{"title":"狄利克雷双相差夹杂的存在性结果","authors":"Nicuşor Costea, Shengda Zeng","doi":"10.24193/subbmath.2023.1.04","DOIUrl":null,"url":null,"abstract":"\"In this paper we consider a class of double phase differential inclusions of the type $$\\left\\{ \\begin{array}{ll} -{\\rm div\\;}\\left(|\\nabla u|^{p-2}\\nabla u+\\mu(x)|\\nabla u|^{q-2}\\nabla u\\right) \\in \\partial_C^2 f(x,u) , & \\mbox{ in }\\Omega,\\\\ u=0, & \\mbox{ on }\\partial\\Omega, \\end{array} \\right.$$ where $\\Omega \\subset \\mathbb{R}^N$, with $N\\ge 2$, is a bounded domain with Lipschitz boundary, $f(x,t)$ is measurable w.r.t. the first variable on $\\Omega$ and locally Lipschitz w.r.t. the second variable and $\\partial_C^2 f(x,\\cdot)$ stands for the Clarke subdifferential of $t\\mapsto f(x,t)$. The variational formulation of the problem gives rise to a so-called hemivariational inequality and the corresponding energy functional is not differentiable, but only locally Lipschitz. We use nonsmooth critical point theory to prove the existence of at least one weak solution, provided the $\\partial_C^2 f(x,\\cdot)$ satisfies an appropriate growth condition.\"","PeriodicalId":30022,"journal":{"name":"Studia Universitatis BabesBolyai Geologia","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Existence results for Dirichlet double phase differential inclusions\",\"authors\":\"Nicuşor Costea, Shengda Zeng\",\"doi\":\"10.24193/subbmath.2023.1.04\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"\\\"In this paper we consider a class of double phase differential inclusions of the type $$\\\\left\\\\{ \\\\begin{array}{ll} -{\\\\rm div\\\\;}\\\\left(|\\\\nabla u|^{p-2}\\\\nabla u+\\\\mu(x)|\\\\nabla u|^{q-2}\\\\nabla u\\\\right) \\\\in \\\\partial_C^2 f(x,u) , & \\\\mbox{ in }\\\\Omega,\\\\\\\\ u=0, & \\\\mbox{ on }\\\\partial\\\\Omega, \\\\end{array} \\\\right.$$ where $\\\\Omega \\\\subset \\\\mathbb{R}^N$, with $N\\\\ge 2$, is a bounded domain with Lipschitz boundary, $f(x,t)$ is measurable w.r.t. the first variable on $\\\\Omega$ and locally Lipschitz w.r.t. the second variable and $\\\\partial_C^2 f(x,\\\\cdot)$ stands for the Clarke subdifferential of $t\\\\mapsto f(x,t)$. The variational formulation of the problem gives rise to a so-called hemivariational inequality and the corresponding energy functional is not differentiable, but only locally Lipschitz. We use nonsmooth critical point theory to prove the existence of at least one weak solution, provided the $\\\\partial_C^2 f(x,\\\\cdot)$ satisfies an appropriate growth condition.\\\"\",\"PeriodicalId\":30022,\"journal\":{\"name\":\"Studia Universitatis BabesBolyai Geologia\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-03-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Studia Universitatis BabesBolyai Geologia\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.24193/subbmath.2023.1.04\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Studia Universitatis BabesBolyai Geologia","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.24193/subbmath.2023.1.04","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Existence results for Dirichlet double phase differential inclusions
"In this paper we consider a class of double phase differential inclusions of the type $$\left\{ \begin{array}{ll} -{\rm div\;}\left(|\nabla u|^{p-2}\nabla u+\mu(x)|\nabla u|^{q-2}\nabla u\right) \in \partial_C^2 f(x,u) , & \mbox{ in }\Omega,\\ u=0, & \mbox{ on }\partial\Omega, \end{array} \right.$$ where $\Omega \subset \mathbb{R}^N$, with $N\ge 2$, is a bounded domain with Lipschitz boundary, $f(x,t)$ is measurable w.r.t. the first variable on $\Omega$ and locally Lipschitz w.r.t. the second variable and $\partial_C^2 f(x,\cdot)$ stands for the Clarke subdifferential of $t\mapsto f(x,t)$. The variational formulation of the problem gives rise to a so-called hemivariational inequality and the corresponding energy functional is not differentiable, but only locally Lipschitz. We use nonsmooth critical point theory to prove the existence of at least one weak solution, provided the $\partial_C^2 f(x,\cdot)$ satisfies an appropriate growth condition."