B. Abdellaoui, K. Biroud, A. Primo, Fernando Soria, Abdelbadie Younes
In this work we address the question of existence and non existence of positive solutions to a class of fractional problems with non local gradient term. More precisely, we consider the problem begin{document}$ left{ begin{array}{rcll} (-Delta )^s u & = &lambda dfrac{u}{|x|^{2s}}+ (mathfrak{F}(u)(x))^p+ rho f & text{ in } Omega, u&>&0 & text{ in }Omega, u& = &0 & text{ in }(mathbb{R}^NsetminusOmega), end{array}right. $end{document} where $ Omegasubset mathbb{R}^N $ is a $ C^{1, 1} $ bounded domain, $ N > 2s, rho > 0 $, $ 0 < s < 1 $, $ 1 < p < infty $ and $ 0 < lambda < Lambda_{N, s} $, the Hardy constant defined below. We assume that $ f $ is a non-negative function with additional hypotheses. Here $ mathfrak{F}(u) $ is a nonlocal "gradient" term. In particular, if $ mathfrak{F}(u)(x) = |(-Delta)^{frac s2}u(x)| $, then we are able to show the existence of a critical exponents $ p_{+}(lambda, s) $ such that: 1) if $ p > p_{+}(lambda, s) $, there is no positive solution, 2) if $ p < p_{+}(lambda, s) $, there exists, at least, a positive supersolution solution for suitable data and $ rho $ small. Moreover, under additional restriction on $ p $, there exists a solution for general datum $ f $.
In this work we address the question of existence and non existence of positive solutions to a class of fractional problems with non local gradient term. More precisely, we consider the problem begin{document}$ left{ begin{array}{rcll} (-Delta )^s u & = &lambda dfrac{u}{|x|^{2s}}+ (mathfrak{F}(u)(x))^p+ rho f & text{ in } Omega, u&>&0 & text{ in }Omega, u& = &0 & text{ in }(mathbb{R}^NsetminusOmega), end{array}right. $end{document} where $ Omegasubset mathbb{R}^N $ is a $ C^{1, 1} $ bounded domain, $ N > 2s, rho > 0 $, $ 0 < s < 1 $, $ 1 < p < infty $ and $ 0 < lambda < Lambda_{N, s} $, the Hardy constant defined below. We assume that $ f $ is a non-negative function with additional hypotheses. Here $ mathfrak{F}(u) $ is a nonlocal "gradient" term. In particular, if $ mathfrak{F}(u)(x) = |(-Delta)^{frac s2}u(x)| $, then we are able to show the existence of a critical exponents $ p_{+}(lambda, s) $ such that: 1) if $ p > p_{+}(lambda, s) $, there is no positive solution, 2) if $ p < p_{+}(lambda, s) $, there exists, at least, a positive supersolution solution for suitable data and $ rho $ small. Moreover, under additional restriction on $ p $, there exists a solution for general datum $ f $.
