89 research outputs found
Anti symmetric solutions of non-linear laminar flow between parallel permeable disks
The equations describing similarity solutions for flow between infinite parallel permeable disks with equal rates of suction or injection at the walls is derived using the stream function. This leads to a fourth order non-linear Ordinary Differential Equation. This equation is shown to admit anti-symmetric solutions using the moving plane method
A threshold phenomenon for embeddings of into Orlicz spaces
We consider a sequence of positive smooth critical points of the
Adams-Moser-Trudinger embedding of into Orlicz spaces. We study its
concentration-compactness behavior and show that if the sequence is not
precompact, then the liminf of the -norms of the functions is greater
than or equal to a positive geometric constant.Comment: 14 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
Existence of solutions to a higher dimensional mean-field equation on manifolds
For we prove an existence result for the equation on a closed Riemannian
manifold of dimension for certain values of .Comment: 15 Page
Asymptotic behaviour of a semilinear elliptic system with a large exponent
Consider the problem \begin{eqnarray*} -\Delta u &=& v^{\frac 2{N-2}},\quad
v>0\quad {in}\quad \Omega, -\Delta v &=& u^{p},\:\:\:\quad u>0\quad {in}\quad
\Omega, u&=&v\:\:=\:\:0 \quad {on}\quad \partial \Omega, \end{eqnarray*} where
is a bounded convex domain in with smooth boundary
We study the asymptotic behaviour of the least energy
solutions of this system as We show that the solution remain
bounded for large and have one or two peaks away form the boundary. When
one peak occurs we characterize its location.Comment: 16 pages, submmited for publicatio
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
A Computer-Assisted Uniqueness Proof for a Semilinear Elliptic Boundary Value Problem
A wide variety of articles, starting with the famous paper (Gidas, Ni and
Nirenberg in Commun. Math. Phys. 68, 209-243 (1979)) is devoted to the
uniqueness question for the semilinear elliptic boundary value problem
-{\Delta}u={\lambda}u+u^p in {\Omega}, u>0 in {\Omega}, u=0 on the boundary of
{\Omega}, where {\lambda} ranges between 0 and the first Dirichlet Laplacian
eigenvalue. So far, this question was settled in the case of {\Omega} being a
ball and, for more general domains, in the case {\lambda}=0. In (McKenna et al.
in J. Differ. Equ. 247, 2140-2162 (2009)), we proposed a computer-assisted
approach to this uniqueness question, which indeed provided a proof in the case
{\Omega}=(0,1)x(0,1), and p=2. Due to the high numerical complexity, we were
not able in (McKenna et al. in J. Differ. Equ. 247, 2140-2162 (2009)) to treat
higher values of p. Here, by a significant reduction of the complexity, we will
prove uniqueness for the case p=3
Sharp constants in weighted trace inequalities on Riemannian manifolds
We establish some sharp weighted trace inequalities
W^{1,2}(\rho^{1-2\sigma}, M)\hookrightarrow L^{\frac{2n}{n-2\sigma}}(\pa M)
on dimensional compact smooth manifolds with smooth boundaries, where
is a defining function of and . This is stimulated
by some recent work on fractional (conformal) Laplacians and related problems
in conformal geometry, and also motivated by a conjecture of Aubin.Comment: 34 page
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