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1. CMB Online first

Steinerberger, Stefan
 An Endpoint Alexandrov Bakelman Pucci estimate in the Plane The classical Alexandrov-Bakelman-Pucci estimate for the Laplacian states $$\max_{x \in \Omega}{ |u(x)|} \leq \max_{x \in \partial \Omega}{|u(x)|} + c_{s,n} \operatorname{diam}(\Omega)^{2-\frac{n}{s}} \left\| \Delta u \right\|_{L^s(\Omega)}$$ where $\Omega \subset \mathbb{R}^n$, $u \in C^2(\Omega) \cap C(\overline{\Omega})$ and $s \gt n/2$. The inequality fails for $s = n/2$. A Sobolev embedding result of Milman and Pustylnik, originally phrased in a slightly different context, implies an endpoint inequality: if $n \geq 3$ and $\Omega \subset \mathbb{R}^n$ is bounded, then $$\max_{x \in \Omega}{ |u(x)|} \leq \max_{x \in \partial \Omega}{|u(x)|} + c_n \left\| \Delta u \right\|_{L^{\frac{n}{2},1}(\Omega)},$$ where $L^{p,q}$ is the Lorentz space refinement of $L^p$. This inequality fails for $n=2$ and we prove a sharp substitute result: there exists $c\gt 0$ such that for all $\Omega \subset \mathbb{R}^2$ with finite measure $$\max_{x \in \Omega}{ |u(x)|} \leq \max_{x \in \partial \Omega}{|u(x)|} + c \max_{x \in \Omega} \int_{y \in \Omega}{ \max \left\{ 1, \log{ \left(\frac{|\Omega|}{\|x-y\|^2} \right)} \right\} \left| \Delta u(y) \right| dy}.$$ This is somewhat dual to the classical Trudinger-Moser inequality; we also note that it is sharper than the usual estimates given in Orlicz spaces, the proof is rearrangement-free. The Laplacian can be replaced by any uniformly elliptic operator in divergence form. Keywords:Alexandrov-Bakelman-Pucci estimate, second order Sobolev inequality, Trudinger-Moser inequalityCategories:35A23, 35B50, 28A75, 49Q20

2. CMB Online first

Gigli, Nicola; Rigoni, Chiara
 A note about the strong maximum principle on RCD spaces We give a direct proof of the strong maximum principle on finite dimensional RCD spaces based on the Laplacian comparison of the squared distance. Keywords:maximum principle, RCD spaceCategories:31E05, 35B50

3. CMB 2018 (vol 61 pp. 738)

Cruz-Uribe, David; Rodney, Scott; Rosta, Emily
 PoincarÃ© Inequalities and Neumann Problems for the $p$-Laplacian We prove an equivalence between weighted PoincarÃ© inequalities and the existence of weak solutions to a Neumann problem related to a degenerate $p$-Laplacian. The PoincarÃ© inequalities are formulated in the context of degenerate Sobolev spaces defined in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the $p$-Laplacian. Keywords:degenerate Sobolev space, $p$-Laplacian, PoincarÃ© inequalitiesCategories:30C65, 35B65, 35J70, 42B35, 42B37, 46E35

4. CMB 2018 (vol 61 pp. 787)

Liu, Yu; Qi, Shuai
 Endpoint Estimates of Riesz Transforms Associated with Generalized SchrÃ¶dinger Operators In this paper we establish the endpoint estimates and Hardy type estimates for the Riesz transform associated with the generalized SchrÃ¶dinger operator. Keywords:SchrÃ¶dinger operator, fundamental solution, Riesz transformCategories:35J10, 42B20, 42B30

5. CMB Online first

Rosales, Leobardo
 Generalizing Hopf's boundary point lemma We give a Hopf boundary point lemma for weak solutions of linear divergence form uniformly elliptic equations, with HÃ¶lder continuous top-order coefficients and lower-order coefficients in a Morrey space. Keywords:partial differential equation, divergence form, Hopf boundary point lemmaCategories:35B50, 35A07

