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The fourth-order total variation flow in $ mathbb{R}^n $ $mathbb｛R｝^n中的四阶总变分流$
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2022-05-16 DOI: 10.3934/mine.2023091
Y. Giga, H. Kuroda, Michal Lasica
{"title":"The fourth-order total variation flow in $ mathbb{R}^n $","authors":"Y. Giga, H. Kuroda, Michal Lasica","doi":"10.3934/mine.2023091","DOIUrl":"https://doi.org/10.3934/mine.2023091","url":null,"abstract":"We define rigorously a solution to the fourth-order total variation flow equation in $ mathbb{R}^n $. If $ ngeq3 $, it can be understood as a gradient flow of the total variation energy in $ D^{-1} $, the dual space of $ D^1_0 $, which is the completion of the space of compactly supported smooth functions in the Dirichlet norm. However, in the low dimensional case $ nleq2 $, the space $ D^{-1} $ does not contain characteristic functions of sets of positive measure, so we extend the notion of solution to a larger space. We characterize the solution in terms of what is called the Cahn-Hoffman vector field, based on a duality argument. This argument relies on an approximation lemma which itself is interesting. We introduce a notion of calibrability of a set in our fourth-order setting. This notion is related to whether a characteristic function preserves its form throughout the evolution. It turns out that all balls are calibrable. However, unlike in the second-order total variation flow, the outside of a ball is calibrable if and only if $ nneq2 $. If $ nneq2 $, all annuli are calibrable, while in the case $ n = 2 $, if an annulus is too thick, it is not calibrable. We compute explicitly the solution emanating from the characteristic function of a ball. We also provide a description of the solution emanating from any piecewise constant, radially symmetric datum in terms of a system of ODEs.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42351934","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
Symmetry of hypersurfaces and the Hopf Lemma 超曲面的对称性与Hopf引理
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2022-05-10 DOI: 10.3934/mine.2023084
Yanyan Li
{"title":"Symmetry of hypersurfaces and the Hopf Lemma","authors":"Yanyan Li","doi":"10.3934/mine.2023084","DOIUrl":"https://doi.org/10.3934/mine.2023084","url":null,"abstract":"A classical theorem of A. D. Alexandrov says that a connected compact smooth hypersurface in Euclidean space with constant mean curvature must be a sphere. We give exposition to some results on symmetry properties of hypersurfaces with ordered mean curvature and associated variations of the Hopf Lemma. Some open problems will be discussed.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42873894","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
On regularity and existence of weak solutions to nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients 粗糙系数非线性Kolmogorov-Fokker-Planck型方程弱解的正则性和存在性
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2022-04-26 DOI: 10.3934/mine.2023043
Prashanta Garain, K. Nystrom
{"title":"On regularity and existence of weak solutions to nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients","authors":"Prashanta Garain, K. Nystrom","doi":"10.3934/mine.2023043","DOIUrl":"https://doi.org/10.3934/mine.2023043","url":null,"abstract":"<abstract><p>We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form</p>\u0000\u0000<p><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ begin{equation*} (partial_t+Xcdotnabla_Y)u = nabla_Xcdot(A(nabla_X u, X, Y, t)). end{equation*} $end{document} </tex-math></disp-formula></p>\u0000<p>The function $ A = A(xi, X, Y, t): mathbb R^mtimes mathbb R^mtimes mathbb R^mtimes mathbb Rto mathbb R^m $ is assumed to be continuous with respect to $ xi $, and measurable with respect to $ X, Y $ and $ t $. $ A = A(xi, X, Y, t) $ is allowed to be nonlinear but with linear growth. We establish higher integrability and local boundedness of weak sub-solutions, weak Harnack and Harnack inequalities, and Hölder continuity with quantitative estimates. In addition we establish existence and uniqueness of weak solutions to a Dirichlet problem in certain bounded $ X $, $ Y $ and $ t $ dependent domains.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46405591","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: 2
