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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":"44 1","pages":""},"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":"26 1","pages":""},"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
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":"90 1","pages":""},"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":"42 1","pages":""},"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":"13 1","pages":""},"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":"440 1","pages":""},"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
Obstacle Problems with Double Boundary Condition for Least Gradient Functions in Metric Measure Spaces 公度量空间中最小梯度函数的双边界条件障碍问题
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-04-03 DOI: 10.1007/s11118-024-10135-7
Josh Kline
{"title":"Obstacle Problems with Double Boundary Condition for Least Gradient Functions in Metric Measure Spaces","authors":"Josh Kline","doi":"10.1007/s11118-024-10135-7","DOIUrl":"https://doi.org/10.1007/s11118-024-10135-7","url":null,"abstract":"<p>In the setting of a metric space equipped with a doubling measure supporting a (1, 1)-Poincaré inequality, we study the problem of minimizing the BV-energy in a bounded domain <span>(Omega )</span> of functions bounded between two obstacle functions inside <span>(Omega )</span>, and whose trace lies between two prescribed functions on the boundary. If the class of candidate functions is nonempty, we show that solutions exist for continuous obstacles and continuous boundary data when <span>(Omega )</span> is a uniform domain whose boundary is of positive mean curvature in the sense of Lahti, Malý, Shanmugalingam, and Speight (2019). While such solutions are not unique in general, we show the existence of unique minimal solutions. Since candidate functions need not agree outside of the domain, standard compactness arguments fail to provide existence of weak solutions as they are defined for the problem with single boundary condition. To overcome this, we introduce a class of <span>( varepsilon )</span>-<i>weak solutions</i> as an intermediate step. Our existence results generalize those of Ziemer and Zumbrun (1999), who studied this problem in the Euclidean setting with a single obstacle and single boundary condition.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"8 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140566749","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
Total Variation Error Bounds for the Approximation of the Invariant Distribution of Parabolic Semilinear SPDEs Using the Standard Euler Scheme 使用标准欧拉方案逼近抛物线半线性 SPDE 的不变分布的总变差误差边界
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-03-19 DOI: 10.1007/s11118-024-10132-w
Charles-Edouard Bréhier
{"title":"Total Variation Error Bounds for the Approximation of the Invariant Distribution of Parabolic Semilinear SPDEs Using the Standard Euler Scheme","authors":"Charles-Edouard Bréhier","doi":"10.1007/s11118-024-10132-w","DOIUrl":"https://doi.org/10.1007/s11118-024-10132-w","url":null,"abstract":"<p>We study the long time behavior of the standard linear implicit Euler scheme for the discretization of a class of erdogic parabolic semilinear SPDEs driven by additive space-time white noise. When the nonlinearity is a gradient, the invariant distribution is of Gibbs form, but it cannot be approximated in the total variation sense by the standard Euler scheme. We prove that the numerical scheme gives an approximation in the total variation sense of a modified Gibbs distribution, which is the invariant distribution of a modified SPDE. The modified distribution and the modified equation depend on the time-step size. This original result goes beyond existing results in the literature where the weak error estimates for the approximation of the invariant distribution do not imply convergence in total variation when the time-step size vanishes. The proof of the main result requires regularity properties of associated infinite dimensional Kolmogorov equations.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"148 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140169087","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
Exponential Contractivity and Propagation of Chaos for Langevin Dynamics of McKean-Vlasov Type with Lévy Noises 带列维噪声的麦金-弗拉索夫型兰万动力学的指数收缩性和混沌传播
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-03-15 DOI: 10.1007/s11118-024-10130-y
{"title":"Exponential Contractivity and Propagation of Chaos for Langevin Dynamics of McKean-Vlasov Type with Lévy Noises","authors":"","doi":"10.1007/s11118-024-10130-y","DOIUrl":"https://doi.org/10.1007/s11118-024-10130-y","url":null,"abstract":"<h3>Abstract</h3> <p>By the probabilistic coupling approach which combines a new refined basic coupling with the synchronous coupling for Lévy processes, we obtain explicit exponential contraction rates in terms of the standard <span> <span>(L^1)</span> </span>-Wasserstein distance for the following Langevin dynamic <span> <span>((X_t,Y_t)_{tge 0})</span> </span> of McKean-Vlasov type on <span> <span>(mathbb R^{2d})</span> </span>: <span> <span>$$begin{aligned} left{ begin{array}{l} dX_t=Y_t,dt, dY_t=left( b(X_t)+displaystyle int _{mathbb R^d}tilde{b}(X_t,z),mu ^X_t(dz)-{gamma }Y_tright) ,dt+dL_t,quad mu ^X_t=textrm{Law}(X_t), end{array} right. end{aligned}$$</span> </span>where <span> <span>({gamma }&gt;0)</span> </span>, <span> <span>(b:mathbb R^drightarrow mathbb R^d)</span> </span> and <span> <span>(tilde{b}:mathbb R^{2d}rightarrow mathbb R^d)</span> </span> are two globally Lipschitz continuous functions, and <span> <span>((L_t)_{tge 0})</span> </span> is an <span> <span>(mathbb R^d)</span> </span>-valued pure jump Lévy process. The proof is also based on a novel distance function, which is designed according to the distance of the marginals associated with the constructed coupling process. Furthermore, by applying the coupling technique above with some modifications, we also provide the propagation of chaos uniformly in time for the corresponding mean-field interacting particle systems with Lévy noises in the standard <span> <span>(L^1)</span> </span>-Wasserstein distance as well as with explicit bounds.</p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"9 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140155562","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 $$L_{p}-$$ Theory for Integro-Differential Operators with Spatially Dependent Coefficients 关于具有空间依赖系数的积分微分算子的 $$L_{p}-$$ 理论
IF 1.1 3区 数学
Potential Analysis Pub Date : 2024-03-14 DOI: 10.1007/s11118-024-10131-x
Sutawas Janreung, Tatpon Siripraparat, Chukiat Saksurakan
{"title":"On $$L_{p}-$$ Theory for Integro-Differential Operators with Spatially Dependent Coefficients","authors":"Sutawas Janreung, Tatpon Siripraparat, Chukiat Saksurakan","doi":"10.1007/s11118-024-10131-x","DOIUrl":"https://doi.org/10.1007/s11118-024-10131-x","url":null,"abstract":"<p>The parabolic integro-differential Cauchy problem with spatially dependent coefficients is considered in generalized Bessel potential spaces where smoothness is defined by Lévy measures with O-regularly varying profile. The coefficients are assumed to be bounded and Hölder continuous in the spatial variable. Our results can cover interesting classes of Lévy measures that go beyond those comparable to <span>(dy/left| yright| ^{d+alpha }.)</span></p>","PeriodicalId":49679,"journal":{"name":"Potential Analysis","volume":"320 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140155557","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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