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Academic literature on the topic 'Liouville theorems'

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Published: 1 April 2023

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Journal articles on the topic "Liouville theorems"

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Korniłowicz, Artur, Adam Naumowicz, and Adam Grabowski. "All Liouville Numbers are Transcendental." Formalized Mathematics 25, no. 1 (March 28, 2017): 49–54. http://dx.doi.org/10.1515/forma-2017-0004.

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Summary In this Mizar article, we complete the formalization of one of the items from Abad and Abad’s challenge list of “Top 100 Theorems” about Liouville numbers and the existence of transcendental numbers. It is item #18 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/. Liouville numbers were introduced by Joseph Liouville in 1844 [15] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and It is easy to show that all Liouville numbers are irrational. The definition and basic notions are contained in [10], [1], and [12]. Liouvile constant, which is defined formally in [12], is the first explicit transcendental (not algebraic) number, another notable examples are e and π [5], [11], and [4]. Algebraic numbers were formalized with the help of the Mizar system [13] very recently, by Yasushige Watase in [23] and now we expand these techniques into the area of not only pure algebraic domains (as fields, rings and formal polynomials), but also for more settheoretic fields. Finally we show that all Liouville numbers are transcendental, based on Liouville’s theorem on Diophantine approximation.
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Glagoleva, R. Ya. "Phragmen-Liouville-type theorems and Liouville theorems for a linear parabolic equation." Mathematical Notes of the Academy of Sciences of the USSR 37, no. 1 (January 1985): 67–70. http://dx.doi.org/10.1007/bf01652519.

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Tuna, Hüseyin, and Aytekin Eryilmaz. "Livšic’s theorem for q-Sturm—Liouville operators." Studia Scientiarum Mathematicarum Hungarica 53, no. 4 (December 2016): 512–24. http://dx.doi.org/10.1556/012.2016.53.4.1348.

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In this paper, we study dissipative q-Sturm—Liouville operators in Weyl’s limit circle case. We describe all maximal dissipative, maximal accretive, self adjoint extensions of q-Sturm—Liouville operators. Using Livšic’s theorems, we prove a theorem on completeness of the system of eigenvectors and associated vectors of the dissipative q-Sturm—Liouville operators.
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Araya, Ataklti, and Ahmed Mohammed. "On Cauchy–Liouville-type theorems." Advances in Nonlinear Analysis 8, no. 1 (August 24, 2017): 725–42. http://dx.doi.org/10.1515/anona-2017-0158.

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Abstract In this paper we explore Liouville-type theorems to solutions of PDEs involving the ϕ-Laplace operator in the setting of Orlicz–Sobolev spaces. Our results extend Liouville-type theorems that have been obtained recently.
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Bear, H. S. "Liouville theorems for heat functions." Communications in Partial Differential Equations 11, no. 14 (January 1986): 1605–25. http://dx.doi.org/10.1080/03605308608820476.

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Jin, Zhiren. "Liouville theorems for harmonic maps." Inventiones Mathematicae 108, no. 1 (December 1992): 1–10. http://dx.doi.org/10.1007/bf02100594.

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Awadalla, Muath, Muthaiah Subramanian, Kinda Abuasbeh, and Murugesan Manigandan. "On the Generalized Liouville–Caputo Type Fractional Differential Equations Supplemented with Katugampola Integral Boundary Conditions." Symmetry 14, no. 11 (October 29, 2022): 2273. http://dx.doi.org/10.3390/sym14112273.

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In this study, we examine the existence and Hyers–Ulam stability of a coupled system of generalized Liouville–Caputo fractional order differential equations with integral boundary conditions and a connection to Katugampola integrals. In the first and third theorems, the Leray–Schauder alternative and Krasnoselskii’s fixed point theorem are used to demonstrate the existence of a solution. The Banach fixed point theorem’s concept of contraction mapping is used in the second theorem to emphasise the analysis of uniqueness, and the results for Hyers–Ulam stability are established in the next theorem. We establish the stability of Ulam–Hyers using conventional functional analysis. Finally, examples are used to support the results. When a generalized Liouville–Caputo (ρ) parameter is modified, asymmetric results are obtained. This study presents novel results that significantly contribute to the literature on this topic.
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Chen, Wenxiong, and Leyun Wu. "Liouville Theorems for Fractional Parabolic Equations." Advanced Nonlinear Studies 21, no. 4 (October 14, 2021): 939–58. http://dx.doi.org/10.1515/ans-2021-2148.