{"title":"Fractional KPZ equations with fractional gradient term and Hardy potential","authors":"B. Abdellaoui, K. Biroud, A. Primo, Fernando Soria, Abdelbadie Younes","doi":"10.3934/mine.2023042","DOIUrl":"https://doi.org/10.3934/mine.2023042","url":null,"abstract":"In this work we address the question of existence and non existence of positive solutions to a class of fractional problems with non local gradient term. More precisely, we consider the problem begin{document}$ left{ begin{array}{rcll} (-Delta )^s u & = &lambda dfrac{u}{|x|^{2s}}+ (mathfrak{F}(u)(x))^p+ rho f & text{ in } Omega, u&>&0 & text{ in }Omega, u& = &0 & text{ in }(mathbb{R}^NsetminusOmega), end{array}right. $end{document} where $ Omegasubset mathbb{R}^N $ is a $ C^{1, 1} $ bounded domain, $ N > 2s, rho > 0 $, $ 0 < s < 1 $, $ 1 < p < infty $ and $ 0 < lambda < Lambda_{N, s} $, the Hardy constant defined below. We assume that $ f $ is a non-negative function with additional hypotheses. Here $ mathfrak{F}(u) $ is a nonlocal \"gradient\" term. In particular, if $ mathfrak{F}(u)(x) = |(-Delta)^{frac s2}u(x)| $, then we are able to show the existence of a critical exponents $ p_{+}(lambda, s) $ such that: 1) if $ p > p_{+}(lambda, s) $, there is no positive solution, 2) if $ p < p_{+}(lambda, s) $, there exists, at least, a positive supersolution solution for suitable data and $ rho $ small. Moreover, under additional restriction on $ p $, there exists a solution for general datum $ f $.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224570","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}
We extend the so-called universal potential estimates of Kuusi-Mingione type (J. Funct. Anal. 262: 4205–4269, 2012) to the singular case $ 1 < pleq 2-1/n $ for the quasilinear equation with measure data
begin{document}$ begin{equation*} -operatorname{div}(A(x,nabla u)) = mu end{equation*} $end{document}
in a bounded open subset $ Omega $ of $ mathbb{R}^n $, $ ngeq 2 $, with a finite signed measure $ mu $ in $ Omega $. The operator $ operatorname{div}(A(x, nabla u)) $ is modeled after the $ p $-Laplacian $ Delta_p u: = {rm div}, (|nabla u|^{p-2}nabla u) $, where the nonlinearity $ A(x, xi) $ ($ x, xi in mathbb{R}^n $) is assumed to satisfy natural growth and monotonicity conditions of order $ p $, as well as certain additional regularity conditions in the $ x $-variable.
We extend the so-called universal potential estimates of Kuusi-Mingione type (J. Funct. Anal. 262: 4205–4269, 2012) to the singular case $ 1 < pleq 2-1/n $ for the quasilinear equation with measure data begin{document}$ begin{equation*} -operatorname{div}(A(x,nabla u)) = mu end{equation*} $end{document} in a bounded open subset $ Omega $ of $ mathbb{R}^n $, $ ngeq 2 $, with a finite signed measure $ mu $ in $ Omega $. The operator $ operatorname{div}(A(x, nabla u)) $ is modeled after the $ p $-Laplacian $ Delta_p u: = {rm div}, (|nabla u|^{p-2}nabla u) $, where the nonlinearity $ A(x, xi) $ ($ x, xi in mathbb{R}^n $) is assumed to satisfy natural growth and monotonicity conditions of order $ p $, as well as certain additional regularity conditions in the $ x $-variable.
{"title":"Universal potential estimates for $ 1 < pleq 2-frac{1}{n} $","authors":"Quoc-Hung Nguyen, N. Phuc","doi":"10.3934/mine.2023057","DOIUrl":"https://doi.org/10.3934/mine.2023057","url":null,"abstract":"<abstract><p>We extend the so-called universal potential estimates of Kuusi-Mingione type (J. Funct. Anal. 262: 4205–4269, 2012) to the singular case $ 1 < pleq 2-1/n $ for the quasilinear equation with measure data</p> <p><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ begin{equation*} -operatorname{div}(A(x,nabla u)) = mu end{equation*} $end{document} </tex-math></disp-formula></p> <p>in a bounded open subset $ Omega $ of $ mathbb{R}^n $, $ ngeq 2 $, with a finite signed measure $ mu $ in $ Omega $. The operator $ operatorname{div}(A(x, nabla u)) $ is modeled after the $ p $-Laplacian $ Delta_p u: = {rm div}, (|nabla u|^{p-2}nabla u) $, where the nonlinearity $ A(x, xi) $ ($ x, xi in mathbb{R}^n $) is assumed to satisfy natural growth and monotonicity conditions of order $ p $, as well as certain additional regularity conditions in the $ x $-variable.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224763","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}
Internal gravity waves are propagating disturbances in stably stratified fluids, and can transport momentum and energy over large spatial extents. From a fundamental viewpoint, internal waves are interesting due to the nature of their dispersion relation, and their linear dynamics are reasonably well-understood. From an oceanographic viewpoint, a qualitative and quantitative understanding of significant internal wave generation in the ocean is emerging, while their dissipation mechanisms are being debated. This paper reviews the current knowledge on instabilities in internal gravity waves, primarily focusing on the growth of small-amplitude disturbances. Historically, wave-wave interactions based on weakly nonlinear expansions have driven progress in this field, to investigate spontaneous energy transfer to various temporal and spatial scales. Recent advances in numerical/experimental modeling and field observations have further revealed noticeable differences between various internal wave spatial forms in terms of their instability characteristics; this in turn has motivated theoretical calculations on appropriately chosen internal wave fields in various settings. After a brief introduction, we present a pedagogical discussion on linear internal waves and their different two-dimensional spatial forms. The general ideas concerning triadic resonance in internal waves are then introduced, before proceeding towards instability characteristics of plane waves, wave beams and modes. Results from various theoretical, experimental and numerical studies are summarized to provide an overall picture of the gaps in our understanding. An ocean perspective is then given, both in terms of the relevant outstanding questions and the various additional factors at play. While the applications in this review are focused on the ocean, several ideas are relevant to atmospheric and astrophysical systems too.
