Advances in Differential Equations最新文献_第7页

Non-divergence operators structured on homogeneous Hörmander vector fields: heat kernels and global Gaussian bounds 结构在齐次Hörmander向量场上的非发散算子:热核和全局高斯边界

IF 1.4 3区数学

Advances in Differential Equations Pub Date : 2020-11-18 DOI: 10.57262/ade026-1112-621

Stefano Biagi, M. Bramanti

{"title":"Non-divergence operators structured on homogeneous Hörmander vector fields: heat kernels and global Gaussian bounds","authors":"Stefano Biagi, M. Bramanti","doi":"10.57262/ade026-1112-621","DOIUrl":"https://doi.org/10.57262/ade026-1112-621","url":null,"abstract":"Let $X_{1},...,X_{m}$ be a family of real smooth vector fields defined in $mathbb{R}^{n}$, $1$-homogeneous with respect to a nonisotropic family of dilations and satisfying Hormander's rank condition at $0$ (and therefore at every point of $mathbb{R}^{n}$). The vector fields are not assumed to be translation invariant with respect to any Lie group structure. Let us consider the nonvariational evolution operator $$ mathcal{H}:=sum_{i,j=1}^{m}a_{i,j}(t,x)X_{i}X_{j}-partial_{t}% $$ where $(a_{i,j}(t,x))_{i,j=1}^{m}$ is a symmetric uniformly positive $mtimes m$ matrix and the entries $a_{ij}$ are bounded Holder continuous functions on $mathbb{R}^{1+n}$, with respect to the \"parabolic\" distance induced by the vector fields. We prove the existence of a global heat kernel $Gamma(cdot;s,y)in C_{X,mathrm{loc}}^{2,alpha}(mathbb{R}^{1+n}setminus{(s,y)})$ for $mathcal{H}$, such that $Gamma$ satisfies two-sided Gaussian bounds and $partial_{t}Gamma, X_{i}Gamma,X_{i}X_{j}Gamma$ satisfy upper Gaussian bounds on every strip $[0,T]timesmathbb{R}^n$. We also prove a scale-invariant parabolic Harnack inequality for $mathcal{H}$, and a standard Harnack inequality for the corresponding stationary operator $$ \u0000mathcal{L}:=sum_{i,j=1}^{m}a_{i,j}(x)X_{i}X_{j}. \u0000$$ \u0000with Holder continuos coefficients.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2020-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42547617","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: 2

On the planar Schrödinger-Poisson system with zero mass and critical exponential growth 具有零质量和临界指数增长的平面Schrödinger-Poisson系统

IF 1.4 3区数学

Advances in Differential Equations Pub Date : 2020-11-01 DOI: 10.57262/ade/1605150119

Sitong Chen, Xianhua Tang

引用次数: 4

Gradient estimates for elliptic oblique derivative problems via the maximum principle 基于极大值原理的椭圆斜导数问题的梯度估计

IF 1.4 3区数学

Advances in Differential Equations Pub Date : 2020-11-01 DOI: 10.57262/ade/1605150120

G. M. Lieberman

引用次数: 0

On the non-existence of compact surfaces of genus one with prescribed, almost constant mean curvature, close to the singular limit 关于一属紧曲面的不存在性，具有规定的，几乎常数的平均曲率，接近奇异极限

IF 1.4 3区数学

Advances in Differential Equations Pub Date : 2020-08-31 DOI: 10.57262/ade027-0304-193

P. Caldiroli, A. Iacopetti

引用次数: 1

Non-local tug-of-war with noise for the geometric 非本地拔河与噪音的几何

IF 1.4 3区数学

Advances in Differential Equations Pub Date : 2020-07-27 DOI: 10.57262/ade027-0102-31

M. Lewicka

{"title":"Non-local tug-of-war with noise for the geometric","authors":"M. Lewicka","doi":"10.57262/ade027-0102-31","DOIUrl":"https://doi.org/10.57262/ade027-0102-31","url":null,"abstract":"This paper concerns the fractional $p$-Laplace operator $Delta_p^s$ in non-divergence form, which has been introduced in [Bjorland, Caffarelli, Figalli (2012)]. For any $pin [2,infty)$ and $sin (frac{1}{2},1)$ we first define two families of non-local, non-linear averaging operators, parametrised by $epsilon$ and defined for all bounded, Borel functions $u:mathbb{R}^Nto mathbb{R}$. We prove that $Delta_p^s u(x)$ emerges as the $epsilon^{2s}$-order coefficient in the expansion of the deviation of each $epsilon$-average from the value $u(x)$, in the limit of the domain of averaging exhausting an appropriate cone in $mathbb{R}^N$ at the rate $epsilonto 0$. \u0000Second, we consider the $epsilon$-dynamic programming principles modeled on the first average, and show that their solutions converge uniformly as $epsilonto 0$, to viscosity solutions of the homogeneous non-local Dirichlet problem for $Delta_p^s$, when posed in a domain $mathcal{D}$ that satisfies the external cone condition and subject to bounded, uniformly continuous data on $mathbb{R}^Nsetminus mathcal{D}$. \u0000Finally, we interpret such $epsilon$-approximating solutions as values to the non-local Tug-of-War game with noise. In this game, players choose directions while the game position is updated randomly within the infinite cone that aligns with the specified direction, whose aperture angle depends on $p$ and $N$, and whose $epsilon$-tip has been removed.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2020-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44558414","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: 3

On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary 具有时型边界的全局双曲流形上friedrich系统的Cauchy问题

IF 1.4 3区数学

Advances in Differential Equations Pub Date : 2020-07-06 DOI: 10.57262/ade027-0708-497

N. Ginoux, S. Murro

引用次数: 14

On the well-posedness and decay characterization of solutions for incompressible electron inertial Hall-MHD equations 不可压缩电子惯性Hall-MHD方程解的适定性和衰减表征

IF 1.4 3区数学

Advances in Differential Equations Pub Date : 2020-07-01 DOI: 10.57262/ade/1594692076

Xiaopeng Zhao

引用次数: 4

A perturbation theorem for abstract linear non-autonomous systems with an application to a mixed hyperbolic problem 抽象线性非自治系统的扰动定理及其在混合双曲问题中的应用

IF 1.4 3区数学

Advances in Differential Equations Pub Date : 2020-07-01 DOI: 10.57262/ade/1594692075

D. Guidetti

引用次数: 0

Multi-bump solutions for fractional Schrödinger equation with electromagnetic fields and critical nonlinearity 具有电磁场和临界非线性的分数阶Schrödinger方程的多碰撞解

IF 1.4 3区数学

Advances in Differential Equations Pub Date : 2020-07-01 DOI: 10.57262/ade/1594692077

Sihua Liang, N. T. Chung, Binlin Zhang

引用次数: 3

On the stable self-similar waves for the Camassa-Holm and Degasperis-Procesi equations Camassa-Holm和Degasperis-Procesi方程的稳定自相似波

IF 1.4 3区数学

Advances in Differential Equations Pub Date : 2020-05-01 DOI: 10.57262/ade/1589594421

Liangchen Li, Hengyan Li, Weiping Yan

引用次数: 0