Pub Date : 2020-11-18DOI: 10.57262/ade026-1112-621
Stefano Biagi, M. Bramanti
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 $$ mathcal{L}:=sum_{i,j=1}^{m}a_{i,j}(x)X_{i}X_{j}. $$ with Holder continuos coefficients.
{"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":null,"pages":null},"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}
This paper is concerned with the following planar Schrodinger-Poisson system with zero mass begin{equation*} begin{cases} -Delta u+lambda phi u=f(x,u), ;; & xin {mathbb R}^{2}, Delta phi=2pi u^2, ;; & xin {mathbb R}^{2}, end{cases} end{equation*} where $lambda > 0$ and $fin mathcal{C}(mathbb R^2timesmathbb R, mathbb R)$ is of subcritical or critical exponential growth in the sense of Trudinger-Moser. By using some new analytical approaches, we prove that the above system has axially symmetric solutions under weak assumptions on $lambda$ and $f$. This seems the first result on the planar Schrodinger-Poisson system with zero mass.
{"title":"On the planar Schrödinger-Poisson system with zero mass and critical exponential growth","authors":"Sitong Chen, Xianhua Tang","doi":"10.57262/ade/1605150119","DOIUrl":"https://doi.org/10.57262/ade/1605150119","url":null,"abstract":"This paper is concerned with the following planar Schrodinger-Poisson system with zero mass begin{equation*} begin{cases} -Delta u+lambda phi u=f(x,u), ;; & xin {mathbb R}^{2}, Delta phi=2pi u^2, ;; & xin {mathbb R}^{2}, end{cases} end{equation*} where $lambda > 0$ and $fin mathcal{C}(mathbb R^2timesmathbb R, mathbb R)$ is of subcritical or critical exponential growth in the sense of Trudinger-Moser. By using some new analytical approaches, we prove that the above system has axially symmetric solutions under weak assumptions on $lambda$ and $f$. This seems the first result on the planar Schrodinger-Poisson system with zero mass.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49545953","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}
{"title":"Gradient estimates for elliptic oblique derivative problems via the maximum principle","authors":"G. M. Lieberman","doi":"10.57262/ade/1605150120","DOIUrl":"https://doi.org/10.57262/ade/1605150120","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45302666","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}
Pub Date : 2020-08-31DOI: 10.57262/ade027-0304-193
P. Caldiroli, A. Iacopetti
In Euclidean 3-space endowed with a Cartesian reference system we consider a class of surfaces, called Delaunay tori, constructed by bending segments of Delaunay cylinders with neck-size $a$ and $n$ lobes along circumferences centered at the origin. Such surfaces are complete and compact, have genus one and almost constant, say 1, mean curvature, when $n$ is large. Considering a class of mappings $Hcolonmathbb{R}^{3}tomathbb{R}$ such that $H(X)to 1$ as $|X|toinfty$ with some decay of inverse-power type, we show that for $n$ large and $|a|$ small, in a suitable neighborhood of any Delaunay torus with $n$ lobes and neck-size $a$ there is no parametric surface constructed as normal graph over the Delaunay torus and whose mean curvature equals $H$ at every point.
{"title":"On the non-existence of compact surfaces of genus one with prescribed, almost constant mean curvature, close to the singular limit","authors":"P. Caldiroli, A. Iacopetti","doi":"10.57262/ade027-0304-193","DOIUrl":"https://doi.org/10.57262/ade027-0304-193","url":null,"abstract":"In Euclidean 3-space endowed with a Cartesian reference system we consider a class of surfaces, called Delaunay tori, constructed by bending segments of Delaunay cylinders with neck-size $a$ and $n$ lobes along circumferences centered at the origin. Such surfaces are complete and compact, have genus one and almost constant, say 1, mean curvature, when $n$ is large. Considering a class of mappings $Hcolonmathbb{R}^{3}tomathbb{R}$ such that $H(X)to 1$ as $|X|toinfty$ with some decay of inverse-power type, we show that for $n$ large and $|a|$ small, in a suitable neighborhood of any Delaunay torus with $n$ lobes and neck-size $a$ there is no parametric surface constructed as normal graph over the Delaunay torus and whose mean curvature equals $H$ at every point.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2020-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47950878","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}
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 $epsilonto 0$. Second, 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 $mathbb{R}^Nsetminus mathcal{D}$. Finally, 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.
{"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":null,"pages":null},"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}
Pub Date : 2020-07-06DOI: 10.57262/ade027-0708-497
N. Ginoux, S. Murro
In this paper, the Cauchy problem for a Friedrichs system on a globally hyperbolic manifold with a timelike boundary is investigated. By imposing admissible boundary conditions, the existence and the uniqueness of strong solutions are shown. Furthermore, if the Friedrichs system is hyperbolic, the Cauchy problem is proved to be well-posed in the sense of Hadamard. Finally, examples of Friedrichs systems with admissible boundary conditions are provided. Keywords: symmetric hyperbolic systems, symmetric positive systems, admissible boundary conditions, Dirac operator, normally hyperbolic operator, Klein-Gordon operator, heat operator, reaction-diffusion operator, globally hyperbolic manifolds with timelike boundary.
{"title":"On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary","authors":"N. Ginoux, S. Murro","doi":"10.57262/ade027-0708-497","DOIUrl":"https://doi.org/10.57262/ade027-0708-497","url":null,"abstract":"In this paper, the Cauchy problem for a Friedrichs system on a globally hyperbolic manifold with a timelike boundary is investigated. By imposing admissible boundary conditions, the existence and the uniqueness of strong solutions are shown. Furthermore, if the Friedrichs system is hyperbolic, the Cauchy problem is proved to be well-posed in the sense of Hadamard. Finally, examples of Friedrichs systems with admissible boundary conditions are provided. Keywords: symmetric hyperbolic systems, symmetric positive systems, admissible boundary conditions, Dirac operator, normally hyperbolic operator, Klein-Gordon operator, heat operator, reaction-diffusion operator, globally hyperbolic manifolds with timelike boundary.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2020-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44931333","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}
{"title":"On the well-posedness and decay characterization of solutions for incompressible electron inertial Hall-MHD equations","authors":"Xiaopeng Zhao","doi":"10.57262/ade/1594692076","DOIUrl":"https://doi.org/10.57262/ade/1594692076","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43441394","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}
{"title":"A perturbation theorem for abstract linear non-autonomous systems with an application to a mixed hyperbolic problem","authors":"D. Guidetti","doi":"10.57262/ade/1594692075","DOIUrl":"https://doi.org/10.57262/ade/1594692075","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45780238","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}
{"title":"Multi-bump solutions for fractional Schrödinger equation with electromagnetic fields and critical nonlinearity","authors":"Sihua Liang, N. T. Chung, Binlin Zhang","doi":"10.57262/ade/1594692077","DOIUrl":"https://doi.org/10.57262/ade/1594692077","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48384335","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}
{"title":"On the stable self-similar waves for the Camassa-Holm and Degasperis-Procesi equations","authors":"Liangchen Li, Hengyan Li, Weiping Yan","doi":"10.57262/ade/1589594421","DOIUrl":"https://doi.org/10.57262/ade/1589594421","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48813860","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}
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