Singular structures in solutions to the Monge-Ampère equation with point masses 具有点质量的Monge-Ampère方程解的奇异结构
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2022-04-24 DOI: 10.3934/mine.2023083
Connor Mooney, Arghya Rakshit
{"title":"Singular structures in solutions to the Monge-Ampère equation with point masses","authors":"Connor Mooney, Arghya Rakshit","doi":"10.3934/mine.2023083","DOIUrl":"https://doi.org/10.3934/mine.2023083","url":null,"abstract":"We construct new examples of Monge-Ampère metrics with polyhedral singular structures, motivated by problems related to the optimal transport of point masses and to mirror symmetry. We also analyze the stability of the singular structures under small perturbations of the data given in the problem under consideration.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46318282","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: 3
A diagrammatic view of differential equations in physics 物理学中微分方程的图解
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2022-04-01 DOI: 10.3934/mine.2023036
Evan Patterson, Andre Baas, Timothy Hosgood, James P. Fairbanks
{"title":"A diagrammatic view of differential equations in physics","authors":"Evan Patterson, Andre Baas, Timothy Hosgood, James P. Fairbanks","doi":"10.3934/mine.2023036","DOIUrl":"https://doi.org/10.3934/mine.2023036","url":null,"abstract":"Presenting systems of differential equations in the form of diagrams has become common in certain parts of physics, especially electromagnetism and computational physics. In this work, we aim to put such use of diagrams on a firm mathematical footing, while also systematizing a broadly applicable framework to reason formally about systems of equations and their solutions. Our main mathematical tools are category-theoretic diagrams, which are well known, and morphisms between diagrams, which have been less appreciated. As an application of the diagrammatic framework, we show how complex, multiphysical systems can be modularly constructed from basic physical principles. A wealth of examples, drawn from electromagnetism, transport phenomena, fluid mechanics, and other fields, is included.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42404902","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: 5
The fractional Malmheden theorem 分数阶Malmheden定理
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2022-03-14 DOI: 10.3934/mine.2023024
S. Dipierro, G. Giacomin, E. Valdinoci
{"title":"The fractional Malmheden theorem","authors":"S. Dipierro, G. Giacomin, E. Valdinoci","doi":"10.3934/mine.2023024","DOIUrl":"https://doi.org/10.3934/mine.2023024","url":null,"abstract":"We provide a fractional counterpart of the classical results by Schwarz and Malmheden on harmonic functions. From that we obtain a representation formula for $ s $-harmonic functions as a linear superposition of weighted classical harmonic functions which also entails a new proof of the fractional Harnack inequality. This proof also leads to optimal constants for the fractional Harnack inequality in the ball.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70223687","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: 2
The vanishing discount problem for monotone systems of Hamilton-Jacobi equations: a counterexample to the full convergence Hamilton-Jacobi方程单调系统的消失折扣问题:完全收敛的一个反例
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2022-02-05 DOI: 10.3934/mine.2023072
H. Ishii
{"title":"The vanishing discount problem for monotone systems of Hamilton-Jacobi equations: a counterexample to the full convergence","authors":"H. Ishii","doi":"10.3934/mine.2023072","DOIUrl":"https://doi.org/10.3934/mine.2023072","url":null,"abstract":"In recent years there has been intense interest in the vanishing discount problem for Hamilton-Jacobi equations. In the case of the scalar equation, B. Ziliotto has recently given an example of the Hamilton-Jacobi equation having non-convex Hamiltonian in the gradient variable, for which the full convergence of the solutions does not hold as the discount factor tends to zero. We give here an explicit example of nonlinear monotone systems of Hamilton-Jacobi equations having convex Hamiltonians in the gradient variable, for which the full convergence of the solutions fails as the discount factor goes to zero.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47363902","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: 10
