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26. CMB 2011 (vol 55 pp. 663)

Zhou, Chunqin
 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

27. CMB 2011 (vol 55 pp. 537)

Kang, Dongsheng
 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 iterationCategories:35B33, 35B40, 35J60

28. CMB 2011 (vol 55 pp. 623)

Pan, Jiaqing
 The Continuous Dependence on the Nonlinearities of Solutions of Fast Diffusion Equations In this paper, we consider the Cauchy problem $$\begin{cases} u_{t}=\Delta(u^{m}), &x\in{}\mathbb{R}^{N}, t>0, N\geq3, \\ % ^^----- here u(x,0)=u_{0}(x), &x\in{}\mathbb{R}^{N}. \end{cases}$$ We will prove that: (i) for $m_{c} Keywords:fast diffusion equations, Cauchy problem, continuous dependence on nonlinearityCategories:35K05, 35K10, 35K15 29. CMB 2011 (vol 55 pp. 249) Chang, Der-Chen; Li, Bao Qin  Description of Entire Solutions of Eiconal Type Equations The paper describes entire solutions to the eiconal type non-linear partial differential equations, which include the eiconal equations$(X_1(u))^2+(X_2(u))^2=1$as special cases, where$X_1=p_1{\partial}/{\partial z_1}+p_2{\partial}/{\partial z_2}$,$X_2=p_3{\partial}/{\partial z_1}+p_4{\partial}/{\partial z_2}$are linearly independent operators with$p_j$being arbitrary polynomials in$\mathbf{C}^2$. Keywords:entire solution, eiconal equation, polynomial, transcendental functionCategories:32A15, 35F20 30. CMB 2011 (vol 55 pp. 3) Agarwal, Ravi P.; Mustafa, Octavian G.  On a Local Theory of Asymptotic Integration for Nonlinear Differential Equations We improve several recent results in the asymptotic integration theory of nonlinear ordinary differential equations via a variant of the method devised by J. K. Hale and N. Onuchic The results are used for investigating the existence of positive solutions to certain reaction-diffusion equations. Keywords:asymptotic integration, Emden-Fowler differential equation, reaction-diffusion equationCategories:34E10, 34C10, 35Q35 31. CMB 2011 (vol 55 pp. 88) Ghanbari, K.; Shekarbeigi, B.  Inequalities for Eigenvalues of a General Clamped Plate Problem Let$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, inequalityCategory:35P15 32. CMB 2011 (vol 54 pp. 249) Dattori da Silva, Paulo L.  A Note about Analytic Solvability of Complex Planar Vector Fields with Degeneracies This paper deals with the analytic solvability of a special class of complex vector fields defined on the real plane, where they are tangent to a closed real curve, while off the real curve, they are elliptic. Keywords:semi-global solvability, analytic solvability, normalization, complex vector fields, condition~($\mathcal P$)Categories:35A01, 58Jxx 33. CMB 2010 (vol 54 pp. 28) Chang, Yu-Hsien; Hong, Cheng-Hong  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,$C_{0}$-semigroupCategories:35K30, 47D03 34. CMB 2010 (vol 54 pp. 126) Jin, Yongyang; Zhang, Genkai  Fundamental Solutions of Kohn Sub-Laplacians on Anisotropic Heisenberg Groups and H-type Groups We prove that the fundamental solutions of Kohn sub-Laplacians$\Delta + i\alpha \partial_t$on the anisotropic Heisenberg groups are tempered distributions and have meromorphic continuation in$\alpha$with simple poles. We compute the residues and find the partial fundamental solutions at the poles. We also find formulas for the fundamental solutions for some matrix-valued Kohn type sub-Laplacians on H-type groups. Categories:22E30, 35R03, 43A80 35. 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 solutionsCategories:58J05, 35P30 36. CMB 2010 (vol 53 pp. 737) Vougalter, Vitali  On the Negative Index Theorem for the Linearized Non-Linear SchrÃ¶dinger Problem A new and elementary proof is given of the recent result of Cuccagna, Pelinovsky, and Vougalter based on the variational principle for the quadratic form of a self-adjoint operator. It is the negative index theorem for a linearized NLS operator in three dimensions. Categories:35Q55, 81Q10 37. 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 so-called asymptotic compactness property of the solution operator by energy equation approach. Keywords:global attractor, Fourier restriction norm, damping system, asymptotic compactnessCategories:35Q53, 35B40, 35B41, 37L30 38. CMB 2009 (vol 53 pp. 163) Taylor, Michael  Variants of Arnold's Stability Results for 2D Euler Equations We establish variants of stability estimates in norms somewhat stronger than the$H^1$-norm under Arnold's stability hypotheses on steady solutions to the Euler equations for fluid flow on planar domains. Category:35Q35 39. CMB 2009 (vol 53 pp. 153) Niu, Pengcheng; Ou, Yafei; Han, Junqiang  Several Hardy Type Inequalities with Weights Related to Generalized Greiner Operator In this paper, we establish several weighted$L^p (1\lt p \lt \infty)$Hardy type inequalities related to the generalized Greiner operator by improving the method of Kombe. Then the best constants in inequalities are discussed by introducing new polar coordinates. Keywords:generalized Greiner operator, polar coordinates, Hardy inequalityCategories:35B05, 35H99 40. 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 Littlewood-type theorem to show that the approach region introduced by Kor\'anyi and Taylor (1983) is best possible. Keywords:boundary behavior, Helmholtz equationCategories:31B25, 35J05 41. 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 42. 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 embeddingCategories:35J65, 35B33 43. 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 TheoremCategories:35J20, 35J60, 35J85 44. 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: $$\tag{LS} \label{LS1} i\partial_t u +\Delta u -Vu=0,\quad u_{\restriction t=0}=u_0\in L^2.$$ 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$(\log|x|)^2/|x|^2$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 quasi-modes of polynomial order whose support concentrates on the pole. Categories:35B65, 35L05, 35Q40, 35Q55 45. 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 $$-\Delta_{p}u-\lambda m(x)|u|^{p-2}u=\mu|u|^{p-2}u \quad \text{in } \Omega,$$ 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, stabilityCategories:35P30, 35P60, 35J70 46. 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 47. CMB 2006 (vol 49 pp. 144) Taylor, Michael  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 48. CMB 2005 (vol 48 pp. 405) Froese, Richard  Liouville's Theorem in the Radially Symmetric Case We present a very short proof of Liouville's theorem for solutions to a non-uniformly elliptic radially symmetric equation. The proof uses the Ricatti equation satisfied by the Dirichlet to Neumann map. Categories:35B05, 34A30 49. 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 50. 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 quasi-lin\'eaire elliptique sont pr\'esent\'ees. Categories:58E05, 35J20
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