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A limiting case in partial regularity for quasiconvex functionals 准凸函数部分正则性的极限情况
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2023-08-10 DOI: 10.3934/mine.2024001
M. Piccinini
{"title":"A limiting case in partial regularity for quasiconvex functionals","authors":"M. Piccinini","doi":"10.3934/mine.2024001","DOIUrl":"https://doi.org/10.3934/mine.2024001","url":null,"abstract":"<abstract><p>Local minimizers of nonhomogeneous quasiconvex variational integrals with standard $ p $-growth of the type</p> <p><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ wmapsto int left[F(Dw)-fcdot wright]{,{{rm{d}}}x} $end{document} </tex-math></disp-formula></p> <p>feature almost everywhere $ mbox{BMO} $-regular gradient provided that $ f $ belongs to the borderline Marcinkiewicz space $ L(n, infty) $.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139351057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Local boundedness of weak solutions to elliptic equations with $ p, q- $growth 具有$ p, q- $增长的椭圆方程弱解的局部有界性
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2023-01-01 DOI: 10.3934/mine.2023065
G. Cupini, Paolo Marcellini, E. Mascolo
{"title":"Local boundedness of weak solutions to elliptic equations with $ p, q- $growth","authors":"G. Cupini, Paolo Marcellini, E. Mascolo","doi":"10.3934/mine.2023065","DOIUrl":"https://doi.org/10.3934/mine.2023065","url":null,"abstract":"This article is dedicated to Giuseppe Mingione for his $ 50^{th} $ birthday, a leading expert in the regularity theory and in particular in the subject of this manuscript. In this paper we give conditions for the local boundedness of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form of the type considered below in (1.1), under $ p, q- $growth assumptions. The novelties with respect to the mathematical literature on this topic are the general growth conditions and the explicit dependence of the differential equation on $ u $, other than on its gradient $ Du $ and on the $ x $ variable.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70225098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 11
Preface to the Special Issue: Nonlinear PDEs and geometric analysis – Dedicated to Neil Trudinger on the occasion of his 80th birthday 《非线性偏微分方程和几何分析》特刊序言——献给尼尔·特鲁丁格，纪念他80岁生日
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2023-01-01 DOI: 10.3934/mine.2023095
J. Clutterbuck, Jiakun Liu
{"title":"Preface to the Special Issue: Nonlinear PDEs and geometric analysis – Dedicated to Neil Trudinger on the occasion of his 80th birthday","authors":"J. Clutterbuck, Jiakun Liu","doi":"10.3934/mine.2023095","DOIUrl":"https://doi.org/10.3934/mine.2023095","url":null,"abstract":"<abstract><p>This contribution is the preface of the Special Issue: Nonlinear PDEs and geometric analysis – Dedicated to Neil Trudinger on the occasion of his 80th birthday.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70225186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Lewy-Stampacchia inequality for noncoercive parabolic obstacle problems 非强制抛物型障碍问题的Lewy-Stampacchia不等式
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2023-01-01 DOI: 10.3934/mine.2023071
F. Farroni, G. Moscariello, Gabriella Zecca
{"title":"Lewy-Stampacchia inequality for noncoercive parabolic obstacle problems","authors":"F. Farroni, G. Moscariello, Gabriella Zecca","doi":"10.3934/mine.2023071","DOIUrl":"https://doi.org/10.3934/mine.2023071","url":null,"abstract":"We investigate the obstacle problem for a class of nonlinear and noncoercive parabolic variational inequalities whose model is a Leray–Lions type operator having singularities in the coefficients of the lower order terms. We prove the existence of a solution to the obstacle problem satisfying a Lewy-Stampacchia type inequality.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70225338","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Approximation of elliptic and parabolic equations with Dirichlet boundary conditions 椭圆型和抛物型方程的Dirichlet边界条件近似
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2023-01-01 DOI: 10.3934/mine.2023079
Youchan Kim, Seungjin Ryu, Pilsoo Shin
{"title":"Approximation of elliptic and parabolic equations with Dirichlet boundary conditions","authors":"Youchan Kim, Seungjin Ryu, Pilsoo Shin","doi":"10.3934/mine.2023079","DOIUrl":"https://doi.org/10.3934/mine.2023079","url":null,"abstract":"We obtain an approximation result of the weak solutions to elliptic and parabolic equations with Dirichlet boundary conditions. We show that the weak solution can be obtained with a limit of approximations by regularizing the nonlinearities and approximating the domains.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70225489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The mixed virtual element discretization for highly-anisotropic problems: the role of the boundary degrees of freedom 高各向异性问题的混合虚元离散化:边界自由度的作用
4区 工程技术
Mathematics in Engineering Pub Date : 2023-01-01 DOI: 10.3934/mine.2023099
Stefano Berrone, Stefano Scialò, Gioana Teora
{"title":"The mixed virtual element discretization for highly-anisotropic problems: the role of the boundary degrees of freedom","authors":"Stefano Berrone, Stefano Scialò, Gioana Teora","doi":"10.3934/mine.2023099","DOIUrl":"https://doi.org/10.3934/mine.2023099","url":null,"abstract":"<abstract><p>In this paper, we discuss the accuracy and the robustness of the mixed Virtual Element Methods when dealing with highly anisotropic diffusion problems. In particular, we analyze the performance of different approaches which are characterized by different sets of both boundary and internal degrees of freedom in the presence of a strong anisotropy of the diffusion tensor with constant or variable coefficients. A new definition of the boundary degrees of freedom is also proposed and tested.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136303863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Gradient estimates for the solutions of higher order curvature equations with prescribed contact angle 给定接触角的高阶曲率方程解的梯度估计
