Remarks on Saddle Points in the Calculus of Variations

1988 ◽  
pp. 217-235
Author(s):  
Kung-Ching Chang
Keyword(s):  
Calculus Of Variations ◽  
Saddle Points

scholarly journals Existence via regularity of solutions for elliptic systems and saddle points of functionals of the calculus of variations

2017 ◽  
Vol 6 (2) ◽  
pp. 99-120 ◽  
Author(s):  
Lucio Boccardo ◽  
Luigi Orsina
Keyword(s):  
Calculus Of Variations ◽  
Weak Solutions ◽  
Saddle Points ◽  
Elliptic Systems ◽  
Regularity Of Solutions ◽  
Integral Functionals ◽  
Content Type ◽  
The Core ◽  
Regularity Of Weak Solutions

AbstractThe core of this paper concerns the existence (via regularity) of weak solutions in ${W_{0}^{1,2}}$ of a class of elliptic systems such as$\left\{\begin{aligned} \displaystyle-\operatorname{div}((A+\varphi)\nabla u)&% \displaystyle=f,\\ \displaystyle-\operatorname{div}(M(x)\nabla\varphi)&\displaystyle=\frac{1}{2}% \lvert\nabla u\rvert^{2},\end{aligned}\right.$deriving from saddle points of integral functionals of the type$J(v,\psi)=\frac{1}{2}\int_{\Omega}(A+\psi_{+})\lvert\nabla v\rvert^{2}-\frac{1% }{2}\int_{\Omega}M(x)\nabla\psi\nabla\psi-\int_{\Omega}fv.$

On Asymmetric Distances

2013 ◽  
Vol 1 ◽  
pp. 200-231 ◽  
Author(s):  
Andrea C.G. Mennucci
Keyword(s):  
Total Variation ◽  
Calculus Of Variations ◽  
Distance Function ◽  
Metric Space ◽  
Metric Spaces ◽  
Geodesic Segment ◽  
Intrinsic Metric ◽  
Typical Application ◽  
Variation Formula

Abstract In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.

Gas-Phase Clustering of NO+ with H2S and H2O

2007 ◽  
Vol 72 (8) ◽  
pp. 1122-1138 ◽  
Author(s):  
Milan Uhlár ◽  
Ivan Černušák
Keyword(s):  
Hydrogen Bonds ◽  
Water Molecule ◽  
Gas Phase ◽  
Saddle Points ◽  
Solvent Molecule ◽  
Local Minima ◽  
Additional Water ◽  
Three Body ◽  
Rain Formation ◽  
Stable Structures

The complex NO+·H2S, which is assumed to be an intermediate in acid rain formation, exhibits thermodynamic stability of ∆Hº300 = -76 kJ mol-1, or ∆Gº300 = -47 kJ mol-1. Its further transformation via H-transfer is associated with rather high barriers. One of the conceivable routes to lower the energy of the transition state is the action of additional solvent molecule(s) that can mediate proton transfer. We have studied several NO+·H2S structures with one or two additional water molecule(s) and have found stable structures (local minima), intermediates and saddle points for the three-body NO+·H2S·H2O and four-body NO+·H2S·(H2O)2 clusters. The hydrogen bonds network in the four-body cluster plays a crucial role in its conversion to thionitrous acid.

scholarly journals Supersymmetric solitons and a degeneracy of solutions in AdS/CFT

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Andrés Anabalón ◽  
Simon F. Ross
Keyword(s):  
Phase Transition ◽  
Riemann Surface ◽  
Magnetic Flux ◽  
Gauge Field ◽  
Wilson Line ◽  
Saddle Points ◽  
Planar Boundary ◽  
Four Dimensions ◽  
Special Value

Abstract We study Lorentzian supersymmetric configurations in D = 4 and D = 5 gauged $$ \mathcal{N} $$ N = 2 supergravity. We show that there are smooth 1/2 BPS solutions which are asymptotically AdS4 and AdS5 with a planar boundary, a compact spacelike direction and with a Wilson line on that circle. There are solitons where the S1 shrinks smoothly to zero in the interior, with a magnetic flux through the circle determined by the Wilson line, which are AdS analogues of the Melvin fluxtube. There is also a solution with a constant gauge field, which is pure AdS. Both solutions preserve half of the supersymmetries at a special value of the Wilson line. There is a phase transition between these two saddle-points as a function of the Wilson line precisely at the supersymmetric point. Thus, the supersymmetric solutions are degenerate, at least at the supergravity level. We extend this discussion to one of the Romans solutions in four dimensions when the Euclidean boundary is S1× Σg where Σg is a Riemann surface with genus g > 0. We speculate that the supersymmetric state of the CFT on the boundary is dual to a superposition of the two degenerate geometries.

scholarly journals Discovery of a weak topological insulating state and van Hove singularity in triclinic RhBi2

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Kyungchan Lee ◽  
Gunnar F. Lange ◽  
Lin-Lin Wang ◽  
Brinda Kuthanazhi ◽  
Thaís V. Trevisan ◽  
...  
Keyword(s):  
Time Reversal ◽  
Topological Insulators ◽  
Dirac Point ◽  
Saddle Points ◽  
Surface States ◽  
Van Hove Singularity ◽  
Space Group ◽  
Topological Materials ◽  
Topological Surface ◽  
Optimal Space

AbstractTime reversal symmetric (TRS) invariant topological insulators (TIs) fullfil a paradigmatic role in the field of topological materials, standing at the origin of its development. Apart from TRS protected strong TIs, it was realized early on that more confounding weak topological insulators (WTI) exist. WTIs depend on translational symmetry and exhibit topological surface states only in certain directions making it significantly more difficult to match the experimental success of strong TIs. We here report on the discovery of a WTI state in RhBi2 that belongs to the optimal space group P$$\bar{1}$$ 1 ¯ , which is the only space group where symmetry indicated eigenvalues enumerate all possible invariants due to absence of additional constraining crystalline symmetries. Our ARPES, DFT calculations, and effective model reveal topological surface states with saddle points that are located in the vicinity of a Dirac point resulting in a van Hove singularity (VHS) along the (100) direction close to the Fermi energy (EF). Due to the combination of exotic features, this material offers great potential as a material platform for novel quantum effects.

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