{"title":"The Minkowski inequalities for \u0000 \u0000 \u0000 (\u0000 p\u0000 ,\u0000 α\u0000 )\u0000 \u0000 $(p,alpha)$\u0000 -torsional rigidity and the extensions on Orlicz spaces","authors":"Zhen-Hui Bu, Meng Qin, Denghui Wu","doi":"10.1002/mana.70032","DOIUrl":"https://doi.org/10.1002/mana.70032","url":null,"abstract":"<p>In this paper, we prove the functional Minkowski inequality for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>p</mi>\u0000 <mo>,</mo>\u0000 <mi>α</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(p,alpha)$</annotation>\u0000 </semantics></math>-torsional rigidity under suitable regularity assumptions, where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>p</mi>\u0000 <mo>,</mo>\u0000 <mi>α</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(p,alpha)$</annotation>\u0000 </semantics></math>-torsional rigidity can be formulated as the weak solution of an elliptic boundary value problem of the <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>-Laplacian. We also establish the Orlicz Brunn–Minkowski and Orlicz Minkowski inequalities for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>p</mi>\u0000 <mo>,</mo>\u0000 <mi>α</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(p,alpha)$</annotation>\u0000 </semantics></math>-torsional rigidity, which are extensions of the Brunn–Minkowski and Minkowski inequalities for torsional rigidity. Finally, we describe the equivalence between the Orlicz Brunn–Minkowski inequality and the Orlicz Minkowski inequality.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 9","pages":"3191-3209"},"PeriodicalIF":0.8,"publicationDate":"2025-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145038107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}