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On the critical points of solutions of Robin boundary problems 论罗宾边界问题解的临界点
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-10 DOI: arxiv-2409.06576
Fabio De Regibus, Massimo Grossi
{"title":"On the critical points of solutions of Robin boundary problems","authors":"Fabio De Regibus, Massimo Grossi","doi":"arxiv-2409.06576","DOIUrl":"https://doi.org/arxiv-2409.06576","url":null,"abstract":"In this paper we prove the uniqueness of the critical point for stable\u0000solutions of the Robin problem [ begin{cases} -Delta u=f(u)&text{in\u0000}Omega u>0&text{in }Omega partial_nu u+beta u=0&text{on\u0000}partialOmega, end{cases} ] where $Omegasubseteqmathbb{R}^2$ is a smooth\u0000and bounded domain with strictly positive curvature of the boundary, $fge0$ is\u0000a smooth function and $beta>0$. Moreover, for $beta$ large the result fails\u0000as soon as the domain is no more convex, even if it is very close to be:\u0000indeed, in this case it is possible to find solutions with an arbitrary large\u0000number of critical points.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Stability of the 3-dimensional catenoid for the hyperbolic vanishing mean curvature equation 双曲平均曲率消失方程的三维 catenoid 的稳定性
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-09 DOI: arxiv-2409.05968
Sung-Jin Oh, Sohrab Shahshahani
{"title":"Stability of the 3-dimensional catenoid for the hyperbolic vanishing mean curvature equation","authors":"Sung-Jin Oh, Sohrab Shahshahani","doi":"arxiv-2409.05968","DOIUrl":"https://doi.org/arxiv-2409.05968","url":null,"abstract":"We prove that the $3$-dimensional catenoid is asymptotically stable as a\u0000solution to the hyperbolic vanishing mean curvature equation in Minkowski\u0000space, modulo suitable translation and boost (i.e., modulation) and with\u0000respect to a codimension one set of initial data perturbations. The modulation\u0000and the codimension one restriction on the initial data are necessary (and\u0000optimal) in view of the kernel and the unique simple eigenvalue, respectively,\u0000of the stability operator of the catenoid. The $3$-dimensional problem is more challenging than the higher\u0000(specifically, $5$ and higher) dimensional case addressed in the previous work\u0000of the authors with J.~L\"uhrmann, due to slower temporal decay of waves and\u0000slower spatial decay of the catenoid. To overcome these issues, we introduce\u0000several innovations, such as a proof of Morawetz- (or local-energy-decay-)\u0000estimates for the linearized operator with slowly decaying kernel elements\u0000based on the Darboux transform, a new method to obtain Price's-law-type bounds\u0000for waves on a moving catenoid, as well as a refined profile construction\u0000designed to capture a crucial cancellation in the wave-catenoid interaction. In\u0000conjunction with our previous work on the higher dimensional case, this paper\u0000outlines a systematic approach for studying other soliton stability problems\u0000for $(3+1)$-dimensional quasilinear wave equations.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Dispersive decay for the mass-critical generalized Korteweg-de Vries equation and generalized Zakharov-Kuznetsov equations 质量临界广义科特韦格-德-弗里斯方程和广义扎哈罗夫-库兹涅佐夫方程的分散衰变
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-09 DOI: arxiv-2409.05550
Minjie Shan
{"title":"Dispersive decay for the mass-critical generalized Korteweg-de Vries equation and generalized Zakharov-Kuznetsov equations","authors":"Minjie Shan","doi":"arxiv-2409.05550","DOIUrl":"https://doi.org/arxiv-2409.05550","url":null,"abstract":"In this paper, we discuss pointwise decay estimate for the solution to the\u0000mass-critical generalized Korteweg-de Vries (gKdV) equation with initial data\u0000$u_0in H^{1/2}(mathbb{R})$. It is showed that nonlinear solution enjoys the\u0000same decay rate as linear one. Moreover, we also quantify the decay for\u0000solutions to the generalized Zakharov-Kuznetsov equation which is a natural\u0000multi-dimensional extension of the gKdV equation. We obtain some decay\u0000estimates for nonlinear solutions to generalized Zakharov-Kuznetsov equations\u0000with small initial data in $H^2(mathbb{R}^d)$.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"109 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Asymptotic symmetry of solutions for reaction-diffusion equations via elliptic geometry 通过椭圆几何实现反应扩散方程解的渐近对称性
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-09 DOI: arxiv-2409.05366
Baiyu Liu, Wenlong Yang