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Abstract In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition u → 0 u\to 0 at infinity to a polynomial growth on 𝑢 by constructing proper auxiliary functions. Then we derive monotonicity for the solutions in a half space R + n × R \mathbb{R}_{+}^{n}\times\mathbb{R} and obtain some new connections between the nonexistence of solutions in a half space R + n × R \mathbb{R}_{+}^{n}\times\mathbb{R} and in the whole space R n - 1 × R \mathbb{R}^{n-1}\times\mathbb{R} and therefore prove the corresponding Liouville type theorems. To overcome the difficulty caused by the nonlocality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of nonlocal parabolic problems.
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Gol'dshtein, Vladimir, and Alexander Ukhlov. "On the functional properties of weak (p,q)-quasiconformal homeomorphisms." Ukrainian Mathematical Bulletin 16, no. 3 (October 21, 2019): 329–44. http://dx.doi.org/10.37069/1810-3200-2019-16-3-2.

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We study the functional properties of weak (p,q)-quasiconformal homeomorphisms such as Liouville-type theorems, the global integrability, and the Hölder continuity. The proof of Liouville-type theorems is based on the duality property of weak (p,q)-quasiconformal homeomorphisms.
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Zhu, Meijun. "Liouville theorems on some indefinite equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 129, no. 3 (1999): 649–61. http://dx.doi.org/10.1017/s0308210500021569.

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Dissertations / Theses on the topic "Liouville theorems"

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Fazly, Mostafa. "m-Liouville theorems and regularity results for elliptic PDEs." Thesis, University of British Columbia, 2012. http://hdl.handle.net/2429/43751.

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This thesis which is a compendium of seven papers, focuses on the study of the semilinear elliptic equations and systems, on both bounded and unbounded domains of dimension n, most importantly the Allen-Cahn equation and the De Giorgi’s conjecture (1978). This conjecture brings together two groups of mathematicians: one specializing in nonlinear partial differential equations and another in differential geometry, more specifically on minimal surfaces and constant mean curvature surfaces. De Giorgi conjectured that the monotone and bounded solutions of the Allen-Cahn equation on the whole space of dimension n ≤ 8 must be 1-dimensional solutions. This is known to be true for n ≤ 3 and with extra (natural) assumptions for 4 ≤ n ≤ 8. Motivated by this conjecture, I have introduced two main concepts: The first concept is the “H-monotone solutions” that allows us to formulate a counterpart of the De Giorgi’s conjecture for system of equations stating that the H-monotone and bounded solutions of the gradient systems on the whole space of dimension n ≤ 8 must be 1-dimensional solutions. This seems to be in the right track to extend the De Giorgi's conjecture to systems. The second concept is the “m-Liouville theorem” for m = 0, · · · , n − 1 that allows us to formulate a counterpart of the De Giorgi’s conjecture for equations but this time for higher-dimensional solutions as opposed to 1-dimensional solutions. We use the induction idea that is to use 0-Liouville theorem (0- dimensional solutions) to prove 1-Liouville theorem (1-dimensional solutions) and then to prove (n − 1)- Liouville theorem ((n − 1)-dimensional solutions). The reason that we call this “m-Liouville theorem” is because of the great mathematician Joseph Liouville (1809-1882) who proved a classical theorem in complex analysis stating that "bounded harmonic functions on the whole space must be constant" and constants are 0-dimensional objects. 0-Liouville theorem is at the heart of this thesis and it includes various 0-Liouville theorems for various equations and system. In particular, we give a positive answer to the Henon-Lane-Emden conjecture in dimension three under an extra boundedness assumption. On the other hand, it is well known that there is a close relationship between the regularity of solutions on bounded domains and 0-Liouville theorem for related “limiting equations” on the whole space, via rescaling and blow up procedures. In this direction, we present regularity of solutions for gradient and twisted-gradient systems as well as the uniqueness results for nonlocal eigenvalue problems. The novelty here is a stability inequality for both gradient and twisted-gradient systems that gives us the chance to adjust the known techniques and ideas (for equations) to systems.
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D'Ambrosio, Lorenzo. "Hardy Inequalities and Liouville Type Theorems Associated to Degenerate Operators." Doctoral thesis, SISSA, 2002. http://hdl.handle.net/20.500.11767/4170.

