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26. CMB 2007 (vol 50 pp. 35)
A Singular Critical Potential for the SchrÃ¶dinger Operator Consider a real potential $V$ on
$\RR^d$, $d\geq 2$, and the Schr\"odinger equation:
\begin{equation}
\tag{LS} \label{LS1} i\partial_t u +\Delta u Vu=0,\quad
u_{\restriction t=0}=u_0\in L^2.
\end{equation}
In this paper, we investigate the minimal local regularity of $V$
needed to get local in time dispersive estimates (such as local in
time Strichartz estimates or local smoothing effect with gain of
$1/2$ derivative) on solutions of \eqref{LS1}. Prior works
show some dispersive properties when $V$ (small at infinity) is in
$L^{d/2}$ or in spaces just a little larger but with a smallness
condition on $V$ (or at least on its negative part).
In this work, we prove the critical character of these results by
constructing a positive potential $V$ which has compact support,
bounded outside $0$ and of the order $(\logx)^2/x^2$ near $0$.
The lack of dispersiveness comes from the existence of a sequence
of quasimodes for the operator $P:=\Delta+V$.
The elementary construction of $V$ consists in sticking together
concentrated, truncated potential wells near $0$. This yields a
potential oscillating with infinite speed and amplitude at $0$,
such that the operator $P$ admits a sequence of quasimodes of
polynomial order whose support concentrates on the pole.
Categories:35B65, 35L05, 35Q40, 35Q55 
27. 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
$$
\Delta_{p}u\lambda m(x)u^{p2}u=\muu^{p2}u \quad
\text{in } \Omega,
$$
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 
28. CMB 2006 (vol 49 pp. 226)
The Spectrum and Isometric Embeddings of Surfaces of Revolution A sharp upper bound on the first $S^{1}$ invariant eigenvalue of the Laplacian
for $S^1$ invariant metrics on $S^2$ is used to find obstructions to the existence
of $S^1$ equivariant isometric embeddings of such metrics in $(\R^3,\can)$. As a
corollary we prove: If the first four distinct eigenvalues have even multiplicities
then the metric cannot be equivariantly, isometrically embedded in $(\R^3,\can)$. This
leads to generalizations of some classical results in the theory of surfaces.
Categories:58J50, 58J53, 53C20, 35P15 
29. CMB 2006 (vol 49 pp. 144)
Scattering Length and the Spectrum of $\Delta+V$ Given a nonnegative, 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 
30. CMB 2005 (vol 48 pp. 405)
Liouville's Theorem in the Radially Symmetric Case We present a very short proof of Liouville's theorem for solutions
to a nonuniformly elliptic radially symmetric equation. The proof uses
the Ricatti equation satisfied by the Dirichlet to Neumann map.
Categories:35B05, 34A30 
31. CMB 2005 (vol 48 pp. 3)
Quantum Ergodicity of Boundary Values of Eigenfunctions: A Control Theory Approach Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a
Riemanian manifold with piecewise smooth boundary and suppose that the
billiard associated to the geodesic flow reflecting on the boundary
according to the laws of geometric optics is ergodic.
We prove that the boundary value of the eigenfunctions of the Laplace
operator with reasonable boundary conditions are asymptotically
equidistributed in the boundary, extending previous results by
G\'erard and Leichtnam as well as Hassell and Zelditch,
obtained under the additional assumption of the convexity of~$M$.
Categories:35Q55, 35BXX, 37K05, 37L50, 81Q20 
32. 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 quasilin\'eaire elliptique sont pr\'esent\'ees.
Categories:58E05, 35J20 
33. 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 LaplaceBeltrami 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 
34. CMB 2004 (vol 47 pp. 407)
Multiplicity of Resonances in Black Box Scattering We apply the method of complex scaling to give a natural
proof of a formula relating the multiplicity of a resonance to the
multiplicity of a pole of the scattering matrix.
Category:35P25 
35. CMB 2004 (vol 47 pp. 417)
A Hopf Type Lemma and a CR Type Inversion for the Generalized Greiner Operator In this paper we establish a Hopf type lemma and a CR type
inversion for the generalized Greiner operator. Some nonlinear
Keywords:Hopf type lemma, CR inversion, Liouville type, theorem generalized Greiner operator 
36. CMB 2003 (vol 46 pp. 323)
Characterizing TwoDimensional Maps Whose Jacobians Have Constant Eigenvalues Recent papers have shown that $C^1$ maps $F\colon \mathbb{R}^2
\rightarrow \mathbb{R}^2$
whose Jacobians have constant eigenvalues can be completely
characterized if either the eigenvalues are equal or $F$ is a
polynomial. Specifically, $F=(u,v)$ must take the form
\begin{gather*}
u = ax + by + \beta \phi(\alpha x + \beta y) + e \\
v = cx + dy  \alpha \phi(\alpha x + \beta y) + f
\end{gather*}
for some constants $a$, $b$, $c$, $d$, $e$, $f$, $\alpha$, $\beta$ and
a $C^1$ function $\phi$ in one variable. If, in addition, the function
$\phi$ is not affine, then
\begin{equation}
\alpha\beta (da) + b\alpha^2  c\beta^2 = 0.
