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Nonlinear Dirichlet Forms Associated with Quasiregular Mappings 与准线性映射相关的非线性 Dirichlet 形式
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-05-22 DOI: 10.1007/s11118-024-10145-5
Camelia Beznea, Lucian Beznea, Michael Röckner
{"title":"Nonlinear Dirichlet Forms Associated with Quasiregular Mappings","authors":"Camelia Beznea, Lucian Beznea, Michael Röckner","doi":"10.1007/s11118-024-10145-5","DOIUrl":"https://doi.org/10.1007/s11118-024-10145-5","url":null,"abstract":"<p>If <span>((mathcal{E}, mathcal{D}))</span> is a symmetric, regular, strongly local Dirichlet form on <span>(L^2 (X,m))</span>, admitting a carré du champ operator <span>(Gamma )</span>, and <span>(p&gt;1)</span> is a real number, then one can define a nonlinear form <span>(mathcal{E}^p)</span> by the formula </p><span>$$ mathcal{E}^p(u,v) = int _{X} Gamma (u)^frac{p-2}{2} Gamma (u,v)dm , $$</span><p>where <i>u</i>, <i>v</i> belong to an appropriate subspace of the domain <span>(mathcal{D})</span>. We show that <span>(mathcal{E}^p)</span> is a nonlinear Dirichlet form in the sense introduced by P. van Beusekom. We then construct the associated Choquet capacity. As a particular case we obtain the nonlinear form associated with the <i>p</i>-Laplace operator on <span>(W_0^{1,p})</span>. Using the above procedure, for each <i>n</i>-dimensional quasiregular mapping <i>f</i> we construct a nonlinear Dirichlet form <span>(mathcal{E}^n)</span> (<span>(p=n)</span>) such that the components of <i>f</i> become harmonic functions with respect to <span>(mathcal{E}^n)</span>. Finally, we obtain Caccioppoli type inequalities in the intrinsic metric induced by <span>(mathcal{E})</span>, for harmonic functions with respect to the form <span>(mathcal{E}^p)</span>.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141145763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Stochastic Partial Differential Equations and Invariant Manifolds in Embedded Hilbert Spaces 嵌入希尔伯特空间中的随机偏微分方程和不变曲率
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-05-11 DOI: 10.1007/s11118-024-10134-8
Rajeev Bhaskaran, Stefan Tappe
{"title":"Stochastic Partial Differential Equations and Invariant Manifolds in Embedded Hilbert Spaces","authors":"Rajeev Bhaskaran, Stefan Tappe","doi":"10.1007/s11118-024-10134-8","DOIUrl":"https://doi.org/10.1007/s11118-024-10134-8","url":null,"abstract":"<p>We provide necessary and sufficient conditions for stochastic invariance of finite dimensional submanifolds for solutions of stochastic partial differential equations (SPDEs) in continuously embedded Hilbert spaces with non-smooth coefficients. Furthermore, we establish a link between invariance of submanifolds for such SPDEs in Hermite Sobolev spaces and invariance of submanifolds for finite dimensional SDEs. This provides a new method for analyzing stochastic invariance of submanifolds for finite dimensional Itô diffusions, which we will use in order to derive new invariance results for finite dimensional SDEs.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140927004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Fundamental Solutions and Gaussian Bounds for Degenerate Parabolic Equations with Time-dependent Coefficients 论具有时变系数的畸变抛物方程的基本解和高斯边界
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-05-06 DOI: 10.1007/s11118-024-10143-7
Alireza Ataei, Kaj Nyström
{"title":"On Fundamental Solutions and Gaussian Bounds for Degenerate Parabolic Equations with Time-dependent Coefficients","authors":"Alireza Ataei, Kaj Nyström","doi":"10.1007/s11118-024-10143-7","DOIUrl":"https://doi.org/10.1007/s11118-024-10143-7","url":null,"abstract":"<p>We consider second order degenerate parabolic equations with real, measurable, and time-dependent coefficients. We allow for degenerate ellipticity dictated by a spatial <span>(A_2)</span>-weight. We prove the existence of a fundamental solution and derive Gaussian bounds. Our construction is based on the original work of Kato (Nagoya Math. J. <b>19</b>, 93–125 1961).</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140886659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Marcinkiewicz Estimates for Solutions of Some Elliptic Problems with Singular Data 具有奇异数据的某些椭圆问题解的 Marcinkiewicz 估计数