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Mastrolia, P. "GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR DIFFUSION-TYPE OPERATORS ON COMPLETE RIEMANNIAN MANIFOLDS." Doctoral thesis, Università degli Studi di Milano, 2011. http://hdl.handle.net/2434/153097.

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The aim of this work is twofold. The first main concern, the analytical one, is to study, using the method of gradient estimates, various Liouville-type theorems which are extensions of the classical Liouville Theorem for harmonic functions. We generalize the setting - from the Euclidean space to complete Riemannian manifolds - and the relevant operator - from the Laplacian to a general diffusion operator - and we also consider ``relaxed'' boundedness conditions (such as non-negativity, controlled growth and so on). The second main concern is geometrical, and is deeply related to the first: we prove some triviality results for Einstein warped products and quasi-Einstein manifolds studying a specific Poisson equation for a particular, and geometrically relevant, diffusion operator.
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Cunha, Antonio Wilson Rodrigues da. "Sobre hipersuperfÃcies mÃnimas, aplicaÃÃes do princÃpio do mÃximo fraco e de teoremas tipo-Liouville." Universidade Federal do CearÃ, 2015. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=15135.

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CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior
Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico
In this work we approach four research lines, where we began with the study of isometrically immersed hypersurfaces in a horoball. Next we studied Liouville type theorems in a complete Riemannian manifold for general operators. After we studied hypersurfaces f-minimal closed on a manifold with density, and nally we studied properly embedded minimal hypersurfaces with free boundary in a n-dimensional compact Riemannian manifold. Continuing, we obtain under a more general class operator than '-Laplacian, a Liouville type theorem for a complete Riemannian manifold, so that, prove a classication theorem for Killing graph of a foliation. Firstly, we are going to assume a weak maximum principle and that immersion is contained in a horoball, i.e., the set of bounded above Bussemann functions . We obtain an estimate for the highest quotient of r-curvatures. Moreover, under certain conditions on sectional curvature and assuming that the immersion is contained in a horoball, we forced the validity of the weak maximum principle and obtain the same estimates. Next, we establish a Choi-Wang type estimate for the rst eigenvalue of the weighter Laplacian on spaces with density in responding partially to Yau's conjecture for the rst eigenvalue weighter Laplacian for spaces with density, and moreover, we obtain an inequality Poincare type. With the estimates obtained, we establish an estimate of volume for a closed surface immersed in a space with density. Still following the study of spaces with density, we obtain a type Hientze-Karcher inequality for a compact manifold with nonempty boundary , so that, we obtain that if holds the equality than the manifold is isometric to a Euclidian ball. As consequence, we obtain under same conditions that if the f-mean curvature satisfy a bounded below than the manifold is isometric to a Euclidian ball. Finally, we obtain an estimate for the nonzero rst Steklov eigenvalue, where we are giving a answer partial to a conjecture by Fraser and Li. Moreover, as a consequence we establish an estimate for the total length of the boundary of the properly embedded minimal surfaces with free boundary in terms of its topology, thus, we proved the same when the surface is embedded in the Euclidean ball 3-dimensional.
