For singular even order symmetric differential operators we find the matrices which determine all symmetric extensions of the minimal operator. And for each of these symmetric operators which is bounded below we find the boundary condition of its Friedrichs extension. The operators of regular problems are bounded below and thus each one has a symmetric extension and thus its symmetric extension has a Friedrichs extension. See also https://ejde.math.txstate.edu/special/02/b1/abstr.html
{"title":"Friedrichs extension of singular symmetric differential operators","authors":"Qinglan Bao, Guangsheng Wei, A. Zettl","doi":"10.58997/ejde.sp.02.b1","DOIUrl":"https://doi.org/10.58997/ejde.sp.02.b1","url":null,"abstract":"For singular even order symmetric differential operators we find the matrices which determine all symmetric extensions of the minimal operator. And for each of these symmetric operators which is bounded below we find the boundary condition of its Friedrichs extension. The operators of regular problems are bounded below and thus each one has a symmetric extension and thus its symmetric extension has a Friedrichs extension.\u0000See also https://ejde.math.txstate.edu/special/02/b1/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44382564","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A review of results and techniques on the existence of regular radial solutions to second order elliptic boundary value problems driven by linear and quasilinear operators is presented. Of particular interest are results where the solvability of a given elliptic problem can be analyzed by the relationship between the spectrum of the operator and the behavior of the nonlinearity near infinity and at zero. Energy arguments and Pohozaev type identities are used extensively in that analysis. An appendix with a proof of the contraction mapping principle best suited for using continuous dependence to ordinary differential equations on initial conditions is presented. Another appendix on the phase plane analysis as needed to take advantage of initial conditions is also included. For studies on singular solutions the reader is referred to Ardila et al., Milan J. Math (2014) and references therein. See also https://ejde.math.txstate.edu/special/02/c2/abstr.html
摘要综述了二阶椭圆型边值问题在线性算子和拟线性算子驱动下正则径向解的存在性的研究结果和技术。特别令人感兴趣的结果是，给定椭圆问题的可解性可以通过算子的谱与非线性在无穷近处和零处的行为之间的关系来分析。能量论证和波霍扎耶夫类型同一性在该分析中被广泛使用。本文给出了最适合于在初始条件下使用常微分方程连续相关的收缩映射原理的证明。另一个附录关于相平面分析，需要利用初始条件也包括在内。对于奇异解的研究，读者可参考Ardila et al.， Milan J. Math(2014)及其参考文献。参见https://ejde.math.txstate.edu/special/02/c2/abstr.html
{"title":"Regular solutions to elliptic equations","authors":"A. Castro, Jon Jacobsen","doi":"10.58997/ejde.sp.02.c2","DOIUrl":"https://doi.org/10.58997/ejde.sp.02.c2","url":null,"abstract":"A review of results and techniques on the existence of regular radial solutions to second order elliptic boundary value problems driven by linear and quasilinear operators is presented. Of particular interest are results where the solvability of a given elliptic problem can be analyzed by the relationship between the spectrum of the operator and the behavior of the nonlinearity near infinity and at zero. Energy arguments and Pohozaev type identities are used extensively in that analysis. An appendix with a proof of the contraction mapping principle best suited for using continuous dependence to ordinary differential equations on initial conditions is presented. Another appendix on the phase plane analysis as needed to take advantage of initial conditions is also included. For studies on singular solutions the reader is referred to Ardila et al., Milan J. Math (2014) and references therein.\u0000See also https://ejde.math.txstate.edu/special/02/c2/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41937642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a sublinear perturbation of an elliptic eigenvalue system with homogeneous Neumann boundary conditions. For oscillatory nonlinearities and using bifurcation from infinity, we prove the existence of an unbounded sequence of turning points and an unbounded sequence of resonant solutions. See also https://ejde.math.txstate.edu/special/02/d1/abstr.html
{"title":"Resonant solutions for elliptic systems with Neumann boundary conditions","authors":"B. B. Delgado, R. Pardo","doi":"10.58997/ejde.sp.02.d1","DOIUrl":"https://doi.org/10.58997/ejde.sp.02.d1","url":null,"abstract":"We consider a sublinear perturbation of an elliptic eigenvalue system with homogeneous Neumann boundary conditions. For oscillatory nonlinearities and using bifurcation from infinity, we prove the existence of an unbounded sequence of turning points and an unbounded sequence of resonant solutions. \u0000See also https://ejde.math.txstate.edu/special/02/d1/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43674256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a class of nonlinear fractional Laplacian problems satisfying the homogeneous Dirichlet condition on the exterior of a bounded domain. We prove the existence of a positive weak solution for classes of nonlinearities which are either sublinear or asymptotically linear at infinity. We use the method of sub-and-supersolutions to establish the results. We also provide numerical bifurcation diagrams, corresponding to the theoretical results, using the finite element method in one dimension. See also https://ejde.math.txstate.edu/special/02/h1/abstr.html