6. CMB 2017 (vol 61 pp. 473)

Awonusika, Richard; Taheri, Ali
 A spectral identity on Jacobi polynomials and its analytic implications The Jacobi coefficients $c^{\ell}_{j}(\alpha,\beta)$ ($1\leq j\leq \ell$, $\alpha,\beta\gt -1$) are linked to the Maclaurin spectral expansion of the Schwartz kernel of functions of the Laplacian on a compact rank one symmetric space. It is proved that these coefficients can be computed by transforming the even derivatives of the the Jacobi polynomials $P_{k}^{(\alpha,\beta)}$ ($k\geq 0, \alpha,\beta\gt -1$) into a spectral sum associated with the Jacobi operator. The first few coefficients are explicitly computed and a direct trace interpretation of the Maclaurin coefficients is presented. Keywords:Jacobi coefficient, Laplace-Beltrami operator, symmetric space, Maclaurin expansion, Jacobi polynomialCategories:33C05, 33C45, 35A08, 35C05, 35C10, 35C15

7. CMB 2017 (vol 61 pp. 353)

Qin, Dongdong; He, Yubo; Tang, Xianhua
 Ground State and Multiple Solutions for Kirchhoff Type Equations with Critical Exponent In this paper, we consider the following critical Kirchhoff type equation: \begin{align*} \left\{ \begin{array}{lll} - \left(a+b\int_{\Omega}|\nabla u|^2 \right)\Delta u=Q(x)|u|^4u + \lambda |u|^{q-1}u,~~\mbox{in}~~\Omega, \\ u=0,\quad \text{on}\quad \partial \Omega, \end{array} \right. \end{align*} By using variational methods that are constrained to the Nehari manifold, we prove that the above equation has a ground state solution for the case when $3\lt q\lt 5$. The relation between the number of maxima of $Q$ and the number of positive solutions for the problem is also investigated. Keywords:Kirchhoff type equation, variational methods, critical exponent, Nehari manifold, ground stateCategories:35J20, 35J60, 35J25

8. CMB 2017 (vol 61 pp. 438)

Zhang, Tao; Zhou, Chunqin
 Classification of Solutions for Harmonic Functions with Neumann Boundary Value In this paper, we classify all solutions of $\left\{ \begin{array}{rcll} -\Delta u &=& 0 \quad &\text{ in }\mathbb{R}^{2}_{+}, \\ \dfrac{\partial u}{\partial t}&=&-c|x|^{\beta}e^{u} \quad &\text{ on }\partial \mathbb{R}^{2}_{+} \backslash \{0\}, \\ \end{array} \right.$ with the finite conditions $\int_{\partial \mathbb{R}^{2}_{+}}|x|^{\beta}e^{u}ds \lt C, \qquad \sup\limits_{\overline{\mathbb{R}^{2}_{+}}}{u(x)}\lt C.$ Here, $c$ is a positive number and $\beta \gt -1$. Keywords:Neumann problem, singular coefficient, classification of solutionsCategories:35A05, 35J65

9. CMB 2017 (vol 61 pp. 376)

Sebbar, Abdellah; Al-Shbeil, Isra
 Elliptic Zeta Functions and Equivariant Functions In this paper we establish a close connection between three notions attached to a modular subgroup. Namely the set of weight two meromorphic modular forms, the set of equivariant functions on the upper half-plane commuting with the action of the modular subgroup and the set of elliptic zeta functions generalizing the Weierstrass zeta functions. In particular, we show that the equivariant functions can be parameterized by modular objects as well as by elliptic objects. Keywords:modular form, equivariant function, elliptic zeta functionCategories:11F12, 35Q15, 32L10

10. CMB 2017 (vol 60 pp. 536)

 The Gradient of a Solution of the Poisson Equation in the Unit Ball and Related Operators In this paper we determine the $L^1\to L^1$ and $L^{\infty}\to L^\infty$ norms of an integral operator $\mathcal{N}$ related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions. Keywords:MÃ¶bius transformation, Poisson equation, Newtonian potential, Cauchy transform, Bessel functionCategories:35J05, 47G10

11. CMB 2017 (vol 61 pp. 142)

Li, Bao Qin
 An Equivalent Form of Picard's Theorem and Beyond This paper gives an equivalent form of Picard's theorem via entire solutions of the functional equation $f^2+g^2=1$, and then its improvements and applications to certain nonlinear (ordinary and partial) differential equations. Keywords:entire function, Picard's Theorem, functional equation, partial differential equationCategories:30D20, 32A15, 35F20