Neste trabalho, abordamos quatro linhas de estudo, onde iniciamos com o estudo de hipersuperfcies isometricamente imersas sobre uma horobola. Em seguida estudamos Teoremas tipo Liouville para uma variedade Riemanniana completa em operadores mais gerais que o Laplaciano. Alem disso, estudamos hipersuperfcies f-mÃnimas fechadas em uma variedade com densidade e, por fim, estudamos hipersuperfÃcies mÃnimas com bordo livre, propriamente imersas em uma variedade Riemanniana compacta n-dimensional. Primeiramente, assumindo um princpio do maximo fraco e que a imersÃo està contida em uma horobola, i.e., um conjunto em que a funcÃo de Busemann à limitada superiormente, obtemos uma estimativa para o supremo do quociente das r-Ãsimas curvaturas. AlÃm disso, sob certas condiÃÃes sobre as curvaturas seccionais e assumindo que a imersÃo està contida em uma horobola, forÃamos a validade do princÃpio do mÃximo fraco e obtemos as mesmas estimativas. Prosseguindo, obtemos, para um operador mais geral que o '-Laplaciano, um teorema tipo-Liouville para uma variedade Riemanniana completa. Como aplicaÃÃo provamos um teorema de classificaÃÃo para grÃficos de Killing de uma folheaÃÃo. Em seguida, estabelecemos uma estimativa tipo Choi e Wang para o primeiro autovalor do f-Laplaciano em espaÃos com densidade, no sentido de responder parcialmente à conjectura de Yau para o primeiro autovalor do Laplaciano; alÃm disso, obtemos uma desigualdade tipo Poincarà para esse operador. Com a estimativa obtida, pudemos estabelecer uma estimativa de volume para uma superfÃcie fechada mergulhada em um espaÃo com densidade. Ainda seguindo o estudo de espaÃos com densidade, obtemos uma desigualdade tipo Heintze-Karcher para uma variedade compacta com bordo e verificamos que, se vale a igualdade, entÃo a variedade à isomÃtrica a uma bola Euclidiana. Como consequÃncia, obtemos que, nas mesmas condiÃÃes, e se a f-curvatura mÃdia satisfizer uma certa limitaÃÃo inferior, entÃo a variedade ainda à isometrica a uma bola Euclidiana. Finalmente, obtemos uma estimativa para o primeiro autovalor de Steklov, dando uma resposta parcial a uma conjectura devida a Fraser e Li. AlÃm disso, como consequÃncia, estabelecemos uma estimativa para o comprimento do bordo de uma superfÃcie mÃnima, compacta e propriamente megulhada com bordo livre em termos de sua topologia; assim, provamos o mesmo resultado quando a superfÃcie està mergulhada em uma bola Euclidiana 3-dimensional.
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Afonso, Rafaela Ferreira. "Um estudo do comportamento dos zeros dos Polinômios de Gegenbauer." Universidade Federal de Uberlândia, 2016. https://repositorio.ufu.br/handle/123456789/16825.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
In this dissertation, we study the Sturm Liouvile's theorems for the zeros of the solutions of linear differential equations of second order. These classical theorems are applied to analysis of the monotonicity of functions involving the zeros of classical orthogonal polynomials. in particular, Gegenbauer polynomials.
Neste trabalho estudamos os Teoremas de Sturm Liouville para zeros de soluções de equações diferenciais lineares de segunda ordem. Estes teoremas clássicos são aplicados para análise do crescimento e decrescimento de certas funções que envolvem os zeros de Polinômios Ortogonais Clássicos, como os Polinômios de Gegenbauer.
Mestre em Matemática
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Tupia, Martín Dionisio Arteaga. "A função de três pontos nas teorias de Liouville e N = 1 super Liouville." Universidade de São Paulo, 2015. http://www.teses.usp.br/teses/disponiveis/43/43134/tde-24092015-135051/.

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Neste trabalho são apresentados alguns conceitos básicos da Teoria de Liouville e N=1 Super Liouville, enfatizando o cálculo das funções de três pontos dessas teorias.Uma introdução a Teoria de Campos Conformes (CFT) e a Supersimetria também sao incluídas, as quais constituem ferramentas básicas da presente pesquisa.
In this dissertation we present some basic features about Liouville and N=1 Super Liouville Theory, and focus in the computation of their three point functions. Additionally, we include an introduction to Conformal Field Theories (CFT) and Supersymmetry, which are the basic tools of the present research.
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Bär, Christian. "Some properties of solutions to weakly hypoelliptic equations." Universität Potsdam, 2012. http://opus.kobv.de/ubp/volltexte/2012/6006/.