{"title":"Positive solutions for nonlinear fractional Laplacian problems","authors":"Elliott Hollifield","doi":"10.58997/ejde.sp.02.h1","DOIUrl":"https://doi.org/10.58997/ejde.sp.02.h1","url":null,"abstract":"We consider a class of nonlinear fractional Laplacian problems satisfying the homogeneous Dirichlet condition on the exterior of a bounded domain. We prove the existence of a positive weak solution for classes of nonlinearities which are either sublinear or asymptotically linear at infinity. We use the method of sub-and-supersolutions to establish the results. We also provide numerical bifurcation diagrams, corresponding to the theoretical results, using the finite element method in one dimension.\u0000See also https://ejde.math.txstate.edu/special/02/h1/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43652336","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Mariani, Peter K. Asante, William Kubin, Osei K. Tweneboah, Maria P. Beccar-Varela
In this work, we determine appropriate background driving processes for the 3-component superposed Ornstein-Uhlenbeck model by analyzing the fractal characteristics of the data sets using the rescaled range analysis (R/S), the detrended fluctuation analysis (DFA), and the diffusion entropy analysis (DEA). See also https://ejde.math.txstate.edu/special/02/m1/abstr.html
{"title":"Determining the background driving process of the Ornstein-Uhlenbeck model","authors":"M. Mariani, Peter K. Asante, William Kubin, Osei K. Tweneboah, Maria P. Beccar-Varela","doi":"10.58997/ejde.sp.02.m1","DOIUrl":"https://doi.org/10.58997/ejde.sp.02.m1","url":null,"abstract":"In this work, we determine appropriate background driving processes for the 3-component superposed Ornstein-Uhlenbeck model by analyzing the fractal characteristics of the data sets using the rescaled range analysis (R/S), the detrended fluctuation analysis (DFA), and the diffusion entropy analysis (DEA). \u0000See also https://ejde.math.txstate.edu/special/02/m1/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44676998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article we establish a double-orthogonal principle, and a local min-orthogonal method with its step size rule, and its convergence under assumptions more general than those in its previous versions. With such a general framework, we justify mathematically the two new algorithms proposed for solving W-type problems. Numerical examples for finding multiple solutions to W-type and to mixed M-W-type problems illustrate the flexibility of this method. See also
{"title":"Local min-orthogonal principle and its applications for solving multiple solution problems","authors":"Meiqin Li, Jianxin Zhou","doi":"10.58997/ejde.sp.02.l1","DOIUrl":"https://doi.org/10.58997/ejde.sp.02.l1","url":null,"abstract":"In this article we establish a double-orthogonal principle, and a local min-orthogonal method with its step size rule, and its convergence under assumptions more general than those in its previous versions. With such a general framework, we justify mathematically the two new algorithms proposed for solving W-type problems. Numerical examples for finding multiple solutions to W-type and to mixed M-W-type problems illustrate the flexibility of this method. \u0000See also","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46132387","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the boundary value problem $$displaylines{ - Delta u + c(x) u = alpha m(x) u^+ - beta m(x) u^- +f(x,u), quad x in Omega, cr frac{partial u}{partial eta} + sigma (x) u =alpha rho (x) u^+- beta rho (x) u^- +g(x,u), quad x in partial Omega, }$$ where ((alpha, beta) in mathbb{R}^2), (c, m in L^infty (Omega)), (sigma, rho in L^infty (partialOmega)), and the nonlinearities f and g are bounded continuous functions. We study the asymmetric (Fucik) spectrum with weights, and prove existence theorems for nonlinear perturbations of this spectrum for both the resonance and non-resonance cases. For the resonance case, we provide a sufficient condition, the so-called generalized Landesman-Lazer condition, for the solvability. The proofs are based on variational methods and rely strongly on the variational characterization of the spectrum. See also https://ejde.math.txstate.edu/special/02/m2/abstr.html
我们考虑边值问题$$displaylines｛-Delta u+c（x）u＝alpha m（x）u^+-beta m（x （c，m in L^infty（Omega）），（sigma，rho in L^ infty，并且非线性f和g是有界连续函数。我们研究了具有权的非对称（Fucik）谱，并证明了该谱在共振和非共振情况下非线性扰动的存在性定理。对于共振情况，我们提供了可解性的一个充分条件，即所谓的广义Landesman-Lazer条件。这些证明基于变分方法，并强烈依赖于谱的变分特征。另请参阅https://ejde.math.txstate.edu/special/02/m2/abstr.html