Spectral stability of the curlcurl operator via uniform Gaffney inequalities on perturbed electromagnetic cavities 扰动电磁腔上基于一致Gaffney不等式的curlcurl算子的谱稳定性
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2022-01-30 DOI: 10.3934/mine.2023018
P. D. Lamberti, Michele Zaccaron
{"title":"Spectral stability of the curlcurl operator via uniform Gaffney inequalities on perturbed electromagnetic cavities","authors":"P. D. Lamberti, Michele Zaccaron","doi":"10.3934/mine.2023018","DOIUrl":"https://doi.org/10.3934/mine.2023018","url":null,"abstract":"We prove spectral stability results for the $ curl curl $ operator subject to electric boundary conditions on a cavity upon boundary perturbations. The cavities are assumed to be sufficiently smooth but we impose weak restrictions on the strength of the perturbations. The methods are of variational type and are based on two main ingredients: the construction of suitable Piola-type transformations between domains and the proof of uniform Gaffney inequalities obtained by means of uniform a priori $ H^2 $-estimates for the Poisson problem of the Dirichlet Laplacian. The uniform a priori estimates are proved by using the results of V. Maz'ya and T. Shaposhnikova based on Sobolev multipliers. Connections to boundary homogenization problems are also indicated.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42914467","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: 3
On an anisotropic fractional Stefan-type problem with Dirichlet boundary conditions 具有Dirichlet边界条件的各向异性分数阶stefan型问题
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2022-01-19 DOI: 10.3934/mine.2023047
Catharine Lo, J. Rodrigues
{"title":"On an anisotropic fractional Stefan-type problem with Dirichlet boundary conditions","authors":"Catharine Lo, J. Rodrigues","doi":"10.3934/mine.2023047","DOIUrl":"https://doi.org/10.3934/mine.2023047","url":null,"abstract":"<abstract><p>In this work, we consider the fractional Stefan-type problem in a Lipschitz bounded domain $ Omegasubsetmathbb{R}^d $ with time-dependent Dirichlet boundary condition for the temperature $ vartheta = vartheta(x, t) $, $ vartheta = g $ on $ Omega^ctimes]0, T[$, and initial condition $ eta_0 $ for the enthalpy $ eta = eta(x, t) $, given in $ Omegatimes]0, T[$ by</p>\u0000\u0000<p><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ frac{partial eta}{partial t} +mathcal{L}_A^s vartheta = fquadtext{ with }etain beta(vartheta), $end{document} </tex-math></disp-formula></p>\u0000\u0000<p>where $ mathcal{L}_A^s $ is an anisotropic fractional operator defined in the distributional sense by</p>\u0000\u0000<p><disp-formula> <label/> <tex-math id=\"FE2\"> begin{document}$ langlemathcal{L}_A^su, vrangle = int_{mathbb{R}^d}AD^sucdot D^sv, dx, $end{document} </tex-math></disp-formula></p>\u0000\u0000<p>$ beta $ is a maximal monotone graph, $ A(x) $ is a symmetric, strictly elliptic and uniformly bounded matrix, and $ D^s $ is the distributional Riesz fractional gradient for $ 0 < s < 1 $. We show the existence of a unique weak solution with its corresponding weak regularity. We also consider the convergence as $ snearrow 1 $ towards the classical local problem, the asymptotic behaviour as $ ttoinfty $, and the convergence of the two-phase Stefan-type problem to the one-phase Stefan-type problem by varying the maximal monotone graph $ beta $.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41476164","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
Fluid instabilities, waves and non-equilibrium dynamics of interacting particles: a short overview 流体不稳定性，波和相互作用粒子的非平衡动力学:一个简短的概述
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2022-01-01 DOI: 10.3934/mine.2023033
Roberta Bianchini, C. Saffirio
{"title":"Fluid instabilities, waves and non-equilibrium dynamics of interacting particles: a short overview","authors":"Roberta Bianchini, C. Saffirio","doi":"10.3934/mine.2023033","DOIUrl":"https://doi.org/10.3934/mine.2023033","url":null,"abstract":"<jats:p xml:lang=\"fr\" />","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224541","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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