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A linear differential operator L is called weakly hypoelliptic if any local solution u of Lu = 0 is smooth. We allow for systems, i.e. the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which covers all elliptic, overdetermined elliptic, subelliptic and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence, and Riemann's first removable singularity theorem. In the case of constant coefficients we show that Liouville's theorem holds, any bounded solution must be constant and any L^p solution must vanish.
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Neves, Rui Gomes Mendona. "Conformal field theories on random surfaces and the non-critical string." Thesis, Durham University, 1997. http://etheses.dur.ac.uk/4750/.

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Recently, it has become increasingly clear that boundaries play a significant role in the understanding of the non-perturbative phase of the dynamics of strings. In this thesis we propose to study the effects of boundaries in non-critical string theory. We thus analyse boundary conformal field theories on random surfaces using the conformal gauge approach of David, Distler and Kawai. The crucial point is the choice of boundary conditions on the Liouville field. We discuss the Weyl anomaly cancellation for Polyakov's non-critical open bosonic string with Neumann, Dirichlet and free boundary conditions. Dirichlet boundary conditions on the Liouville field imply that the metric is discontinuous as the boundary is approached. We consider the semi-classical limit and argue how it singles out the free boundary conditions for the Liouville held. We define the open string susceptibility, the anomalous gravitational scaling dimensions and a new Yang-Mills Feynman mass critical exponent. Finally, we consider an application to the theory of non-critical dual membranes. We show that the strength of the leading stringy non-perturbative effects is of the order e(^-o(1/βst)), a result that mimics those found in critical string theory and in matrix models. We show how this restricts the space of consistent theories. We also identify non-critical one dimensional D-instantons as dynamical objects which exchange closed string states and calculate the order of their size. The extension to the minimal c ≤ 1 boundary conformal models is also briefly discussed.
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Lima, Jalman Alves de. "Teoremas Tipo Liouville e Desigualdades Tipo Harnack para Equações Elípticas Semilineares via Método Moving Spheres." Universidade Federal da Paraí­ba, 2011. http://tede.biblioteca.ufpb.br:8080/handle/tede/7400.

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Made available in DSpace on 2015-05-15T11:46:10Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 440995 bytes, checksum: d194a6a60d04b251160ec2e62f106e77 (MD5) Previous issue date: 2011-06-10
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In this work, we will do some applications of the Moving Spheres method, a variant of the method of Moving Planes, in order to obtain some Liouville-type theorems and some Harnack-type inequalities in Rn, as well as in the Euclidian half space Rn +. Our study focuses on, mostly, in the article written by Yan Yan Li and Lei Zhan [32], as well as some references of the same article. We concentrate in studying some properties of positive solutions of some semilinear elliptic partial differential equations with critical exponent and giving different proofs, improvements, and extensions of some previously established Liouville-type theorems and Harnack-type inequalities.
Neste trabalho, faremos algumas aplicações do método Moving Spheres, uma variante do método Moving Planes, na obtenção de alguns teoremas tipo Liouville e de algumas desigualdades tipo Harnack em Rn, bem como no semi-espaço euclidiano Rn +. Nosso estudo se concentra, marjoritariamente, no artigo do Yan Yan Li e Lei Zhang [32], bem como algumas referências do mesmo. Nos concentramos em estudar propriedades de soluções positivas de algumas equações diferenciais parciais elípticas semilineares com expoente crítico e dar provas diversificadas, refinamentos e extensões de alguns Teoremas tipo Liouville e desigualdades tipo Harnack já estabelecidos.
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COLOMBO, GIULIO. "GLOBAL GRADIENT BOUNDS FOR SOLUTIONS OF PRESCRIBED MEAN CURVATURE EQUATIONS ON RIEMANNIAN MANIFOLDS." Doctoral thesis, Università degli Studi di Milano, 2021. http://hdl.handle.net/2434/813095.