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-05-03 DOI: 10.1007/s11118-024-10140-w
Lucio Boccardo, Luigi Orsina
{"title":"Marcinkiewicz Estimates for Solutions of Some Elliptic Problems with Singular Data","authors":"Lucio Boccardo, Luigi Orsina","doi":"10.1007/s11118-024-10140-w","DOIUrl":"https://doi.org/10.1007/s11118-024-10140-w","url":null,"abstract":"<p>In this paper we prove regularity result for solutions of the boundary value problem </p><span>$$ left{ begin{array}{cl} -{{,textrm{div},}}(M(x),nabla u) + u = -{{,textrm{div},}}(u,E(x)) + f(x),, &amp;{} text{ in },, Omega , u = 0,, &amp;{} text{ on },,partial Omega , end{array} right. $$</span><p>with the vector field <i>E</i>(<i>x</i>) and the function <i>f</i>(<i>x</i>) belonging to some Marcinkiewicz spaces.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140886572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Capacities and Density Conditions in Metric Spaces 公制空间中的容量和密度条件
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-04-30 DOI: 10.1007/s11118-024-10137-5
Javier Canto, Lizaveta Ihnatsyeva, Juha Lehrbäck, Antti V. Vähäkangas
{"title":"Capacities and Density Conditions in Metric Spaces","authors":"Javier Canto, Lizaveta Ihnatsyeva, Juha Lehrbäck, Antti V. Vähäkangas","doi":"10.1007/s11118-024-10137-5","DOIUrl":"https://doi.org/10.1007/s11118-024-10137-5","url":null,"abstract":"<p>We examine the relations between different capacities in the setting of a metric measure space. First, we prove a comparability result for the Riesz <span>((beta ,p))</span>-capacity and the relative Hajłasz <span>((beta ,p))</span>-capacity, for <span>(1&lt;p&lt;infty )</span> and <span>(0&lt;beta le 1)</span>, under a suitable kernel estimate related to the Riesz potential. Then we show that in geodesic spaces the corresponding capacity density conditions are equivalent even without assuming the kernel estimate. In the last part of the paper, we compare the relative Hajłasz (1, <i>p</i>)-capacity to the relative variational <i>p</i>-capacity.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140835320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Intrinsic Harnack’s Inequality for a General Nonlinear Parabolic Equation in Non-divergence Form 非发散形式一般非线性抛物方程的本征哈纳克不等式
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-04-25 DOI: 10.1007/s11118-024-10141-9
Tapio Kurkinen, Jarkko Siltakoski
{"title":"Intrinsic Harnack’s Inequality for a General Nonlinear Parabolic Equation in Non-divergence Form","authors":"Tapio Kurkinen, Jarkko Siltakoski","doi":"10.1007/s11118-024-10141-9","DOIUrl":"https://doi.org/10.1007/s11118-024-10141-9","url":null,"abstract":"<p>We prove the intrinsic Harnack’s inequality for a general form of a parabolic equation that generalizes both the standard parabolic <i>p</i>-Laplace equation and the normalized version arising from stochastic game theory. We prove each result for the optimal range of exponents and ensure that we get stable constants.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140800515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Spectral Asymptotics of the Cauchy Operator and its Product with Bergman’s Projection on a Doubly Connected Domain 双连域上考奇算子及其与伯格曼投影的乘积的谱渐近性
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-04-19 DOI: 10.1007/s11118-024-10139-3
Djordjije Vujadinović
{"title":"Spectral Asymptotics of the Cauchy Operator and its Product with Bergman’s Projection on a Doubly Connected Domain","authors":"Djordjije Vujadinović","doi":"10.1007/s11118-024-10139-3","DOIUrl":"https://doi.org/10.1007/s11118-024-10139-3","url":null,"abstract":"<p>We found the exact asymptotics of the singular numbers for the Cauchy transform and its product with Bergman’s projection over the space <span>(L^{2}(Omega ),)</span> where <span>(Omega )</span> is a doubly-connected domain in the complex plane.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140628580","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Inclusion Relations Among Fractional Orlicz-Sobolev Spaces and a Littlewood-Paley Characterization 分数奥利兹-索博廖夫空间之间的包含关系和 Littlewood-Paley 特征