12. CMB 2017 (vol 60 pp. 422)

Tang, Xianhua
 New Super-quadratic Conditions for Asymptotically Periodic SchrÃ¶dinger Equations This paper is dedicated to studying the semilinear SchrÃ¶dinger equation $$\left\{ \begin{array}{ll} -\triangle u+V(x)u=f(x, u), \ \ \ \ x\in {\mathbf{R}}^{N}, \\ u\in H^{1}({\mathbf{R}}^{N}), \end{array} \right.$$ where $f$ is a superlinear, subcritical nonlinearity. It focuses on the case where $V(x)=V_0(x)+V_1(x)$, $V_0\in C(\mathbf{R}^N)$, $V_0(x)$ is 1-periodic in each of $x_1, x_2, \ldots, x_N$ and $\sup[\sigma(-\triangle +V_0)\cap (-\infty, 0)]\lt 0\lt \inf[\sigma(-\triangle +V_0)\cap (0, \infty)]$, $V_1\in C(\mathbf{R}^N)$ and $\lim_{|x|\to\infty}V_1(x)=0$. A new super-quadratic condition is obtained, which is weaker than some well known results. Keywords:SchrÃ¶dinger equation, superlinear, asymptotically periodic, ground state solutions of Nehari-Pankov typeCategories:35J20, 35J60

13. CMB 2017 (vol 60 pp. 436)

Weng, Peixuan; Liu, Li
 Globally Asymptotic Stability of a Delayed Integro-Differential Equation with Nonlocal Diffusion We study a population model with nonlocal diffusion, which is a delayed integro-differential equation with double nonlinearity and two integrable kernels. By comparison method and analytical technique, we obtain globally asymptotic stability of the zero solution and the positive equilibrium. The results obtained reveal that the globally asymptotic stability only depends on the property of nonlinearity. As application, an example for a population model with age structure is discussed at the end of the article. Keywords:integro-differential equation, nonlocal diffusion, equilibrium, globally asymptotic stability, population model with age structureCategories:45J05, 35K57, 92D25

14. CMB 2016 (vol 59 pp. 606)

Mihăilescu, Mihai; Moroşanu, Gheorghe
 Eigenvalues of $-\Delta_p -\Delta_q$ Under Neumann Boundary Condition The eigenvalue problem $-\Delta_p u-\Delta_q u=\lambda|u|^{q-2}u$ with $p\in(1,\infty)$, $q\in(2,\infty)$, $p\neq q$ subject to the corresponding homogeneous Neumann boundary condition is investigated on a bounded open set with smooth boundary from $\mathbb{R}^N$ with $N\geq 2$. A careful analysis of this problem leads us to a complete description of the set of eigenvalues as being a precise interval $(\lambda_1, +\infty )$ plus an isolated point $\lambda =0$. This comprehensive result is strongly related to our framework which is complementary to the well-known case $p=q\neq 2$ for which a full description of the set of eigenvalues is still unavailable. Keywords:eigenvalue problem, Sobolev space, Nehari manifold, variational methodsCategories:35J60, 35J92, 46E30, 49R05

15. CMB 2016 (vol 59 pp. 734)

Dimassi, Mouez
 Semi-classical Asymptotics for SchrÃ¶dinger Operator with Oscillating Decaying Potential We study the distribution of the discrete spectrum of the SchrÃ¶dinger operator perturbed by a fast oscillating decaying potential depending on a small parameter $h$. Keywords:periodic SchrÃ¶dinger operator, semi-classical asymptotics, effective Hamiltonian, asymptotic expansion, spectral shift functionCategories:81Q10, 35P20, 47A55, 47N50, 81Q15