{"title":"Fucik spectrum with weights and existence of solutions for nonlinear elliptic equations with nonlinear boundary conditions","authors":"N. Mavinga, Q. Morris, S. Robinson","doi":"10.58997/ejde.sp.02.m2","DOIUrl":"https://doi.org/10.58997/ejde.sp.02.m2","url":null,"abstract":"We consider the boundary value problem $$displaylines{ - Delta u + c(x) u = alpha m(x) u^+ - beta m(x) u^- +f(x,u), quad x in Omega, cr frac{partial u}{partial eta} + sigma (x) u =alpha rho (x) u^+- beta rho (x) u^- +g(x,u), quad x in partial Omega, }$$ where ((alpha, beta) in mathbb{R}^2), (c, m in L^infty (Omega)), (sigma, rho in L^infty (partialOmega)), and the nonlinearities f and g are bounded continuous functions. We study the asymmetric (Fucik) spectrum with weights, and prove existence theorems for nonlinear perturbations of this spectrum for both the resonance and non-resonance cases. For the resonance case, we provide a sufficient condition, the so-called generalized Landesman-Lazer condition, for the solvability. The proofs are based on variational methods and rely strongly on the variational characterization of the spectrum.\u0000See also https://ejde.math.txstate.edu/special/02/m2/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48767063","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide alternate necessary and/or sufficient conditions on the sign-changing coefficient p(t) for the maximum principle for the second-order periodic problem (u''=p(t) u + q(t) ) to hold, i.e., for nonnegative (q) to yield a nonpositive periodic solution (u). See also https://ejde.math.txstate.edu/special/02/h2/abstr.html
我们为二阶周期问题(u''=p(t) u + q(t) )的最大值原理提供了换号系数p(t)的必要和/或充分条件，即，对于非负的(q)产生非正的周期解(u)。参见https://ejde.math.txstate.edu/special/02/h2/abstr.html
{"title":"Optimal conditions for the maximum principle for second-order periodic problems","authors":"G. Holubová","doi":"10.58997/ejde.sp.02.h2","DOIUrl":"https://doi.org/10.58997/ejde.sp.02.h2","url":null,"abstract":"We provide alternate necessary and/or sufficient conditions on the sign-changing coefficient p(t) for the maximum principle for the second-order periodic problem (u''=p(t) u + q(t) ) to hold, i.e., for nonnegative (q) to yield a nonpositive periodic solution (u).\u0000See also https://ejde.math.txstate.edu/special/02/h2/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48234376","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a survey on the compactness of the set of solutions for the Yamabe problem on manifolds with boundary. The stability of the problem is also discussed. See also https://ejde.math.txstate.edu/special/02/g1/abstr.html
{"title":"Yamabe boundary problem with scalar-flat manifolds target","authors":"Marco Ghimenti, A. Micheletti","doi":"10.58997/ejde.sp.02.g1","DOIUrl":"https://doi.org/10.58997/ejde.sp.02.g1","url":null,"abstract":"We present a survey on the compactness of the set of solutions for the Yamabe problem on manifolds with boundary. The stability of the problem is also discussed.\u0000See also https://ejde.math.txstate.edu/special/02/g1/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43969984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a new nonlinear version of the well-known Black-Scholes model for option pricing in financial mathematics. The nonlinear Black-Scholes partial differential equation is based on the quasilinear diffusion term with the p-Laplace operator (Delta_p) for (1 < p < infty). The existence and uniqueness of a weak solution in a weighted Sobolev space is proved, first, by methods for nonlinear parabolic problems using the Gel'fand triplet and, alternatively, by a method based on nonlinear semigroups. Finally, possible choices of other weighted Sobolev spaces are discussed to produce a function space setting more realistic in financial mathematics. See also https://ejde.math.txstate.edu/special/02/t1/abstr.html
{"title":"Nonlinear diffusion with the p-Laplacian in a Black-Scholes-type model","authors":"P. Takáč","doi":"10.58997/ejde.sp.02.t1","DOIUrl":"https://doi.org/10.58997/ejde.sp.02.t1","url":null,"abstract":"We present a new nonlinear version of the well-known Black-Scholes model for option pricing in financial mathematics. The nonlinear Black-Scholes partial differential equation is based on the quasilinear diffusion term with the p-Laplace operator (Delta_p) for (1 < p < infty). The existence and uniqueness of a weak solution in a weighted Sobolev space is proved, first, by methods for nonlinear parabolic problems using the Gel'fand triplet and, alternatively, by a method based on nonlinear semigroups. Finally, possible choices of other weighted Sobolev spaces are discussed to produce a function space setting more realistic in financial mathematics.\u0000See also https://ejde.math.txstate.edu/special/02/t1/abstr.html","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49404393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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