{"title":"Instabilities in internal gravity waves","authors":"Dheeraj Varma, M. Mathur, T. Dauxois","doi":"10.3934/mine.2023016","DOIUrl":"https://doi.org/10.3934/mine.2023016","url":null,"abstract":"Internal gravity waves are propagating disturbances in stably stratified fluids, and can transport momentum and energy over large spatial extents. From a fundamental viewpoint, internal waves are interesting due to the nature of their dispersion relation, and their linear dynamics are reasonably well-understood. From an oceanographic viewpoint, a qualitative and quantitative understanding of significant internal wave generation in the ocean is emerging, while their dissipation mechanisms are being debated. This paper reviews the current knowledge on instabilities in internal gravity waves, primarily focusing on the growth of small-amplitude disturbances. Historically, wave-wave interactions based on weakly nonlinear expansions have driven progress in this field, to investigate spontaneous energy transfer to various temporal and spatial scales. Recent advances in numerical/experimental modeling and field observations have further revealed noticeable differences between various internal wave spatial forms in terms of their instability characteristics; this in turn has motivated theoretical calculations on appropriately chosen internal wave fields in various settings. After a brief introduction, we present a pedagogical discussion on linear internal waves and their different two-dimensional spatial forms. The general ideas concerning triadic resonance in internal waves are then introduced, before proceeding towards instability characteristics of plane waves, wave beams and modes. Results from various theoretical, experimental and numerical studies are summarized to provide an overall picture of the gaps in our understanding. An ocean perspective is then given, both in terms of the relevant outstanding questions and the various additional factors at play. While the applications in this review are focused on the ocean, several ideas are relevant to atmospheric and astrophysical systems too.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70223239","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}
In this paper, we present two type of contributions to the study of two-phases problems. In such problems, the main focus is on optimising a diffusion function $ a $ under $ L^infty $ and $ L^1 $ constraints, this function $ a $ appearing in a diffusive term of the form $ -{{nabla}} cdot(a{{nabla}}) $ in the model, in order to maximise a certain criterion. We provide a parabolic Talenti inequality and a partial bang-bang property in radial geometries for a general class of elliptic optimisation problems: namely, if a radial solution exists, then it must saturate, at almost every point, the $ L^infty $ constraints defining the admissible class. This is done using an oscillatory method.