\end{equation}
This paper shows how these theorems cannot be extended by constructing
a realanalytic map whose Jacobian eigenvalues are $\pm 1/2$ and does
not fit the previous form. This example is also used to construct
nonobvious solutions to nonlinear PDEs, including the MongeAmp\`ere
equation.
Keywords:Jacobian Conjecture, injectivity, MongeAmpÃ¨re equation Categories:26B10, 14R15, 35L70 
37. 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 
38. CMB 2001 (vol 44 pp. 210)
Growth Estimates on Positive Solutions of the Equation $\Delta u+K u^{\frac{n+2}{n2}}=0$ 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 $g=u^{4/(n2)}g_o$
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 $L^{2n/(n2)}$norm
of the solution and show that it has slow decay.
Keywords:positive solution, conformal scalar curvature equation, growth estimate Categories:35J60, 58G03 
39. CMB 2000 (vol 43 pp. 477)
The Dynamics of Localized Solutions of Nonlocal ReactionDiffusion Equations Many classes of singularly perturbed reactiondiffusion equations
possess localized solutions where the gradient of the solution is
large only in the vicinity of certain points or interfaces in the
domain. The problems of this type that are considered are an
interface propagation model from materials science and an
activatorinhibitor model of morphogenesis. These two models are
formulated as nonlocal partial differential equations. Results
concerning the existence of equilibria, their stability, and the
dynamical behavior of localized structures in the interior and on
the boundary of the domain are surveyed for these two models. By
examining the spectrum associated with the linearization of these
problems around certain canonical solutions, it is shown that the
nonlocal term can lead to the existence of an exponentially small
principal eigenvalue for the linearized problem. This eigenvalue
is then responsible for an exponentially slow, or metastable,
motion of the localized structure.
Categories:35Q35, 35C20, 35K60 
40. CMB 2000 (vol 43 pp. 51)
Eigenfunction Decay For the Neumann Laplacian on HornLike Domains The growth properties at infinity for eigenfunctions corresponding to
embedded eigenvalues of the Neumann Laplacian on hornlike domains
are studied. For domains that pinch at polynomial rate, it is shown
that the eigenfunctions vanish at infinity faster than the reciprocal
of any polynomial. For a class of domains that pinch at an exponential
rate, weaker, $L^2$ bounds are proven. A corollary is that eigenvalues
can accumulate only at zero or infinity.
Keywords:Neumann Laplacian, hornlike domain, spectrum Categories:35P25, 58G25 
41. CMB 1999 (vol 42 pp. 169)
Heat Kernels of Lorentz Cones We obtain an explicit formula for heat kernels of Lorentz cones, a
family of classical symmetric cones. By this formula, the heat
kernel of a Lorentz cone is expressed by a function of time $t$ and
two eigenvalues of an element in the cone. We obtain also upper and
lower bounds for the heat kernels of Lorentz cones.
Keywords:Lorentz cone, symmetric cone, Jordan algebra, heat kernel, heat equation, LaplaceBeltrami operator, eigenvalues Categories:35K05, 43A85, 35K15, 80A20 
42. 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 nonlinearity $g$ 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
LandesmanLazer 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:LandesmanLazer condition, Leray Schauder degree Categories:35J65, 47H11, 47H15 
43. CMB 1997 (vol 40 pp. 174)
Nonuniqueness for the $p$harmonic flow If $f_0\colon\Omega\subset \R^m\to S^n$ is a weakly $p$harmonic map from
a bounded smooth domain $\Omega$ in $\R^m$ (with $2

44. CMB 1997 (vol 40 pp. 244)
Nonexistence results of positive entire solutions for quasilinear elliptic inequalities This paper treats the quasilinear elliptic inequality
$$
\div (Du^{m2}Du) \geq p(x)u^{\sigma},
\quad x \in \Rs^N,
$$
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 
45. CMB 1997 (vol 40 pp. 60)
Cauchy's problem for harmonic functions with entire data on a sphere We give an elementary potentialtheoretic proof of a theorem of
G.~Johnsson: all solutions of Cauchy's problems for the Laplace
equations with an entire data on a sphere extend harmonically to
the whole space ${\bf R}^N$ except, perhaps, for the center of the
sphere.
Keywords:harmonic functions, Cauchy's problem, homogeneous harmonics Categories:35B60, 31B20 