16. CMB 2016 (vol 59 pp. 417)

Song, Hongxue; Chen, Caisheng; Yan, Qinglun
 Existence of Multiple Solutions for a $p$-Laplacian System in $\textbf{R}^{N}$ with Sign-changing Weight Functions In this paper, we consider the quasi-linear elliptic problem \left\{ \begin{aligned} & -M \left(\int_{\mathbb{R}^{N}}|x|^{-ap}|\nabla u|^{p}dx \right){\rm div} \left(|x|^{-ap}|\nabla u|^{p-2}\nabla u \right) \\ & \qquad=\frac{\alpha}{\alpha+\beta}H(x)|u|^{\alpha-2}u|v|^{\beta}+\lambda h_{1}(x)|u|^{q-2}u, \\ & -M \left(\int_{\mathbb{R}^{N}}|x|^{-ap}|\nabla v|^{p}dx \right){\rm div} \left(|x|^{-ap}|\nabla v|^{p-2}\nabla v \right) \\ & \qquad=\frac{\beta}{\alpha+\beta}H(x)|v|^{\beta-2}v|u|^{\alpha}+\mu h_{2}(x)|v|^{q-2}v, \\ &u(x)\gt 0,\quad v(x)\gt 0, \quad x\in \mathbb{R}^{N} \end{aligned} \right. where $\lambda, \mu\gt 0$, $1\lt p\lt N$, $1\lt q\lt p\lt p(\tau+1)\lt \alpha+\beta\lt p^{*}=\frac{Np}{N-p}$, $0\leq a\lt \frac{N-p}{p}$, $a\leq b\lt a+1$, $d=a+1-b\gt 0$, $M(s)=k+l s^{\tau}$, $k\gt 0$, $l, \tau\geq0$ and the weight $H(x), h_{1}(x), h_{2}(x)$ are continuous functions which change sign in $\mathbb{R}^{N}$. We will prove that the problem has at least two positive solutions by using the Nehari manifold and the fibering maps associated with the Euler functional for this problem. Keywords:Nehari manifold, quasilinear elliptic system, $p$-Laplacian operator, concave and convex nonlinearitiesCategory:35J66

17. CMB 2016 (vol 59 pp. 553)

Kachmar, Ayman
 A New Formula for the Energy of Bulk Superconductivity The energy of a type II superconductor submitted to an external magnetic field of intensity close to the second critical field is given by the celebrated Abrikosov energy. If the external magnetic field is comparable to and below the second critical field, the energy is given by a reference function obtained as a special (thermodynamic) limit of a non-linear energy. In this note, we give a new formula for this reference energy. In particular, we obtain it as a special limit of a linear energy defined over configurations normalized in the $L^4$-norm. Keywords:Ginzburg-Landau functionalCategories:35B40, 35P15, 35Q56

18. CMB 2016 (vol 59 pp. 542)

Jiang, Yongxin; Wang, Wei; Feng, Zhaosheng
 Spatial Homogenization of Stochastic Wave Equation with Large Interaction A dynamical approximation of a stochastic wave equation with large interaction is derived. A random invariant manifold is discussed. By a key linear transformation, the random invariant manifold is shown to be close to the random invariant manifold of a second-order stochastic ordinary differential equation. Keywords:stochastic wave equation, homogeneous system, approximation, random invariant manifold, Neumann boundary conditionCategories:60F10, 60H15, 35Q55

19. CMB 2015 (vol 59 pp. 73)

Gasiński, Leszek; Papageorgiou, Nikolaos S.
 Positive Solutions for the Generalized Nonlinear Logistic Equations We consider a nonlinear parametric elliptic equation driven by a nonhomogeneous differential operator with a logistic reaction of the superdiffusive type. Using variational methods coupled with suitable truncation and comparison techniques, we prove a bifurcation type result describing the set of positive solutions as the parameter varies. Keywords:positive solution, bifurcation type result, strong comparison principle, nonlinear regularity, nonlinear maximum principleCategories:35J25, 35J92

20. CMB 2015 (vol 58 pp. 723)

Castro, Alfonso; Fischer, Emily
 Infinitely Many Rotationally Symmetric Solutions to a Class of Semilinear Laplace-Beltrami Equations on Spheres We show that a class of semilinear Laplace-Beltrami equations on the unit sphere in $\mathbb{R}^n$ has infinitely many rotationally symmetric solutions. The solutions to these equations are the solutions to a two point boundary value problem for a singular ordinary differential equation. We prove the existence of such solutions using energy and phase plane analysis. We derive a Pohozaev-type identity in order to prove that the energy to an associated initial value problem tends to infinity as the energy at the singularity tends to infinity. The nonlinearity is allowed to grow as fast as $|s|^{p-1}s$ for $|s|$ large with $1 \lt p \lt (n+5)/(n-3)$. Keywords:Laplace-Beltrami operator, semilinear equation, rotational solution, superlinear nonlinearity, sub-super critical nonlinearityCategories:58J05, 35A24