{"title":"Asymptotic symmetry of solutions for reaction-diffusion equations via elliptic geometry","authors":"Baiyu Liu, Wenlong Yang","doi":"arxiv-2409.05366","DOIUrl":"https://doi.org/arxiv-2409.05366","url":null,"abstract":"In this paper, we investigate the asymptotic symmetry and monotonicity of\u0000positive solutions to a reaction-diffusion equation in the unit ball, utilizing\u0000techniques from elliptic geometry. Firstly, we discuss the properties of\u0000solutions in the elliptic space. Then, we establish crucial principles,\u0000including the asymptotic narrow region principle.Finally, we employ the method\u0000of moving planes to demonstrate the asymptotic symmetry of the solutions.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
SBV regularity of Entropy Solutions for Hyperbolic Systems of Balance Laws with General Flux function 具有一般通量函数的双曲平衡律系统熵解的 SBV 正则性
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-09 DOI: arxiv-2409.06087
Fabio Ancona, Laura Caravenna, Andrea Marson
{"title":"SBV regularity of Entropy Solutions for Hyperbolic Systems of Balance Laws with General Flux function","authors":"Fabio Ancona, Laura Caravenna, Andrea Marson","doi":"arxiv-2409.06087","DOIUrl":"https://doi.org/arxiv-2409.06087","url":null,"abstract":"We prove that vanishing viscosity solutions to smooth non-degenerate systems\u0000of balance laws having small bounded variation, in one space dimension, must be\u0000functions of special bounded variation. For more than one equation, this is new\u0000also in the case of systems of conservation laws out of the context of genuine\u0000nonlinearity. For general smooth strictly hyperbolic systems of balance laws,\u0000this regularity fails, as known for systems of balance laws: we generalize the\u0000SBV-like regularity of the eigenvalue functions of the Jacobian matrix of flux\u0000from conservation to balance laws. Proofs are based on extending Oleinink-type\u0000balance estimates, with the introduction of new source measures, localization\u0000arguments, and observations in real analysis. Preliminary version.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Well-posedness of the time-periodic Jordan-Moore-Gibson-Thompson equation 时间周期性约旦-摩尔-吉布森-汤普森方程的好拟性
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-09 DOI: arxiv-2409.05355
Barbara Kaltenbacher
{"title":"Well-posedness of the time-periodic Jordan-Moore-Gibson-Thompson equation","authors":"Barbara Kaltenbacher","doi":"arxiv-2409.05355","DOIUrl":"https://doi.org/arxiv-2409.05355","url":null,"abstract":"Motivated by applications of nonlinear ultrasonics under continuous wave\u0000excitation, we study the Jordan-Moore-Gibson-Thompson (JMGT) equation -- a\u0000third order in time quasilinear PDE -- under time periodicity conditions. Here\u0000the coefficient of the third order time derivative is the so-called relaxation\u0000time and a thorough understanding of the limiting behaviour for vanishing\u0000relaxation time is essential to link these JMGT equations to classical second\u0000order models in nonlinear acoustics, As compared to the meanwhile well understood initial value problem for JMGT,\u0000the periodic setting poses substantial challenges due to a loss of temporal\u0000regularity, while the analysis still requires an $L^infty$ control of\u0000solutions in space and time in order to maintain stability or equivalently, to\u0000avoid degeneracy of the second time derivative coefficient. We provide a full well-posedness analysis with and without gradient\u0000nonlinearity, as relevant for modelling non-cumulative nonlinear effects, under\u0000practically relevant mixed boundary conditions. The source-to-state map is thus\u0000well-defined and we additionally show it to be Lipschitz continuously\u0000differentiable, a result that is useful for inverse problems applications such\u0000as acoustic nonlinearity tomography. The energy bounds derived for the\u0000well-posedness analysis of periodic JMGT equations also allow to fully justify\u0000the singular limit for vanishing relaxation time.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"95 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179060","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
SBV-like regularity of Entropy Solutions for a Scalar Balance Law 标量平衡定律熵解的 SBV 类正则性
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-09 DOI: arxiv-2409.06095
Fabio Ancona, Laura Caravenna, Andrea Marson