在本文中，我们对两相问题的研究提出了两种类型的贡献。在这类问题中，主要重点是在$ L^infty $和$ L^1 $约束下优化扩散函数$ a $，该函数$ a $在模型中以形式为$ -{{nabla}} cdot(a{{nabla}}) $的扩散项出现，以最大化某个准则。我们为一类椭圆优化问题提供了一个抛物线Talenti不等式和径向几何中的部分bang-bang性质:即，如果存在径向解，那么它必须在几乎每个点上饱和，$ L^infty $约束定义了可接受的类。这是用振荡法完成的。
{"title":"Some comparison results and a partial bang-bang property for two-phases problems in balls","authors":"Idriss Mazari","doi":"10.3934/mine.2023010","DOIUrl":"https://doi.org/10.3934/mine.2023010","url":null,"abstract":"In this paper, we present two type of contributions to the study of two-phases problems. In such problems, the main focus is on optimising a diffusion function $ a $ under $ L^infty $ and $ L^1 $ constraints, this function $ a $ appearing in a diffusive term of the form $ -{{nabla}} cdot(a{{nabla}}) $ in the model, in order to maximise a certain criterion. We provide a parabolic Talenti inequality and a partial bang-bang property in radial geometries for a general class of elliptic optimisation problems: namely, if a radial solution exists, then it must saturate, at almost every point, the $ L^infty $ constraints defining the admissible class. This is done using an oscillatory method.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70223437","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}
A bounded domain $ Omega $ in a Riemannian manifold $ M $ is said to have the Pompeiu property if the only continuous function which integrates to zero on $ Omega $ and on all its congruent images is the zero function. In some respects, the Pompeiu property can be viewed as an overdetermined problem, given its relation with the Schiffer problem. It is well-known that every Euclidean ball fails to have the Pompeiu property while spherical balls have the property for almost all radii (Ungar's Freak theorem). In the present paper we discuss the Pompeiu property when $ M $ is compact and admits an isoparametric foliation. In particular, we identify precise conditions on the spectrum of the Laplacian on $ M $ under which the level domains of an isoparametric function fail to have the Pompeiu property. Specific calculations are carried out when the ambient manifold is the round sphere, and some consequences are derived. Moreover, a detailed discussion of Ungar's Freak theorem and its generalizations is also carried out.
黎曼流形M中的一个有界定义域我们说它具有庞培性质如果唯一的连续函数在它的所有同余像上积分为零是零函数。在某些方面，鉴于庞培性质与希弗问题的关系，它可以被看作是一个超定问题。众所周知，每个欧几里得球都不具有庞培性质，而球形球几乎在所有半径上都具有庞培性质(Ungar’s Freak theorem)。本文讨论了$ M $紧致并允许等参叶理时的庞培性质。特别地，我们在M上的拉普拉斯谱上确定了等参函数的水平域不具有庞培性质的精确条件。对环境流形为圆球时进行了具体的计算，并得出了一些结论。此外，还详细讨论了Ungar的反常定理及其推广。
{"title":"Isoparametric foliations and the Pompeiu property","authors":"L. Provenzano, A. Savo","doi":"10.3934/mine.2023031","DOIUrl":"https://doi.org/10.3934/mine.2023031","url":null,"abstract":"A bounded domain $ Omega $ in a Riemannian manifold $ M $ is said to have the Pompeiu property if the only continuous function which integrates to zero on $ Omega $ and on all its congruent images is the zero function. In some respects, the Pompeiu property can be viewed as an overdetermined problem, given its relation with the Schiffer problem. It is well-known that every Euclidean ball fails to have the Pompeiu property while spherical balls have the property for almost all radii (Ungar's Freak theorem). In the present paper we discuss the Pompeiu property when $ M $ is compact and admits an isoparametric foliation. In particular, we identify precise conditions on the spectrum of the Laplacian on $ M $ under which the level domains of an isoparametric function fail to have the Pompeiu property. Specific calculations are carried out when the ambient manifold is the round sphere, and some consequences are derived. Moreover, a detailed discussion of Ungar's Freak theorem and its generalizations is also carried out.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70223695","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}
We consider positive solutions to semilinear elliptic problems with Hardy potential and a first order term in bounded smooth domain $ Omega $ with $ 0in overline Omega $. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure under suitable assumptions on the nonlinearity.
