Generic Existence of Solutions and Generic Well-Posedness of Optimization Problems

2013 ◽  
pp. 445-453
Author(s):  
P. S. Kenderov ◽  
J. P. Revalski
Keyword(s):  
Existence Of Solutions ◽  
Optimization Problems ◽  
Well Posedness ◽  
Generic Existence

scholarly journals Generic Existence of Solutions of Symmetric Optimization Problems

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2004
Author(s):  
Alexander J. Zaslavski
Keyword(s):  
Objective Function ◽  
Optimization Problem ◽  
Existence Of Solutions ◽  
Optimization Problems ◽  
Baire Category ◽  
Objective Functions ◽  
Countable Intersection ◽  
Symmetric Optimization ◽  
Dense Sets ◽  
Generic Existence

In this paper we study a class of symmetric optimization problems which is identified with a space of objective functions, equipped with an appropriate complete metric. Using the Baire category approach, we show the existence of a subset of the space of functions, which is a countable intersection of open and everywhere dense sets, such that for every objective function from this intersection the corresponding symmetric optimization problem possesses a solution.

scholarly journals The Muskat problem with surface tension and equal viscosities in subcritical $$L_p$$-Sobolev spaces

2021 ◽  
Author(s):  
Anca-Voichita Matioc ◽  
Bogdan-Vasile Matioc
Keyword(s):  
Mathematical Model ◽  
Surface Tension ◽  
Global Existence ◽  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Well Posedness ◽  
Global Existence Of Solutions ◽  
Muskat Problem ◽  
The Mathematical Model ◽  
Quasilinear Parabolic

AbstractIn this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces $$W^s_p(\mathbb {R})$$ W p s ( R ) , where $${p\in (1,2]}$$ p ∈ ( 1 , 2 ] and $${s\in (1+1/p,2)}$$ s ∈ ( 1 + 1 / p , 2 ) . This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in $$W^{\overline{s}-2}_p(\mathbb {R})$$ W p s ¯ - 2 ( R ) , where $${\overline{s}\in (1+1/p,s)}$$ s ¯ ∈ ( 1 + 1 / p , s ) . Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.

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