scholarly journals Nonexistence for hyperbolic problems on Riemannian manifolds1

2020 ◽  
Vol 120 (1-2) ◽  
pp. 87-101
Author(s):  
Dario D. Monticelli ◽  
Fabio Punzo ◽  
Marco Squassina
Keyword(s):  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Wave Operator ◽  
Existence Of Solutions ◽  
Necessary Conditions ◽  
Hyperbolic Problems ◽  
Power Nonlinearity ◽  
General Weight ◽  
Noncompact Riemannian Manifolds ◽  
General Weight Function

We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems defined on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on the whole Euclidean space. A general weight function depending on spacetime is allowed in front of the power nonlinearity.

A review: The Laplacian and the d’Alembertian

2017 ◽  
Author(s):  
Christopher D. Sogge
Keyword(s):  
Wave Equation ◽  
Euclidean Space ◽  
Fundamental Solution ◽  
Minkowski Space ◽  
Riemannian Manifolds ◽  
Wave Operator ◽  
Wave Equations ◽  
Fundamental Solutions ◽  
Specific Function ◽  
Solutions Of Wave Equations

This chapter reviews the Laplacian and the d'Alembertian. It begins with a brief discussion on the solution of wave equation both in Euclidean space and on manifolds and how this knowledge can be used to derive properties of eigenfunctions on Riemannian manifolds. A key step in understanding properties of solutions of wave equations on manifolds is to compute the types of distributions that include the fundamental solution of the wave operator in Minkowski space (d'Alembertian), with a specific function for the Euclidean Laplacian on Rn. The chapter also reviews another equation involving the Laplacian, before discussing the fundamental solutions of the d'Alembertian in R1+n.

SMOLYANOV–WEIZSÄCKER SURFACE MEASURES GENERATED BY DIFFUSIONS ON THE SET OF TRAJECTORIES IN RIEMANNIAN MANIFOLDS

2008 ◽  
Vol 11 (01) ◽  
pp. 21-31 ◽  
Author(s):  
ILYA V. TELYATNIKOV
Keyword(s):  
Brownian Motion ◽  
Euclidean Space ◽  
Diffusion Process ◽  
Riemannian Manifolds ◽  
Marginal Distribution ◽  
Diffusion Processes ◽  
Ambient Space ◽  
Time Interval ◽  
Standard Brownian Motion ◽  
Limit Surface

We consider surface measures on the set of trajectories in a smooth compact Riemannian submanifold of Euclidean space generated by diffusion processes in the ambient space. A construction of surface measures on the path space of a smooth compact Riemannian submanifold of Euclidean space was introduced by Smolyanov and Weizsäcker for the case of the standard Brownian motion. The result presented in this paper extends the result of Smolyanov and Weizsäcker to the case when we consider measures generated by diffusion processes in the ambient space with nonidentical correlation operators. For every partition of the time interval, we consider the marginal distribution of the diffusion process in the ambient space under the condition that it visits the manifold at all times of the partition, when the mesh of the partition tends to zero. We prove the existence of some limit surface measures and the equivalence of the above measures to the distribution of some diffusion process on the manifold.

scholarly journals Local and Nonlocal Singular Liouville Equations in Euclidean Spaces

2019 ◽  
Author(s):  
Ali Hyder ◽  
Gabriele Mancini ◽  
Luca Martinazzi
Keyword(s):  
Asymptotic Behavior ◽  
Euclidean Space ◽  
Finite Volume ◽  
Existence Of Solutions ◽  
Euclidean Spaces ◽  
Open Problems ◽  
Supercritical Regime

AbstractWe study the metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation \begin{equation*} (-\Delta)^{\frac{n}{2}}w=e^{nw}-c\delta_{0} \ \textrm{on}\ {\mathbb{R}}^n, \end{equation*}under a finite volume condition. We analyze the asymptotic behavior at infinity and the existence of solutions for every $n\ge 3$ also in a supercritical regime. Finally, we state some open problems.

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