Extremal solutions of some constrained control problems∗

Optimization ◽  
1995 ◽  
Vol 35 (4) ◽  
pp. 345-355 ◽  
Author(s):  
H. X. Phu ◽  
H. G. Bock ◽  
J. P. Schölder
Keyword(s):  
Extremal Solutions ◽  
Control Problems ◽  
Constrained Control ◽  
Constrained Control Problems

Regularity of almost extremal solutions of Monge–Ampère equations

1991 ◽  
Vol 117 (1-2) ◽  
pp. 21-29 ◽  
Author(s):  
John I. E. Urbas
Keyword(s):  
Large Class ◽  
Convex Domain ◽  
Gauss Curvature ◽  
Extremal Solutions ◽  
Classical Solutions ◽  
Uniformly Convex ◽  
Suitable Sense ◽  
Monge Ampere

SynopsisWe show that for a large class of Monge-Ampère equations, generalised solutions on a uniformly convex domain Ω⊂ℝn are classical solutions on any pre-assigned subdomain Ω′⋐Ω, provided the solution is almost extremal in a suitable sense. Alternatively, classical regularity holds on subdomains of Ω which are sufficiently distant from ∂Ω. We also show that classical regularity may fail to hold near ∂Ω in the nonextremal case. The main example of the class of equations considered is the equation of prescribed Gauss curvature.

scholarly journals On the extremal solutions of superlinear Helmholtz problems

2022 ◽  
Vol 40 ◽  
pp. 1-8
Author(s):  
Makkia Dammak ◽  
Majdi El Ghord ◽  
Saber Ali Kharrati
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Weak Sense ◽  
Critical Value ◽  
Extremal Solutions ◽  
Regular Solutions ◽  
Smooth Bounded Domain ◽  
Bifurcation Phenomena ◽  
Stable Solutions

Abstract: In this note, we deal with the Helmholtz equation −∆u+cu = λf(u) with Dirichlet boundary condition in a smooth bounded domain Ω of R n , n > 1. The nonlinearity is superlinear that is limt−→∞ f(t) t = ∞ and f is a positive, convexe and C 2 function defined on [0,∞). We establish existence of regular solutions for λ small enough and the bifurcation phenomena. We prove the existence of critical value λ ∗ such that the problem does not have solution for λ > λ∗ even in the weak sense. We also prove the existence of a type of stable solutions u ∗ called extremal solutions. We prove that for f(t) = e t , Ω = B1 and n ≤ 9, u ∗ is regular.

scholarly journals Extremal solutions of the S3 model and nilpotent orbits of G2(2)

2010 ◽  
Vol 2010 (8) ◽  
Author(s):  
Sung-Soo Kim ◽  
Josef Lindman Hörnlund ◽  
Jakob Palmkvist ◽  
Amitabh Virmani
Keyword(s):  
Extremal Solutions ◽  
Nilpotent Orbits
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