A smooth solution to singular ODEs

2012 ◽  
Author(s):  
G. Vainikko
Keyword(s):  
Smooth Solution ◽  
Singular Odes

Gluing Solutions

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Stress Tensor ◽  
Compact Support ◽  
Smooth Solution ◽  
Continuous Solution ◽  
Total Momentum ◽  
Frame Of Reference ◽  
Galilean Transformation ◽  
Hölder Continuous ◽  
Pressure Form ◽  
Original Frame

This chapter deals with the gluing of solutions and the relevant theorem (Theorem 12.1), which states the condition for a Hölder continuous solution to exist. By taking a Galilean transformation if necessary, the solution can be assumed to have zero total momentum. The cut off velocity and pressure form a smooth solution to the Euler-Reynolds equations with compact support when coupled to a smooth stress tensor. The proof of Theorem (12.1) proceeds by iterating Lemma (10.1) just as in the proof of Theorem (10.1). Applying another Galilean transformation to return to the original frame of reference, the theorem is obtained.

scholarly journals Boundary conditions in topological AdS4/CFT3

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Pietro Benetti Genolini ◽  
Matan Grinberg ◽  
Paul Richmond
Keyword(s):  
Field Theory ◽  
Free Energy ◽  
Boundary Conditions ◽  
Smooth Solution ◽  
Three Dimensional ◽  
Field Theories ◽  
Legendre Transformation ◽  
Generating Functional ◽  
Holographic Duals

Abstract We revisit the construction in four-dimensional gauged Spin(4) supergravity of the holographic duals to topologically twisted three-dimensional $$ \mathcal{N} $$ N = 4 field theories. Our focus in this paper is to highlight some subtleties related to preserving supersymmetry in AdS/CFT, namely the inclusion of finite counterterms and the necessity of a Legendre transformation to find the dual to the field theory generating functional. Studying the geometry of these supergravity solutions, we conclude that the gravitational free energy is indeed independent from the metric of the boundary, and it vanishes for any smooth solution.

Periodic Problems with the Scalar p-Laplacian Resonant at any Eigenvalue via Critical Point Methods

2013 ◽  
Vol 13 (3) ◽  
Author(s):  
Sophia Th. Kyritsi ◽  
Donal O’ Regan ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Critical Point ◽  
Morse Theory ◽  
Critical Point Theory ◽  
Smooth Solution ◽  
Point Theory ◽  
Reaction Term ◽  
Variational Techniques ◽  
Periodic Problems

AbstractWe consider nonlinear periodic problems driven by the scalar p-Laplacian with a Carathéodory reaction term. Under conditions which permit resonance at infinity with respect to any eigenvalue, we show that the problem has a nontrivial smooth solution. Our approach combines variational techniques based on critical point theory with Morse theory.

Gradient estimates for Monge–Ampère type equations on compact almost Hermitian manifolds with boundary

Analysis ◽  
2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Masaya Kawamura
Keyword(s):  
Nonlinear Equations ◽  
Smooth Solution ◽  
A Priori ◽  
Gradient Estimates ◽  
Manifolds With Boundary ◽  
Fully Nonlinear Equations ◽  
Fully Nonlinear ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds ◽  
Monge Ampere

Abstract We investigate Monge–Ampère type fully nonlinear equations on compact almost Hermitian manifolds with boundary and show a priori gradient estimates for a smooth solution of these equations.

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