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Games associated with products of eigenvalues of the Hessian 与黑森特征值乘积相关的博弈
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2023-01-01 DOI: 10.3934/mine.2023066
P. Blanc, Fernando Charro, J. Manfredi, J. Rossi
{"title":"Games associated with products of eigenvalues of the Hessian","authors":"P. Blanc, Fernando Charro, J. Manfredi, J. Rossi","doi":"10.3934/mine.2023066","DOIUrl":"https://doi.org/10.3934/mine.2023066","url":null,"abstract":"We introduce games associated with second-order partial differential equations given by arbitrary products of eigenvalues of the Hessian. We prove that, as a parameter that controls the step length goes to zero, the value functions of the games converge uniformly to a viscosity solution of the partial differential equation. The classical Monge-Ampère equation is an important example under consideration.","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":"70224671","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
Nanoparticle-based organic polymer retinal prostheses: modeling, solution map and simulation 基于纳米粒子的有机聚合物视网膜假体:建模、溶液图和仿真
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2023-01-01 DOI: 10.3934/mine.2023075
G. Chiaravalli, G. Lanzani, R. Sacco, S. Salsa
{"title":"Nanoparticle-based organic polymer retinal prostheses: modeling, solution map and simulation","authors":"G. Chiaravalli, G. Lanzani, R. Sacco, S. Salsa","doi":"10.3934/mine.2023075","DOIUrl":"https://doi.org/10.3934/mine.2023075","url":null,"abstract":"In this article we investigate a mathematical model for a retinal prosthesis made of organic polymer nanoparticles (NP) in the stationary regime. The model consists of a Drift-Diffusion system to describe free charge transport in the NP bulk; a Poisson-Nernst-Planck system to describe ion electrodiffusion in the solution surrounding the NP; and nonlinear transmission conditions at the NP-solution interface. To solve the model we use an iteration procedure for which we prove the existence and briefly comment the uniqueness of a fixed point under suitable smallness assumptions on model parameters. For system discretization we use a stabilized finite element method to prevent unphysical oscillations in the electric potential, carrier number densities and ion molar densities. Model predictions describe the amount of active chemical molecule accumulating at the neuron surface and highlight electrostatic effects induced by the sole presence of the nanoparticle. These results support the use of mathematical modeling as a virtual laboratory for the optimal design of bio-hybrid systems, whose investigation may be impervious due to experimental limits.","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":"70225458","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
Equilibrium of thin shells under large strains without through-the-thickness shear and self-penetration of matter 大应变下薄壳的平衡，无物质的穿厚剪切和自穿透
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2023-01-01 DOI: 10.3934/mine.2023092
P. M. Mariano, D. Mucci
{"title":"Equilibrium of thin shells under large strains without through-the-thickness shear and self-penetration of matter","authors":"P. M. Mariano, D. Mucci","doi":"10.3934/mine.2023092","DOIUrl":"https://doi.org/10.3934/mine.2023092","url":null,"abstract":"We consider elastic thin shells without through-the-thickness shear and depict them as Gauss graphs of parametric surfaces. (We use the term shells to include plates and thin films therein.) We consider an energy depending on the first derivative of the Gauss map (so, it involves curvatures) and its second-rank minors. For it we prove existence of minimizers in terms of currents carried by Gauss graphs. In the limiting process we adopt sequences of competitors that satisfy a condition that prevents self-penetration of matter.","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":"70225639","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
Polyconvex functionals and maximum principle 多凸泛函和极大原理
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2023-01-01 DOI: 10.3934/mine.2023077
M. Carozza, L. Esposito, Raffaella Giova, F. Leonetti
{"title":"Polyconvex functionals and maximum principle","authors":"M. Carozza, L. Esposito, Raffaella Giova, F. Leonetti","doi":"10.3934/mine.2023077","DOIUrl":"https://doi.org/10.3934/mine.2023077","url":null,"abstract":"<abstract><p>Let us consider continuous minimizers $ u : bar Omega subset mathbb{R}^n to mathbb{R}^n $ of</p> <p><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ mathcal{F}(v) = int_{Omega} [|Dv|^p , + , |{rm det},Dv|^r] dx, $end{document} </tex-math></disp-formula></p> <p>with $ p > 1 $ and $ r > 0 $; then it is known that every component $ u^alpha $ of $ u = (u^1, ..., u^n) $ enjoys maximum principle: the set of interior points $ x $, for which the value $ u^alpha(x) $ is greater than the supremum on the boundary, has null measure, that is, $ mathcal{L}^n({ x in Omega: u^alpha (x) > sup_{partial Omega} u^alpha }) = 0 $. If we change the structure of the functional, it might happen that the maximum principle fails, as in the case</p> <p><disp-formula> <label/> <tex-math id=\"FE2\"> begin{document}$ mathcal{F}(v) = int_{Omega}[max{(|Dv|^p - 1); 0 } , + , |{rm det},Dv|^r] dx, $end{document} </tex-math></disp-formula></p> <p>with $ p > 1 $ and $ r > 0 $. Indeed, for a suitable boundary value, the set of the interior points $ x $, for which the value $ u^alpha(x) $ is greater than the supremum on the boundary, has a positive measure, that is $ mathcal{L}^n({ x in Omega: u^alpha (x) > sup_{partial Omega} u^alpha }) > 0 $. In this paper we show that the measure of the image of these bad points is zero, that is $ mathcal{L}^n(u({ x in Omega: u^alpha (x) > sup_{partial Omega} u^alpha })) = 0 $, provided $ p > n $. This is a particular case of a more general theorem.</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":"70225428","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 "nonlinear duality" approach to $ W_0^{1, 1} $ solutions in elliptic systems related to the Keller-Segel model Keller-Segel模型下椭圆系统$ W_0^{1,1} $解的“非线性对偶”方法
