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1. CMB Online first
Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic SchrÃ¶dinger Operators |
Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic SchrÃ¶dinger Operators Let $A:=-(\nabla-i\vec{a})\cdot(\nabla-i\vec{a})+V$ be a
magnetic SchrÃ¶dinger operator on $\mathbb{R}^n$,
where $\vec{a}:=(a_1,\dots, a_n)\in L^2_{\mathrm{loc}}(\mathbb{R}^n,\mathbb{R}^n)$
and $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ satisfy some reverse
HÃ¶lder conditions.
Let $\varphi\colon \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be such that
$\varphi(x,\cdot)$ for any given $x\in\mathbb{R}^n$ is an Orlicz function,
$\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ for all $t\in (0,\infty)$
(the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index
$I(\varphi)\in(0,1]$. In this article, the authors prove that
second-order Riesz transforms $VA^{-1}$ and
$(\nabla-i\vec{a})^2A^{-1}$ are bounded from the
Musielak-Orlicz-Hardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$,
to the Musielak-Orlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, the authors
establish the boundedness of $VA^{-1}$ on $H_{\varphi, A}(\mathbb{R}^n)$. As applications, some
maximal inequalities associated with $A$ in the scale of $H_{\varphi,
A}(\mathbb{R}^n)$ are obtained.
Keywords:Musielak-Orlicz-Hardy space, magnetic SchrÃ¶dinger operator, atom, second-order Riesz transform, maximal inequality Categories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30 |
2. CMB 2011 (vol 55 pp. 663)
An Onofri-type Inequality on the Sphere with Two Conical Singularities In this paper, we give a new proof of the Onofri-type inequality
\begin{equation*}
\int_S e^{2u} \,ds^2 \leq 4\pi(\beta+1) \exp \biggl\{
\frac{1}{4\pi(\beta+1)} \int_S |\nabla u|^2 \,ds^2 +
\frac{1}{2\pi(\beta+1)} \int_S u \,ds^2 \biggr\}
\end{equation*}
on the sphere $S$ with Gaussian curvature $1$ and with conical
singularities divisor $\mathcal A = \beta\cdot p_1 + \beta \cdot p_2$ for
$\beta\in (-1,0)$; here $p_1$ and $p_2$ are antipodal.
Categories:53C21, 35J61, 53A30 |
3. CMB 2011 (vol 55 pp. 537)
Asymptotic Properties of Solutions to Semilinear Equations Involving Multiple Critical Exponents In this paper, we investigate
a semilinear elliptic equation that involves multiple
Hardy-type terms and critical Hardy-Sobolev exponents. By the
Moser iteration method and analytic techniques, the asymptotic
properties of its nontrivial solutions at the singular points are
investigated.
Keywords:elliptic problem, solution, Hardy-Sobolev inequality, singularity, Moser iteration Categories:35B33, 35B40, 35J60 |
4. CMB 2009 (vol 52 pp. 555)
Boundary Behavior of Solutions of the Helmholtz Equation This paper is concerned with the boundary behavior of solutions of
the Helmholtz equation in $\mathbb{R}^\di$.
In particular, we give a Littlewood-type theorem to show that
the approach region introduced by Kor\'anyi and Taylor (1983) is best possible.
Keywords:boundary behavior, Helmholtz equation Categories:31B25, 35J05 |
5. CMB 2008 (vol 51 pp. 140)
First Variations of the Best Sobolev Trace Constant with Respect to the Domain In this paper we study the best constant of the Sobolev trace
embedding $H^{1}(\Omega)\to L^{2}(\partial\Omega)$, where $\Omega$
is a bounded smooth domain in $\RR^N$. We find a formula for the
first variation of the best constant with respect to the domain.
As a consequence, we prove that the ball is a critical domain when
we consider deformations that preserve volume.
Keywords:nonlinear boundary conditions, Sobolev trace embedding Categories:35J65, 35B33 |
6. CMB 2007 (vol 50 pp. 356)
Existence of Positive Solutions for Nonlinear Noncoercive Hemivariational Inequalities In this paper we investigate the existence of positive solutions
for nonlinear elliptic problems driven by the $p$-Laplacian with a
nonsmooth potential (hemivariational inequality). Under asymptotic
conditions that make the Euler functional indefinite and
incorporate in our framework the asymptotically linear problems,
using a variational approach based on nonsmooth critical point
theory, we obtain positive smooth solutions. Our analysis also
leads naturally to multiplicity results.
Keywords:$p$-Laplacian, locally Lipschitz potential, nonsmooth critical point theory, principal eigenvalue, positive solutions, nonsmooth Mountain Pass Theorem Categories:35J20, 35J60, 35J85 |
7. CMB 2006 (vol 49 pp. 358)
On the Principal Eigencurve of the $p$-Laplacian: Stability Phenomena We show that each point of the principal eigencurve of the
nonlinear problem
$$
-\Delta_{p}u-\lambda m(x)|u|^{p-2}u=\mu|u|^{p-2}u \quad
\text{in } \Omega,
$$
is stable (continuous) with respect to the exponent $p$ varying in
$(1,\infty)$; we also prove some convergence results
of the principal eigenfunctions corresponding.