21. CMB 2015 (vol 58 pp. 651)

Tang, Xianhua
 Ground State Solutions of Nehari-Pankov Type for a Superlinear Hamiltonian Elliptic System on ${\mathbb{R}}^{N}$ This paper is concerned with the following elliptic system of Hamiltonian type $\left\{ \begin{array}{ll} -\triangle u+V(x)u=W_{v}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N}, \\ -\triangle v+V(x)v=W_{u}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N}, \\ u, v\in H^{1}({\mathbb{R}}^{N}), \end{array} \right.$ where the potential $V$ is periodic and $0$ lies in a gap of the spectrum of $-\Delta+V$, $W(x, s, t)$ is periodic in $x$ and superlinear in $s$ and $t$ at infinity. We develop a direct approach to find ground state solutions of Nehari-Pankov type for the above system. Especially, our method is applicable for the case when $W(x, u, v)=\sum_{i=1}^{k}\int_{0}^{|\alpha_iu+\beta_iv|}g_i(x, t)t\mathrm{d}t +\sum_{j=1}^{l}\int_{0}^{\sqrt{u^2+2b_juv+a_jv^2}}h_j(x, t)t\mathrm{d}t,$ where $\alpha_i, \beta_i, a_j, b_j\in \mathbb{R}$ with $\alpha_i^2+\beta_i^2\ne 0$ and $a_j\gt b_j^2$, $g_i(x, t)$ and $h_j(x, t)$ are nondecreasing in $t\in \mathbb{R}^{+}$ for every $x\in \mathbb{R}^N$ and $g_i(x, 0)=h_j(x, 0)=0$. Keywords:Hamiltonian elliptic system, superlinear, ground state solutions of Nehari-Pankov type, strongly indefinite functionalsCategories:35J50, 35J55

22. CMB 2015 (vol 58 pp. 471)

Demirbas, Seckin
 Almost Sure Global Well-posedness for the Fractional Cubic SchrÃ¶dinger Equation on Torus In a previous paper, we proved that $1$-d periodic fractional SchrÃ¶dinger equation with cubic nonlinearity is locally well-posed in $H^s$ for $s\gt \frac{1-\alpha}{2}$ and globally well-posed for $s\gt \frac{10\alpha-1}{12}$. In this paper we define an invariant probability measure $\mu$ on $H^s$ for $s\lt \alpha-\frac{1}{2}$, so that for any $\epsilon\gt 0$ there is a set $\Omega\subset H^s$ such that $\mu(\Omega^c)\lt \epsilon$ and the equation is globally well-posed for initial data in $\Omega$. We see that this fills the gap between the local well-posedness and the global well-posedness range in almost sure sense for $\frac{1-\alpha}{2}\lt \alpha-\frac{1}{2}$, i.e. $\alpha\gt \frac{2}{3}$ in almost sure sense. Keywords:NLS, fractional Schrodinger equation, almost sure global wellposednessCategory:35Q55

23. CMB 2015 (vol 58 pp. 486)

Duc, Dinh Thanh; Nhan, Nguyen Du Vi; Xuan, Nguyen Tong
 Inequalities for Partial Derivatives and their Applications We present various weighted integral inequalities for partial derivatives acting on products and compositions of functions which are applied to establish some new Opial-type inequalities involving functions of several independent variables. We also demonstrate the usefulness of our results in the field of partial differential equations. Keywords:inequality for integral, Opial-type inequality, HÃ¶lder's inequality, partial differential operator, partial differential equationCategories:26D10, 35A23

24. CMB 2014 (vol 58 pp. 432)

Yang, Dachun; Yang, Sibei
 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 inequalityCategories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30

25. CMB 2013 (vol 56 pp. 827)

Petridis, Yiannis N.; Raulf, Nicole; Risager, Morten S.
 Erratum to Quantum Limits of Eisenstein Series and Scattering States'' This paper provides an erratum to Y. N. Petridis, N. Raulf, and M. S. Risager, Quantum Limits of Eisenstein Series and Scattering States.'' Canad. Math. Bull., published online 2012-02-03, http://dx.doi.org/10.4153/CMB-2011-200-2. Keywords:quantum limits, Eisenstein series, scattering polesCategories:11F72, 8G25, 35P25
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