26. CMB 2011 (vol 55 pp. 88)
 Ghanbari, K.; Shekarbeigi, B.

Inequalities for Eigenvalues of a General Clamped Plate Problem
Let $D$ be a
connected bounded domain in $\mathbb{R}^n$. Let
$0<\mu_1\leq\mu_2\leq\dots\leq\mu_k\leq\cdots$ be the eigenvalues
of the following Dirichlet
problem:
$$
\begin{cases}\Delta^2u(x)+V(x)u(x)=\mu\rho(x)u(x),\quad x\in
D
u_{\partial D}=\frac{\partial u}{\partial n}_{\partial
D}=0,
\end{cases}
$$
where $V(x)$ is a nonnegative potential,
and $\rho(x)\in C(\bar{D})$ is positive.
We prove the following inequalities:
$$\mu_{k+1}\leq\frac{1}{k}\sum_{i=1}^k\mu_i+\Bigl[\frac{8(n+2)}{n^2}\Bigl(\frac{\rho_{\max}}
{\rho_{\min}}\Bigr)^2\Bigr]^{1/2}\times
\frac{1}{k}\sum_{i=1}^k[\mu_i(\mu_{k+1}\mu_i)]^{1/2},
$$
$$\frac{n^2k^2}{8(n+2)}\leq
\Bigl(\frac{\rho_{\max}}{\rho_{\min}}\Bigr)^2\Bigl[\sum_{i=1}^k\frac{\mu_i^{1/2}}{\mu_{k+1}\mu_i}\Bigr]
\times\sum_{i=1}^k\mu_i^{1/2}.
$$
Keywords:biharmonic operator, eigenvalue, eigenvector, inequality Category:35P15 

27. CMB 2011 (vol 54 pp. 249)
28. CMB 2010 (vol 54 pp. 28)
 Chang, YuHsien; Hong, ChengHong

Generalized Solution of the Photon Transport Problem
The purpose of this paper is to show the existence of a
generalized solution of the photon transport problem. By means of the theory of
equicontinuous $C_{0}$semigroup on a sequentially complete locally convex
topological vector space we show that the perturbed abstract Cauchy problem
has a unique solution when the perturbation operator and the forcing term
function satisfy certain conditions. A consequence of the abstract result is
that it can be directly applied to obtain a generalized solution of the photon
transport problem.
Keywords:photon transport, $C_{0}$semigroup Categories:35K30, 47D03 

29. CMB 2010 (vol 54 pp. 126)
30. CMB 2010 (vol 53 pp. 674)
 Kristály, Alexandru; Papageorgiou, Nikolaos S.; Varga, Csaba

Multiple Solutions for a Class of Neumann Elliptic Problems on Compact Riemannian Manifolds with Boundary
We study a semilinear elliptic problem on a compact Riemannian
manifold with boundary, subject to an inhomogeneous Neumann
boundary condition. Under various hypotheses on the nonlinear
terms, depending on their behaviour in the origin and infinity, we
prove multiplicity of solutions by using variational arguments.
Keywords:Riemannian manifold with boundary, Neumann problem, sublinearity at infinity, multiple solutions Categories:58J05, 35P30 

31. CMB 2010 (vol 53 pp. 737)
32. CMB 2009 (vol 53 pp. 163)
33. CMB 2009 (vol 53 pp. 153)
34. CMB 2009 (vol 53 pp. 295)
 Guo, Boling; Huo, Zhaohui

The Global Attractor of a Damped, Forced Hirota Equation in $H^1$
The existence of the global attractor of a damped
forced Hirota equation in the phase space $H^1(\mathbb R)$ is proved. The
main idea is to establish the socalled asymptotic compactness
property of the solution operator by energy equation approach.
Keywords:global attractor, Fourier restriction norm, damping system, asymptotic compactness Categories:35Q53, 35B40, 35B41, 37L30 

35. CMB 2009 (vol 52 pp. 555)
 Hirata, Kentaro

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 Littlewoodtype 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 

36. CMB 2008 (vol 51 pp. 249)
 Mangoubi, Dan

On the Inner Radius of a Nodal Domain
Let $M$ be a closed Riemannian manifold.
We consider the inner radius of a nodal domain for a large eigenvalue $\lambda$.
We give upper and lower bounds on the inner radius of the type
$C/\lambda^\alpha(\log\lambda)^\beta$. Our proof is based on
a local behavior of eigenfunctions discovered by Donnelly and
Fefferman and a Poincar\'{e} type inequality proved by Maz'ya.
Sharp lower bounds are known
only in dimension two. We give an account of this case too.
Categories:58J50, 35P15, 35P20 

37. CMB 2008 (vol 51 pp. 140)
 Rossi, Julio D.

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 

38. CMB 2007 (vol 50 pp. 356)
 Filippakis, Michael E.; Papageorgiou, Nikolaos S.

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 

39. CMB 2007 (vol 50 pp. 35)
 Duyckaerts, Thomas

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 

40. CMB 2006 (vol 49 pp. 358)
 Khalil, Abdelouahed El; Manouni, Said El; Ouanan, Mohammed

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 

41. CMB 2006 (vol 49 pp. 226)
 Engman, Martin

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 

42. CMB 2006 (vol 49 pp. 144)
43. CMB 2005 (vol 48 pp. 405)
44. CMB 2005 (vol 48 pp. 3)
 Burq, N.

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 

45. CMB 2004 (vol 47 pp. 515)
 Frigon, M.

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 

46. CMB 2004 (vol 47 pp. 504)
47. CMB 2004 (vol 47 pp. 407)
48. CMB 2004 (vol 47 pp. 417)
49. CMB 2003 (vol 46 pp. 323)
 Chamberland, Marc

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 

50. CMB 2001 (vol 44 pp. 346)