{"title":"SBV-like regularity of Entropy Solutions for a Scalar Balance Law","authors":"Fabio Ancona, Laura Caravenna, Andrea Marson","doi":"arxiv-2409.06095","DOIUrl":"https://doi.org/arxiv-2409.06095","url":null,"abstract":"In this note we discuss the SBV-regularity for a scalar balance law in one\u0000space dimension as a case study in order to explain the strategy that we apply\u0000in a separate paper to general hyperbolic systems of balance laws in one space\u0000dimension. While for a single balance law the more general work by Robyr is\u0000already available, the case of 1d-systems presents new behaviors that require a\u0000different strategy. This is why in this note we make the effort to introduce\u0000the notation and tools that are required for the case of more equations. When\u0000the flux presents linear degeneracies, it is know that entropy solutions can\u0000present nasty fractal Cantor-like behaviors, although f'(u) is still SBV: we\u0000thus discuss SBV-like regularity generalizing the work by Bianchini-Yu as\u0000SBV-regularity fails.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179053","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Construction of multi-soliton solutions for the energy critical wave equation in dimension 3 构建三维能量临界波方程的多孑L解
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-09 DOI: arxiv-2409.05267
Istvan Kadar
{"title":"Construction of multi-soliton solutions for the energy critical wave equation in dimension 3","authors":"Istvan Kadar","doi":"arxiv-2409.05267","DOIUrl":"https://doi.org/arxiv-2409.05267","url":null,"abstract":"We study the energy-critical wave equation in three dimensions, focusing on\u0000its ground state soliton, denoted by $W$. Using the Poincar'e symmetry\u0000inherent in the equation, boosting $W$ along any timelike geodesic yields\u0000another solution. The slow decay behavior of $W$, $Wsim r^{-1}$, indicates a\u0000strong interaction among potential multi-soliton solutions. In this paper, for arbitrary $Ngeq0$, we provide an algorithmic procedure to\u0000construct approximate solutions to the energy critical wave equation that: (1)\u0000converge to a superposition of solitons, (2) have no outgoing radiation, (3)\u0000their error to solve the equation decays like $(t-r)^{-N}$. Then, we show that\u0000this approximate solution can be corrected to a real solution.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"137 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Remarks on comparison principles for p-Laplacian with extension to (p,q)-Laplacian 关于扩展到 (p,q) 拉普拉斯的 p 拉普拉斯比较原则的评论
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-09 DOI: arxiv-2409.05999
A. Mohammed, A. Vitolo
{"title":"Remarks on comparison principles for p-Laplacian with extension to (p,q)-Laplacian","authors":"A. Mohammed, A. Vitolo","doi":"arxiv-2409.05999","DOIUrl":"https://doi.org/arxiv-2409.05999","url":null,"abstract":"Our purpose is to generalize some recent comparison principles for operators\u0000driven by p-Laplacian to a wide class of quasilinear equations including (p,\u0000q)-Laplacian. It turns out, in particular, that adding a q-Laplacian to\u0000p-Laplacian allows to weaken the assumptions needed on the Hamiltonian of lower\u0000order terms. The results are specialized in the case that the Hamiltonian has\u0000at most polynomial growth in the gradient with coefficients depending on x and\u0000u.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Gradient estimates for the conductivity problem with imperfect bonding interfaces 具有不完美结合界面的传导性问题的梯度估计
arXiv - MATH - Analysis of PDEs Pub Date : 2024-09-09 DOI: arxiv-2409.05652
Hongjie Dong, Zhuolun Yang, Hanye Zhu
{"title":"Gradient estimates for the conductivity problem with imperfect bonding interfaces","authors":"Hongjie Dong, Zhuolun Yang, Hanye Zhu","doi":"arxiv-2409.05652","DOIUrl":"https://doi.org/arxiv-2409.05652","url":null,"abstract":"We study the field concentration phenomenon between two closely spaced\u0000perfect conductors with imperfect bonding interfaces of low conductivity type.\u0000The boundary condition on these interfaces is given by a Robin-type boundary\u0000condition. A previous conjecture suggested that the gradient of solutions\u0000remains bounded regardless of $varepsilon$, the distance between two\u0000inclusions. In this article, we establish gradient estimates, indicating that\u0000the conjecture is true only when the bonding parameter $gamma$ is sufficiently\u0000small, and the gradient could blow up when $gamma$ is large and the boundary\u0000data is not aligned with shortest line connecting the two inclusions. Moreover,\u0000we derive the optimal blow-up rates under certain symmetry assumptions.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142179058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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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