location:  Publications → journals
Search results

Search: MSC category 35 ( Partial differential equations )

 Expand all        Collapse all Results 26 - 50 of 56

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. CMB 2004 (vol 47 pp. 504)

Cardoso, Fernando; Vodev, Georgi
 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

45. CMB 2004 (vol 47 pp. 407)

Nedelec, L.
 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

46. CMB 2004 (vol 47 pp. 417)

Niu, Pengcheng; Han, Yanwu; Han, Junqiang
 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

47. CMB 2003 (vol 46 pp. 323)

Chamberland, Marc
 Characterizing Two-Dimensional 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 $$\alpha\beta (d-a) + b\alpha^2 - c\beta^2 = 0.$$ This paper shows how these theorems cannot be extended by constructing a real-analytic map whose Jacobian eigenvalues are $\pm 1/2$ and does not fit the previous form. This example is also used to construct non-obvious solutions to nonlinear PDEs, including the Monge--Amp\ere equation. Keywords:Jacobian Conjecture, injectivity, Monge--AmpÃ¨re equationCategories:26B10, 14R15, 35L70

48. CMB 2001 (vol 44 pp. 346)

Wang, Wei
 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, extremalsCategories:35J20, 35J60

49. CMB 2001 (vol 44 pp. 210)

Leung, Man Chun
 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 estimateCategories:35J60, 58G03

50. CMB 2000 (vol 43 pp. 477)

Ward, Michael J.
 The Dynamics of Localized Solutions of Nonlocal Reaction-Diffusion Equations Many classes of singularly perturbed reaction-diffusion 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 activator-inhibitor 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
 Page Previous 1 2 3 Next
 top of page | contact us | privacy | site map |