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-04-16 DOI: 10.1007/s11118-024-10136-6
Dominic Breit, Andrea Cianchi
{"title":"Inclusion Relations Among Fractional Orlicz-Sobolev Spaces and a Littlewood-Paley Characterization","authors":"Dominic Breit, Andrea Cianchi","doi":"10.1007/s11118-024-10136-6","DOIUrl":"https://doi.org/10.1007/s11118-024-10136-6","url":null,"abstract":"<p>Embeddings among fractional Orlicz-Sobolev spaces with different smoothness are characterized. In particular, besides recovering standard embeddings for classical fractional Sobolev spaces, novel results are derived in borderline situations where the latter fail. For instance, limiting embeddings of Pohozhaev-Trudinger-Yudovich type into exponential spaces are offered. The equivalence of Gagliardo-Slobodeckij norms in fractional Orlicz-Sobolev spaces to norms defined via Littlewood-Paley decompositions, oscillations, or Besov type difference quotients is established as well. This equivalence, of independent interest, is a key tool in the proof of the relevant embeddings. They also rest upon a new optimal inequality for convolutions in Orlicz spaces.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140566728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Non-uniqueness in law of three-dimensional magnetohydrodynamics system forced by random noise 受随机噪声强迫的三维磁流体力学系统定律中的非唯一性
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-04-13 DOI: 10.1007/s11118-024-10128-6
Kazuo Yamazaki
{"title":"Non-uniqueness in law of three-dimensional magnetohydrodynamics system forced by random noise","authors":"Kazuo Yamazaki","doi":"10.1007/s11118-024-10128-6","DOIUrl":"https://doi.org/10.1007/s11118-024-10128-6","url":null,"abstract":"<p>We prove non-uniqueness in law of the three-dimensional magnetohydrodynamics system that is forced by random noise of an additive and a linear multiplicative type and has viscous and magnetic diffusion, both of which are weaker than a full Laplacian. We apply convex integration to both equations of velocity and magnetic fields in order to obtain the non-uniqueness in law in the class of probabilistically strong solutions.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140566951","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Bakry-Émery Approach to Lipschitz Transportation on Manifolds 积分榜上的 Lipschitz Transportation 的 Bakry-Émery 方法
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-04-11 DOI: 10.1007/s11118-024-10138-4
Pablo López-Rivera
{"title":"A Bakry-Émery Approach to Lipschitz Transportation on Manifolds","authors":"Pablo López-Rivera","doi":"10.1007/s11118-024-10138-4","DOIUrl":"https://doi.org/10.1007/s11118-024-10138-4","url":null,"abstract":"<p>On weighted Riemannian manifolds we prove the existence of globally Lipschitz transport maps between the weight (probability) measure and log-Lipschitz perturbations of it, via Kim and Milman’s diffusion transport map, assuming that the curvature-dimension condition <span>(varvec{textrm{CD}(rho _{1}, infty )})</span> holds, as well as a second order version of it, namely <span>(varvec{Gamma _{3} ge rho _{2} Gamma _{2}})</span>. We get new results as corollaries to this result, as the preservation of Poincaré’s inequality for the exponential measure on <span>(varvec{(0,+infty )})</span> when perturbed by a log-Lipschitz potential and a new growth estimate for the Monge map pushing forward the gamma distribution on <span>(varvec{(0,+infty )})</span> (then getting as a particular case the exponential one), via Laguerre’s generator.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140603154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","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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