Sobre una conjetura de de Giorgi y algunas variantes
Sobre una conjetura de de Giorgi y algunas variantes
Director
Osorio Acevedo, Luis Eduardo
Autor corporativo
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Otros/Desconocido
Director audiovisual
Editor/Compilador
Editores
Universidad Tecnológica de Pereira
Tipo de Material
Fecha
2022
Cita bibliográfica
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Resumen
Este trabajo presenta el problema en Ecuaciones Diferenciales Parciales denominado
conjetura de De Giorgi, el cual pregunta sobre la clasificaci´on de las soluciones globales
de la ecuaci´on ∆u = u 3 − u teniendo en cuenta que u ∈ C 2 (R n ), |u| ≤ 1, ∂nu > 0, la
conjetura afirma que los conjuntos de nivel de la soluci´on llamada u para n = 2 son
l´ıneas rectas, para 3 ≤ n ≤ 8 son hiperplanos. Este problema est´a probado para n = 2, 3,
utilizando teoremas de regularidad el´ıptica, minimalidad local y teorema de tipo de
Liouville (1997,[17]). Para los resultados parciales con n = 4 se hace mucho ´enfasis en la
simetr´ıa de la soluci´on. En las conclusiones obtenidas hasta el momento para 4 ≤ n ≤ 8 se
usa fuertemente la Γ-convergencia, la idea es obtener una funci´on que converge en L 1
loc a la funci´on caracter´ıstica y que sea ortogonal a un vector unitario a. Como es bien sabido,
a´un no se tiene una demostraci´on para 4 ≤ n ≤ 8, sin embargo, si existen contraejemplos
para n ≥ 9.
Posteriormente, se hace menci´on de algunos trabajos que datan de algunas variantes que
ha tenido la conjetura. Inicialmente entre las variantes est´a la soluci´on de la conjetura
con hip´otesis adicionales sobre los l´ımites en la direcci´on mon´otona, luego el an´alisis de
los minimizadores globales, la estabilidad de las soluciones en la conjetura, los extremos
de las soluciones que son finitas y el ´ındice de Morse par para n = 2 y un avance para
n = 3, la conjetura fraccionaria de De Giorgi; para esta variante se usa el laplaciano
fraccionario y para terminar, se muestran avances de la conjetura de De Giorgi para la
ecuaci´on de Caffarelli-Berestycki-Nirenberg y la ecuaci´on de Lane-Emden teniendo en
cuenta los exponente de Sobolev y Joseph-Lundgren.
This work presents the problem in Partial Differential Equations which has been called De Giorgi’s conjecture, about classification of global solutions of the equation ∆u = u 3 − u with u ∈ C 2 (R n ), |u| ≤ 1, ∂nu > 0, the conjecture states that the level sets of the solution named u for n = 2 are straight lines and for 3 ≤ n ≤ 8 are hyperplanes. This result is tested for n = 2, 3, using elliptic regularity theorems, local minimality and Liouville type Theorems. For n = 4 is need emphasis on the symmetry of the solution. For the conclusion in this moments to 4 ≤ n ≤ 8 the Γ-convergence is strongly used to obtain a function that converges on L 1 loc to the characteristic function that is orthogonal to a unit vector a. It is well known there is not yet a proof for 4 ≤ n ≤ 8, but there are counterexamples for n ≥ 9. Subsequently, some works about the variants of De Giorgi’s conjecture are made. Initially, the solution of the conjecture with the additional hypothesis about the limits in the monotone direction, afterward analysis of the global minimizers, the stability of the solutions in the conjecture, the extremes of the solutions which are finite, and the even Morse index for n = 2 and progress for n = 3, the fractional De Giorgi conjecture; for this variant, the fractional Laplacian is used, and finally, advances of the De Giorgi conjecture are shown for the Caffarelli-Berestycki-Nirenberg equation and the Lane-Emden equation taking into account the Sobolev and Joseph-Lundgren exponent.
This work presents the problem in Partial Differential Equations which has been called De Giorgi’s conjecture, about classification of global solutions of the equation ∆u = u 3 − u with u ∈ C 2 (R n ), |u| ≤ 1, ∂nu > 0, the conjecture states that the level sets of the solution named u for n = 2 are straight lines and for 3 ≤ n ≤ 8 are hyperplanes. This result is tested for n = 2, 3, using elliptic regularity theorems, local minimality and Liouville type Theorems. For n = 4 is need emphasis on the symmetry of the solution. For the conclusion in this moments to 4 ≤ n ≤ 8 the Γ-convergence is strongly used to obtain a function that converges on L 1 loc to the characteristic function that is orthogonal to a unit vector a. It is well known there is not yet a proof for 4 ≤ n ≤ 8, but there are counterexamples for n ≥ 9. Subsequently, some works about the variants of De Giorgi’s conjecture are made. Initially, the solution of the conjecture with the additional hypothesis about the limits in the monotone direction, afterward analysis of the global minimizers, the stability of the solutions in the conjecture, the extremes of the solutions which are finite, and the even Morse index for n = 2 and progress for n = 3, the fractional De Giorgi conjecture; for this variant, the fractional Laplacian is used, and finally, advances of the De Giorgi conjecture are shown for the Caffarelli-Berestycki-Nirenberg equation and the Lane-Emden equation taking into account the Sobolev and Joseph-Lundgren exponent.