{"title":"Qualitative properties of solutions to the Dirichlet problem for a Laplace equation involving the Hardy potential with possibly boundary singularity","authors":"L. Montoro, B. Sciunzi","doi":"10.3934/mine.2023017","DOIUrl":"https://doi.org/10.3934/mine.2023017","url":null,"abstract":"We consider positive solutions to semilinear elliptic problems with Hardy potential and a first order term in bounded smooth domain $ Omega $ with $ 0in overline Omega $. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure under suitable assumptions on the nonlinearity.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70223329","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}
In the paper [20], the authors introduced a Gauss curvature flow to study the Aleksandrov problem and the dual Minkowski problem. The paper [20] treated the cases when one can establish the uniform estimate for the Gauss curvature flow. In this paper, we study the $ L_p $ dual Minkowski problem, an extension of the dual Minkowski problem. We deal with some cases in which there is no uniform estimate for the Gauss curvature flow. We adopt the topological method from [13] to find a special initial condition such that the Gauss curvature flow converges to a solution of the $ L_p $ dual Minkowski problem.
{"title":"Flow by Gauss curvature to the $ L_p $ dual Minkowski problem","authors":"Qiang Guang, Qi-Rui Li, Xu-jia Wang","doi":"10.3934/mine.2023049","DOIUrl":"https://doi.org/10.3934/mine.2023049","url":null,"abstract":"<abstract><p>In the paper <sup>[<xref ref-type=\"bibr\" rid=\"b20\">20</xref>]</sup>, the authors introduced a Gauss curvature flow to study the Aleksandrov problem and the dual Minkowski problem. The paper <sup>[<xref ref-type=\"bibr\" rid=\"b20\">20</xref>]</sup> treated the cases when one can establish the uniform estimate for the Gauss curvature flow. In this paper, we study the $ L_p $ dual Minkowski problem, an extension of the dual Minkowski problem. We deal with some cases in which there is no uniform estimate for the Gauss curvature flow. We adopt the topological method from <sup>[<xref ref-type=\"bibr\" rid=\"b13\">13</xref>]</sup> to find a special initial condition such that the Gauss curvature flow converges to a solution of the $ L_p $ dual Minkowski problem.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224308","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}
{"title":"A monotonicity approach to Pogorelov's Hessian estimates for Monge- Ampère equation","authors":"Yu Yuan","doi":"10.3934/mine.2023037","DOIUrl":"https://doi.org/10.3934/mine.2023037","url":null,"abstract":"<abstract><p>We present an integral approach to Pogorelov's Hessian estimates for the Monge-Ampère equation, originally obtained via a pointwise argument.</p></abstract>","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224609","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}
Partial differential equations are a classical and very active field of research. One of its salient features is to break the rigid distinction between the evolution of the theory and the applications to real world phenomena, since the two are intimately intertwined in the harmonious development of such a fascinating and multifaceted topic of investigation.
{"title":"Partial differential equations from theory to applications: Dedicated to Alberto Farina, on the occasion of his 50th birthday","authors":"S. Dipierro, L. Lombardini","doi":"10.3934/mine.2023050","DOIUrl":"https://doi.org/10.3934/mine.2023050","url":null,"abstract":"Partial differential equations are a classical and very active field of research. One of its salient features is to break the rigid distinction between the evolution of the theory and the applications to real world phenomena, since the two are intimately intertwined in the harmonious development of such a fascinating and multifaceted topic of investigation.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224336","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}
P. Gironimo, S. Leonardi, F. Leonetti, Marta Macrì, Pier Vincenzo Petricca
We prove the existence of a solution to a quasilinear system of degenerate equations, when the datum is in a Marcinkiewicz space. The main assumption asks the off-diagonal coefficients to have support in the union of a geometric progression of squares.
{"title":"Existence of solutions to some quasilinear degenerate elliptic systems with right hand side in a Marcinkiewicz space","authors":"P. Gironimo, S. Leonardi, F. Leonetti, Marta Macrì, Pier Vincenzo Petricca","doi":"10.3934/mine.2023055","DOIUrl":"https://doi.org/10.3934/mine.2023055","url":null,"abstract":"We prove the existence of a solution to a quasilinear system of degenerate equations, when the datum is in a Marcinkiewicz space. The main assumption asks the off-diagonal coefficients to have support in the union of a geometric progression of squares.","PeriodicalId":54213,"journal":{"name":"Mathematics in Engineering","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70224420","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}
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