Keywords:$p$-Laplacian with indefinite weight, principal eigencurve, principal eigenvalue, principal eigenfunction, stability Categories:35P30, 35P60, 35J70 |
8. CMB 2006 (vol 49 pp. 144)
Scattering Length and the Spectrum of $-\Delta+V$ Given a non-negative, locally integrable function $V$
on $\RR^n$, we give a necessary and sufficient condition
that $-\Delta+V$ have purely discrete spectrum,
in terms of the scattering length
of $V$ restricted to boxes.
Category:35J10 |
9. CMB 2004 (vol 47 pp. 515)
Remarques sur l'enlacement en thÃ©orie des points critiques pour des fonctionnelles continues Dans cet article, \`a partir de la notion d'enlacement introduite
dans ~\cite{F} entre des paires d'ensembles $(B,A)$ et $(Q,P)$,
nous \'etablissons l'existence d'un point critique d'une
fonctionnelle continue sur un espace m\'etrique lorsqu'une de ces
paires enlace l'autre. Des renseignements sur la localisation du
point critique sont aussi obtenus. Ces r\'esultats conduisent \`a
une g\'en\'eralisation du th\'eor\`eme des trois points critiques.
Finalement, des applications \`a des probl\`emes aux limites pour
une \'equation quasi-lin\'eaire elliptique sont pr\'esent\'ees.
Categories:58E05, 35J20 |
10. CMB 2004 (vol 47 pp. 504)
High Frequency Resolvent Estimates and Energy Decay of Solutions to the Wave Equation We prove an uniform H\"older continuity of the resolvent of
the Laplace-Beltrami operator on the real axis for a class
of asymptotically Euclidean Riemannian manifolds. As an application we
extend a result of Burq on the behaviour of the
local energy of solutions to the wave equation.
Categories:35B37, 35J15, 47F05 |
11. CMB 2001 (vol 44 pp. 346)
Positive Solution of a Subelliptic Nonlinear Equation on the Heisenberg Group In this paper, we establish the existence of positive solution of a
nonlinear subelliptic equation involving the critical Sobolev exponent on
the Heisenberg group, which generalizes a result of Brezis and
Nirenberg in the Euclidean case.
Keywords:Heisenberg group, subLapacian, critical Sobolev exponent, extremals Categories:35J20, 35J60 |
12. CMB 2001 (vol 44 pp. 210)
Growth Estimates on Positive Solutions of the Equation $\Delta u+K u^{\frac{n+2}{n-2}}=0$ in $\R^n$ We construct unbounded positive $C^2$-solutions of the equation
$\Delta u + K u^{(n + 2)/(n - 2)} = 0$ in $\R^n$ (equipped
with Euclidean metric $g_o$) such that $K$ is bounded between two
positive numbers in $\R^n$, the conformal metric $g=u^{4/(n-2)}g_o$
is complete, and the volume growth of $g$ can be arbitrarily fast
or reasonably slow according to the constructions. By imposing natural
conditions on $u$, we obtain growth estimate on the $L^{2n/(n-2)}$-norm
of the solution and show that it has slow decay.
Keywords:positive solution, conformal scalar curvature equation, growth estimate Categories:35J60, 58G03 |
13. CMB 1997 (vol 40 pp. 464)
On the solvability of a Neumann boundary value problem at resonance We study the existence of solutions of the semilinear equations (1)
$\triangle u + g(x,u)=h$, ${\partial u \over \partial n} = 0$ on
$\partial \Omega$ in which the non-linearity $g$ may grow
superlinearly in $u$ in one of directions $u \to \infty$ and $u \to
-\infty$, and (2) $-\triangle u + g(x,u)=h$, ${\partial u \over
\partial n} = 0$ on $\partial \Omega$ in which the nonlinear term $g$
may grow superlinearly in $u$ as $|u| \to \infty$. The purpose of this
paper is to obtain solvability theorems for (1) and (2) when the
Landesman-Lazer condition does not hold. More precisely, we require
that $h$ may satisfy $\int g^\delta_- (x) \, dx < \int h(x) \, dx = 0<
\int g^\gamma_+ (x)\,dx$, where $\gamma, \delta$ are arbitrarily
nonnegative constants, $g^\gamma_+ (x) = \lim_{u \to \infty} \inf
g(x,u) |u|^\gamma$ and $g^\delta_- (x)=\lim_{u \to -\infty} \sup
g(x,u)|u|^\delta$. The proofs are based upon degree theoretic arguments.
Keywords:Landesman-Lazer condition, Leray Schauder degree Categories:35J65, 47H11, 47H15 |
14. CMB 1997 (vol 40 pp. 244)
Nonexistence results of positive entire solutions for quasilinear elliptic inequalities This paper treats the quasilinear elliptic inequality
$$
\div (|Du|^{m-2}Du) \geq p(x)u^{\sigma},
\quad x \in \Rs^N,
$$
where $N \geq 2$, $m > 1$, $ \sigma > m - 1$, and $p \colon \Rs^N
\rightarrow (0, \infty)$ is continuous. Sufficient conditions are
given for this inequality to have no positive entire solutions. When
$p$ has radial symmetry, the existence of positive entire solutions can
be characterized by our results and some known results.
Categories:35J70, 35B05 |