions are discussed, and existence results are given for fluxes satisfying
some additional conditions. These are based on convergence results either for
the vanishing viscosity method (with standard viscosity or with specific
viscosities "adapted" to the choice of a germ), or for specific germ-adapted
finite volume schemes
Estimates for the Sobolev trace constant with critical exponent and applications
In this paper we find estimates for the optimal constant in the critical
Sobolev trace inequality S\|u\|^p_{L^{p_*}(\partial\Omega) \hookrightarrow
\|u\|^p_{W^{1,p}(\Omega)} that are independent of . This estimates
generalized those of [3] for general . Here is the
critical exponent for the immersion and is the space dimension. Then we
apply our results first to prove existence of positive solutions to a nonlinear
elliptic problem with a nonlinear boundary condition with critical growth on
the boundary, generalizing the results of [16]. Finally, we study an optimal
design problem with critical exponent.Comment: 22 pages, submitte
Positive Least Energy Solutions and Phase Separation for Coupled Schrodinger Equations with Critical Exponent: Higher Dimensional Case
We study the following nonlinear Schr\"{o}dinger system which is related to
Bose-Einstein condensate: {displaymath} {cases}-\Delta u +\la_1 u = \mu_1
u^{2^\ast-1}+\beta u^{\frac{2^\ast}{2}-1}v^{\frac{2^\ast}{2}}, \quad x\in
\Omega, -\Delta v +\la_2 v =\mu_2 v^{2^\ast-1}+\beta v^{\frac{2^\ast}{2}-1}
u^{\frac{2^\ast}{2}}, \quad x\in \om, u\ge 0, v\ge 0 \,\,\hbox{in \om},\quad
u=v=0 \,\,\hbox{on \partial\om}.{cases}{displaymath} Here \om\subset \R^N
is a smooth bounded domain, is the Sobolev critical
exponent, -\la_1(\om)0 and , where
\lambda_1(\om) is the first eigenvalue of with the Dirichlet
boundary condition. When \bb=0, this is just the well-known Brezis-Nirenberg
problem. The special case N=4 was studied by the authors in (Arch. Ration.
Mech. Anal. 205: 515-551, 2012). In this paper we consider {\it the higher
dimensional case }. It is interesting that we can prove the existence
of a positive least energy solution (u_\bb, v_\bb) {\it for any } (which can not hold in the special case N=4). We also study the limit
behavior of (u_\bb, v_\bb) as and phase separation is
expected. In particular, u_\bb-v_\bb will converge to {\it sign-changing
solutions} of the Brezis-Nirenberg problem, provided . In case
\la_1=\la_2, the classification of the least energy solutions is also
studied. It turns out that some quite different phenomena appear comparing to
the special case N=4.Comment: 48 pages. This is a revised version of arXiv:1209.2522v1 [math.AP
On exact number of solutions at infinity for Ambrosetti-Prodi class of problems
This article does not have an abstract
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