In this paper, we consider the discrete fourth-order Schr\"{o}dinger equation on lattice $h\mathbb{Z}^2$. Uniform Strichartz estimates are established by analyzing frequency localized oscillatory integrals with method of stationary phase and applying Littlewood-Paley inequalities. As an application, obtain precise rate $L^2$ convergence from solutions semilinear equations to those corresponding Euclidean plane $\mathbb{R}^2$ in contimuum limit $h \rightarrow 0$.

In this paper, we establish the first existence result for solutions to critical semilinear equation involving logarithmic Schr\"odinger operator with subcritical nonlinearities. Additionally, present least-energy Brezis-Nirenberg type problem fractional pseudo-relativistic and Specifically, demonstrate that of converge, up a subsequence, nontrivial solution limiting operator. Furthermore, provide regularity sublinear Our approach relies on uniform positive bounds elements in Nehari...

Abstract A salami is a connected, locally finite, weighted graph with non-negative Ollivier Ricci curvature and at least two ends of infinite volume. We show that every has exactly no vertices positive curvature. moreover recurrent admits harmonic functions constant gradient. The proofs are based on extremal Lipschitz extensions, variational principle the study functions. Assuming lower bound edge weight, we prove salamis quasi-isometric to line, space all finite dimension subexponentially...

In the present paper, we apply Alexandrov geometry methods to study geometric analysis aspects of infinite semiplanar graphs with nonnegative combinatorial curvature in sense Higuchi. We obtain metric classification these and construct embedded projective plane minus one point. Moreover, show volume doubling property Poincar\'e inequality on such graphs. The quadratic growth implies parabolicity. addition, prove polynomial harmonic function theorem analogous case Riemannian manifolds.

In this note, we study the Liouville equation $\Delta u=-e^u$ on a graph $G$ satisfying certain isoperimetric inequality. Following idea of W. Ding, prove that there exists uniform lower bound for energy, $\sum _G e^u,$ any solution $u$ to equation. particular, 2-dimensional lattice $\mathbb {Z}^2,$ is given by $4.$

In this paper, we study harmonic functions on metric measure spaces with Riemannian Ricci curvature bounded from below, which were introduced by Ambrosio-Gigli-Savar\'e. We prove a Cheng-Yau type local gradient estimate for these spaces. Furthermore, derive various optimal dimension estimates of polynomial growth nonnegative curvature.

By assigning a probability measure via the spectrum of normalized Laplacian to each graph and using Lp Wasserstein distances between measures, we define corresponding spectral dp on set all graphs. This approach can even be extended measuring infinite We prove that diameter graphs, as pseudo-metric space equipped with d1, is one. further study behavior d1 when size graphs tends infinity by interlacing inequalities aiming at exploring large real networks. A monotonic relation evolutionary...

In the present paper, we will derive Liouville theorem and finite dimension for polynomial growth harmonic functions defined on Alexandrov spaces with nonnegative Ricci curvature in sense of Kuwae-Shioya Sturm-Lott-Villani.

We study the 3-dimensional combinatorial Yamabe flow in hyperbolic background geometry. For a triangulation of 3-manifold, we prove that if number tetrahedra incident to each vertex is at least 23, then there exist real or virtual ball packings with vanishing (extended) scalar curvature, i.e., total solid angle equal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4 pi"> <mml:semantics> <mml:mrow> <mml:mn>4</mml:mn>...

We prove a variant of the Davies-Gaffney-Grigor'yan Lemma for continuous time heat kernel on graphs.We use it together with Li-Yau inequality, to obtain strong estimates graphs satisfying exponential curvature dimension inequality. 1 Introduction and main results 1032 1.1 1.2 Main organization paper 1033 1.3 Setting 1035 2 The inequality 1036 2.1 1037 2.2 Gradient Harnack 1038 2.3 Applications 1041 3 1043

In a previous paper, we applied Alexandrov geometry methods to study infinite semiplanar graphs with nonnegative combinatorial curvature. We proved the weak relative volume comparison and Poincaré inequality on these obtain dimension estimate for polynomial growth harmonic functions which is asymptotically quadratic in rate. present instead of using graph, translate problem polygonal surface by filling polygons into graph edge lengths 1. This then an space From function construct that not...