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2023-01-01 DOI: 10.3934/mine.2023093
Bin Deng, Xinan Ma
{"title":"Gradient estimates for the solutions of higher order curvature equations with prescribed contact angle","authors":"Bin Deng, Xinan Ma","doi":"10.3934/mine.2023093","DOIUrl":"https://doi.org/10.3934/mine.2023093","url":null,"abstract":"<abstract><p>In this paper, we use the maximum principle and moving frame technique to prove the global gradient estimates for the higher-order curvature equations with prescribed contact angle problems.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70225175","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
A volume constraint problem for the nonlocal doubly nonlinear parabolic equation 非局部双非线性抛物方程的体积约束问题
4区 工程技术
Mathematics in Engineering Pub Date : 2023-01-01 DOI: 10.3934/mine.2023098
Masashi Misawa, Kenta Nakamura, Yoshihiko Yamaura
{"title":"A volume constraint problem for the nonlocal doubly nonlinear parabolic equation","authors":"Masashi Misawa, Kenta Nakamura, Yoshihiko Yamaura","doi":"10.3934/mine.2023098","DOIUrl":"https://doi.org/10.3934/mine.2023098","url":null,"abstract":"<abstract><p>We consider a volume constraint problem for the nonlocal doubly nonlinear parabolic equation, called the nonlocal $ p $-Sobolev flow, and introduce a nonlinear intrinsic scaling, converting a prototype nonlocal doubly nonlinear parabolic equation into the nonlocal $ p $-Sobolev flow. This paper is dedicated to Giuseppe Mingione on the occasion of his 50th birthday, who is a maestro in the regularity theory of PDEs.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135556559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A guide to the design of the virtual element methods for second- and fourth-order partial differential equations 二阶和四阶偏微分方程虚元法的设计指南
4区 工程技术
Mathematics in Engineering Pub Date : 2023-01-01 DOI: 10.3934/mine.2023100
Yu Leng, Lampros Svolos, Dibyendu Adak, Ismael Boureima, Gianmarco Manzini, Hashem Mourad, Jeeyeon Plohr
{"title":"A guide to the design of the virtual element methods for second- and fourth-order partial differential equations","authors":"Yu Leng, Lampros Svolos, Dibyendu Adak, Ismael Boureima, Gianmarco Manzini, Hashem Mourad, Jeeyeon Plohr","doi":"10.3934/mine.2023100","DOIUrl":"https://doi.org/10.3934/mine.2023100","url":null,"abstract":"<abstract><p>We discuss the design and implementation details of two conforming virtual element methods for the numerical approximation of two partial differential equations that emerge in phase-field modeling of fracture propagation in elastic material. The two partial differential equations are: (i) a linear hyperbolic equation describing the momentum balance and (ii) a fourth-order elliptic equation modeling the damage of the material. Inspired by <sup>[<xref ref-type=\"bibr\" rid=\"b1\">1</xref>,<xref ref-type=\"bibr\" rid=\"b2\">2</xref>,<xref ref-type=\"bibr\" rid=\"b3\">3</xref>]</sup>, we develop a new conforming VEM for the discretization of the two equations, which is implementation-friendly, i.e., different terms can be implemented by exploiting a single projection operator. We use $ C^0 $ and $ C^1 $ virtual elements for the second-and fourth-order partial differential equation, respectively. For both equations, we review the formulation of the virtual element approximation and discuss the details pertaining the implementation.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135561706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A mixed operator approach to peridynamics 周动力学的混合算子方法
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2023-01-01 DOI: 10.3934/mine.2023082
F. Cluni, V. Gusella, Dimitri Mugnai, Edoardo Proietti Lippi, P. Pucci
{"title":"A mixed operator approach to peridynamics","authors":"F. Cluni, V. Gusella, Dimitri Mugnai, Edoardo Proietti Lippi, P. Pucci","doi":"10.3934/mine.2023082","DOIUrl":"https://doi.org/10.3934/mine.2023082","url":null,"abstract":"In the present paper we propose a model describing the nonlocal behavior of an elastic body using a peridynamical approach. Indeed, peridynamics is a suitable framework for problems where discontinuities appear naturally, such as fractures, dislocations, or, in general, multiscale materials. In particular, the regional fractional Laplacian is used as the nonlocal operator. Moreover, a combination of the fractional and classical Laplacian operators is used to obtain a better description of the phenomenological response in elasticity. We consider models with linear and nonlinear perturbations. In the linear case, we prove the existence and uniqueness of the solution, while in the nonlinear case the existence of at least two nontrivial solutions of opposite sign is proved. The linear and nonlinear problems are also solved by a numerical approach which estimates the regional fractional Laplacian by means of its singular integral representation. In both cases, a numerical estimation of the solutions is obtained, using in the nonlinear case an approach involving a random variation of an initial guess of the solution. Moreover, in the linear case a parametric analysis is made in order to study the effects of the parameters involved in the model, such as the order of the fractional Laplacian and the mixture law between local and nonlocal behavior.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70225541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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