In this paper, we study a variational problem with nonconstant gradient constraints. Several aspects related to problems with gradient constraints have been studied in the literature and have seen new developments in recent years. In the case of constant gradient constraint, the problem is the well-known elastic–plastic torsion problem. A relevant issue in this type of problem is the existence of Lagrange multipliers. Here, we consider the equivalent Lagrange multiplier formulation of a nonconstant gradient-constrained problem, and we investigate the class of solutions having a radial symmetry. We rewrite the problem in the radial symmetry case, and we analyse the different situations that may arise. In particular, in the planar case, we derive a condition characterizing the free boundary and obtain the explicit radial solution to the problem and the (Formula presented.) Lagrange multiplier. Some examples support the results.

On the Existence of Radial Solutions to a Nonconstant Gradient-Constrained Problem / Giuffre', S.; Marciano', A.. - In: SYMMETRY. - ISSN 2073-8994. - 14:7(2022). [10.3390/sym14071423]

On the Existence of Radial Solutions to a Nonconstant Gradient-Constrained Problem

GIUFFRE' S.
;
MARCIANO' A.
2022-01-01

Abstract

In this paper, we study a variational problem with nonconstant gradient constraints. Several aspects related to problems with gradient constraints have been studied in the literature and have seen new developments in recent years. In the case of constant gradient constraint, the problem is the well-known elastic–plastic torsion problem. A relevant issue in this type of problem is the existence of Lagrange multipliers. Here, we consider the equivalent Lagrange multiplier formulation of a nonconstant gradient-constrained problem, and we investigate the class of solutions having a radial symmetry. We rewrite the problem in the radial symmetry case, and we analyse the different situations that may arise. In particular, in the planar case, we derive a condition characterizing the free boundary and obtain the explicit radial solution to the problem and the (Formula presented.) Lagrange multiplier. Some examples support the results.
2022
Lagrange multipliers
nonconstant gradient constraints
radial solutions
File in questo prodotto:
File Dimensione Formato  
Giuffrè_2022_symmetry_existence_editor.pdf

accesso aperto

Descrizione: Versione editoriale
Tipologia: Versione Editoriale (PDF)
Licenza: Creative commons
Dimensione 290.26 kB
Formato Adobe PDF
290.26 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12318/129746
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 0
social impact