{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:48:32Z","timestamp":1760230112763,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,11]],"date-time":"2022-07-11T00:00:00Z","timestamp":1657497600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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\u2013plastic 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 Lp Lagrange multiplier. Some examples support the results.<\/jats:p>","DOI":"10.3390\/sym14071423","type":"journal-article","created":{"date-parts":[[2022,7,12]],"date-time":"2022-07-12T03:50:36Z","timestamp":1657597836000},"page":"1423","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Existence of Radial Solutions to a Nonconstant Gradient-Constrained Problem"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8503-9630","authenticated-orcid":false,"given":"Sofia","family":"Giuffr\u00e8","sequence":"first","affiliation":[{"name":"D.I.I.E.S, Mediterranea University of Reggio Calabria, Loc. Feo di Vito, 89122 Reggio Calabria, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7229-5464","authenticated-orcid":false,"given":"Attilio","family":"Marcian\u00f2","sequence":"additional","affiliation":[{"name":"D.I.I.E.S, Mediterranea University of Reggio Calabria, Loc. Feo di Vito, 89122 Reggio Calabria, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"20140281","DOI":"10.1098\/rsta.2014.0281","article-title":"An overview of unconstrained free boundary problems","volume":"373","author":"Figalli","year":"2015","journal-title":"Philos. Trans. Roy. Soc. A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"592","DOI":"10.1016\/j.matpur.2017.05.004","article-title":"Lagrange multipliers and transport densities","volume":"108","author":"Santos","year":"2017","journal-title":"J. Math. Pures Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1080\/03605307908820103","article-title":"A second order elliptic equation with gradient constraint","volume":"4","author":"Evans","year":"1979","journal-title":"Comm. Part. Diff. Equ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/BF02106188","article-title":"The plane stress problem of Ghizetti in elastoplasticity","volume":"3","author":"Cimatti","year":"1976","journal-title":"Appl. Math. Optim"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1080\/03605308308820271","article-title":"Boundary regularity and uniqueness for an elliptic equation with gradient constraint","volume":"8","author":"Ishii","year":"1983","journal-title":"Comm. Partial Diff. Equ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1512\/iumj.1983.32.32030","article-title":"Regularity for elastoplastic type variational inequalities","volume":"32","author":"Jensen","year":"1983","journal-title":"Indiana Univ. Math. J."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1080\/03605308108820181","article-title":"The C1,1-character of solutions of second order elliptic equations with gradient constraint","volume":"6","author":"Wiegner","year":"1981","journal-title":"Comm. Part. Diff. Equ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"542","DOI":"10.1016\/j.jde.2019.12.013","article-title":"Lagrange multipliers and non-constant gradient constrained problem","volume":"269","year":"2020","journal-title":"J. Differ. Equ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1007\/s10898-015-0382-4","article-title":"Functional inequalities, regularity and computation of the deficit and surplus variables in the financial equilibrium problem","volume":"65","author":"Daniele","year":"2016","journal-title":"J. Glob. Optim."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"e1655","DOI":"10.1016\/j.na.2004.12.019","article-title":"Duality Theory and Optimality Conditions for Generalized Complementary Problems","volume":"63","author":"Idone","year":"2005","journal-title":"Nonlinear Anal."},{"key":"ref_11","first-page":"353","article-title":"Duality Minimax and Applications","volume":"6","year":"2021","journal-title":"Minimax Theory Appl."},{"key":"ref_12","first-page":"205","article-title":"Variational problems with non-constant gradient constraints","volume":"59","author":"Santos","year":"2002","journal-title":"Port. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1007\/BF00281472","article-title":"Moltiplicateur de Lagrange en Torsion Elasto-Plastique","volume":"10","author":"Brezis","year":"1972","journal-title":"Arch. Rational Mech. Anal."},{"key":"ref_14","first-page":"1","article-title":"Probl\u00e8mes Unilat\u00e9raux","volume":"51","author":"Brezis","year":"1972","journal-title":"J. Math. Pures Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"153","DOI":"10.24033\/bsmf.1663","article-title":"Sur la r\u00e9gularit\u00e9 de la solution d\u2019in\u00e9quations elliptiques","volume":"96","author":"Brezis","year":"1968","journal-title":"Bull. Soc. Math. France"},{"key":"ref_16","unstructured":"Von Mises, R. (1949). Three remarks on the theory of the ideal plastic body. Reissner Anniversary Volume, Edwards."},{"key":"ref_17","unstructured":"Rodrigues, J.F. (1987). Obstacle Problems in Mathematical Physics, Elsevier Science Publishers B.V.. Mathematics Studies n. 134."},{"key":"ref_18","first-page":"529","article-title":"Radial solutions and free boundary of the elastic-plastic torsion problem","volume":"25","author":"Pratelli","year":"2018","journal-title":"J. Convex Anal."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2013\/502635","article-title":"The Limit Behavior of a Stochastic Logistic Model with Individual Time-Dependent Rates","volume":"2013","author":"Shang","year":"2013","journal-title":"J. Math."},{"key":"ref_20","unstructured":"Lions, P.L. (1982). Generalized Solutions for Hamilton-Jacobi Equations, Pitman Advanced Publishing Program. Research Notes in Mathematics, Volume 69."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/BF02415085","article-title":"Soluzioni a simmetria assiale di equazioni ellittiche","volume":"73","author":"Talenti","year":"1966","journal-title":"Ann. Mat. Pura Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/7\/1423\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:47:54Z","timestamp":1760140074000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/7\/1423"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,11]]},"references-count":21,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["sym14071423"],"URL":"https:\/\/doi.org\/10.3390\/sym14071423","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,7,11]]}}}