Minimum action solutions of some vector field equations

Haim Brezis; Elliott H. Lieb

doi:10.1007/BF01217349

Minimum action solutions of some vector field equations

Haim Brezis, Elliott H. Lieb

Mathematics

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135 Scopus citations

Abstract

The system of equations studied in this paper is -Δu_i=gⁱ(u) on ℝ^d, d≧2, with u:ℝ^d→ℝⁿ and gⁱ(u)=∂G/∂u_i. Associated with this system is the action, S(u)=ε{1/2|∇u|²-G(u)}. Under appropriate conditions on G (which differ for d=2 and d≧3) it is proved that the system has a solution, u ≢0, of finite action and that this solution also minimizes the action within the class {v is a solution, v has finite action, v ≢0}.

Original language	English (US)
Pages (from-to)	97-113
Number of pages	17
Journal	Communications In Mathematical Physics
Volume	96
Issue number	1
DOIs	https://doi.org/10.1007/BF01217349
State	Published - Mar 1984

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Mathematical Physics

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10.1007/BF01217349

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title = "Minimum action solutions of some vector field equations",

abstract = "The system of equations studied in this paper is -Δui=gi(u) on ℝd, d≧2, with u:ℝd→ℝn and gi(u)=∂G/∂ui. Associated with this system is the action, S(u)=ε{1/2|∇u|2-G(u)}. Under appropriate conditions on G (which differ for d=2 and d≧3) it is proved that the system has a solution, u ≢0, of finite action and that this solution also minimizes the action within the class {v is a solution, v has finite action, v ≢0}.",

author = "Haim Brezis and Lieb, {Elliott H.}",

year = "1984",

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AU - Lieb, Elliott H.

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N2 - The system of equations studied in this paper is -Δui=gi(u) on ℝd, d≧2, with u:ℝd→ℝn and gi(u)=∂G/∂ui. Associated with this system is the action, S(u)=ε{1/2|∇u|2-G(u)}. Under appropriate conditions on G (which differ for d=2 and d≧3) it is proved that the system has a solution, u ≢0, of finite action and that this solution also minimizes the action within the class {v is a solution, v has finite action, v ≢0}.

AB - The system of equations studied in this paper is -Δui=gi(u) on ℝd, d≧2, with u:ℝd→ℝn and gi(u)=∂G/∂ui. Associated with this system is the action, S(u)=ε{1/2|∇u|2-G(u)}. Under appropriate conditions on G (which differ for d=2 and d≧3) it is proved that the system has a solution, u ≢0, of finite action and that this solution also minimizes the action within the class {v is a solution, v has finite action, v ≢0}.

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JO - Communications In Mathematical Physics

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Minimum action solutions of some vector field equations

Abstract

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