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2023-01-01 DOI: 10.3934/mine.2023085
L. Boccardo
{"title":"A \"nonlinear duality\" approach to $ W_0^{1, 1} $ solutions in elliptic systems related to the Keller-Segel model","authors":"L. Boccardo","doi":"10.3934/mine.2023085","DOIUrl":"https://doi.org/10.3934/mine.2023085","url":null,"abstract":"In this paper, we prove existence of distributional solutions of a nonlinear elliptic system, related to the Keller-Segel model. Our starting point is the boundedness theorem (for solutions of elliptic equations) proved by Guido Stampacchia and Neil Trudinger.","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":"70225588","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
$ L^{p} $ compactness criteria with an application to variational convergence of some nonlocal energy functionals L^{p} $紧性准则及其在一些非局部能量泛函变分收敛中的应用
4区 工程技术
Mathematics in Engineering Pub Date : 2023-01-01 DOI: 10.3934/mine.2023097
Qiang Du, Tadele Mengesha, Xiaochuan Tian
{"title":"$ L^{p} $ compactness criteria with an application to variational convergence of some nonlocal energy functionals","authors":"Qiang Du, Tadele Mengesha, Xiaochuan Tian","doi":"10.3934/mine.2023097","DOIUrl":"https://doi.org/10.3934/mine.2023097","url":null,"abstract":"<abstract><p>Motivated by some variational problems from a nonlocal model of mechanics, this work presents a set of sufficient conditions that guarantee a compact inclusion in the function space of $ L^{p} $ vector fields defined on a domain $ Omega $ that is either a bounded domain in $ mathbb{R}^{d} $ or $ mathbb{R}^{d} $ itself. The criteria are nonlocal and are given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. Moreover, these criteria for vector fields are also different from those given for scalar fields in that the conditions are based on nonlocal interactions involving only parts of the components of the vector fields. The $ L^{p} $ compactness criteria are utilized in demonstrating the convergence of minimizers of parameterized nonlocal energy functionals.</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":"135440895","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
Layered solutions for a nonlocal Ginzburg-Landau model with periodic modulation 具有周期调制的非局部金兹堡-朗道模型的分层解
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2023-01-01 DOI: 10.3934/mine.2023090
Ko-Shin Chen, C. Muratov, Xiaodong Yan
{"title":"Layered solutions for a nonlocal Ginzburg-Landau model with periodic modulation","authors":"Ko-Shin Chen, C. Muratov, Xiaodong Yan","doi":"10.3934/mine.2023090","DOIUrl":"https://doi.org/10.3934/mine.2023090","url":null,"abstract":"We study layered solutions in a one-dimensional version of the scalar Ginzburg-Landau equation that involves a mixture of a second spatial derivative and a fractional half-derivative, together with a periodically modulated nonlinearity. This equation appears as the Euler-Lagrange equation of a suitably renormalized fractional Ginzburg-Landau energy with a double-well potential that is multiplied by a 1-periodically varying nonnegative factor $ g(x) $ with $ int_0^1 frac{1}{g(x)} dx < infty. $ A priori this energy is not bounded below due to the presence of a nonlocal term in the energy. Nevertheless, through a careful analysis of a minimizing sequence we prove existence of global energy minimizers that connect the two wells at infinity. These minimizers are shown to be the classical solutions of the associated nonlocal Ginzburg-Landau type equation.","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":"70225627","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
Fibonacci signals with timing jitter 具有时序抖动的斐波那契信号
IF 1 4区 工程技术
Mathematics in Engineering Pub Date : 2023-01-01 DOI: 10.3934/mine.2023076
D. Citrin
{"title":"Fibonacci signals with timing jitter","authors":"D. Citrin","doi":"10.3934/mine.2023076","DOIUrl":"https://doi.org/10.3934/mine.2023076","url":null,"abstract":"The power spectral density of a signal comprised of a sequence of Dirac $ delta $-functions at successive times determined by a Fibonacci sequence is the temporal analog of the well known structure factor for a Fibonacci chain. Such a signal is quasi-periodic and, under suitable choice of parameters, is the temporal analog of a one-dimensional quasicrystal. While the effects of disorder in the spatial case of Fibonacci chains has been studied numerically, having an analytically tractable stochastic model is needed both for the spatial and temporal cases to be able to study these effects as model parameters are varied. Here, we consider the effects of errors in where the $ delta $-functions defining the signal in the temporal case occur, i.e., timing jitter. In this work, we present an analytically tractable theory of how timing jitter affects the power spectral density of Fibonacci signals.","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":"70225345","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 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
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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