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This thesis is concerned with the study of qualitative properties of solutions of the minimal surface equation and of a class of prescribed mean curvature equations on complete Riemannian manifolds. We derive global gradient bounds for non-negative solutions of such equations on manifolds satisfying a uniform Ricci lower bound and we obtain Liouville-type theorems and other rigidity results on Riemannian manifolds with non-negative Ricci curvature. The proof of the aforementioned global gradient bounds for non-negative solutions u is based on the application of the maximum principle to an elliptic differential inequality satisfied by a suitable auxiliary function z=f(u,|Du|), in the spirit of Bernstein’s method of a priori estimates for nonlinear PDEs and of Yau’s proof of global gradient bounds for harmonic functions on complete Riemannian manifolds. The particular choice of the auxiliary function z parallels the one in Korevaar’s proof of a priori gradient estimates for the prescribed mean curvature equation in Euclidean space. The rigidity results obtained in the last part of the thesis include a Liouville theorem for positive solutions of the minimal surface equation on complete Riemannian manifolds with non-negative Ricci curvature, a splitting theorem for complete parabolic manifolds of non-negative sectional curvature supporting non-constant solutions with linear growth of the minimal surface equation, and a splitting theorem for domains of complete parabolic manifolds with non-negative Ricci curvature supporting non-constant solutions of overdetermined problems involving the mean curvature operator.
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Books on the topic "Liouville theorems"

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Kha, Minh, and Peter Kuchment. Liouville-Riemann-Roch Theorems on Abelian Coverings. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-67428-1.

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Gwynne, Ewain. Percolation on uniform quadrangulations and SLE6 on √8/3-Liouville quantum gravity. Paris: Société mathématique de France, 2021.

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Kha, Minh, and Peter Kuchment. Liouville-Riemann-Roch Theorems on Abelian Coverings. Springer International Publishing AG, 2021.

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Mann, Peter. Hamilton-Jacobi Theory. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0019.

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This chapter focuses on Liouville’s theorem and classical statistical mechanics, deriving the classical propagator. The terms ‘phase space volume element’ and ‘Liouville operator’ are defined and an n-particle phase space probability density function is constructed to derive the Liouville equation. This is deconstructed into the BBGKY hierarchy, and radial distribution functions are used to develop n-body correlation functions. Koopman–von Neumann theory is investigated as a classical wavefunction approach. The chapter develops an operatorial mechanics based on classical Hilbert space, and discusses the de Broglie–Bohm formulation of quantum mechanics. Partition functions, ensemble averages and the virial theorem of Clausius are defined and Poincaré’s recurrence theorem, the Gibbs H-theorem and the Gibbs paradox are discussed. The chapter also discusses commuting observables, phase–amplitude decoupling, microcanonical ensembles, canonical ensembles, grand canonical ensembles, the Boltzmann factor, Mayer–Montroll cluster expansion and the equipartition theorem and investigates symplectic integrators, focusing on molecular dynamics.
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Horii, Zene. Nonlinear Lattice Statistical Mechanics: The Liouville-Horii Theorem. BookSurge Publishing, 2007.

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Horii, Zene. Nonlinear Lattice Statistical Mechanics: The Liouville-Horii Theorem. BookSurge Publishing, 2007.

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Mann, Peter. Liouville’s Theorem & Classical Statistical Mechanics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0020.

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This chapter returns to the discussion of constrained Hamiltonian dynamics, now in the canonical setting, including topics such as regular Lagrangians, constraint surfaces, Hessian conditions and the constrained action principle. The standard approach to Hamiltonian mechanics is to treat all the variables as being independent; in the constrained case, a constraint function links the variables so they are no longer independent. In this chapter, the Dirac–Bergmann theory for singular Lagrangians is developed, using an action-based approach. The chapter then investigates consistency conditions and Dirac’s different types of constraints (i.e. first-class constraints, second-class constraints, primary constraints and secondary constraints) before deriving the Dirac bracket from simple arguments. The Jackiw–Fadeev constraint formulation is then discussed before the chapter closes with the Güler formulation for a constrained Hamilton–Jacobi theory.
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Mann, Peter. Classical Electromagnetism. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0027.

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In this chapter, Noether’s theorem as a classical field theory is presented and the properties of variations are again discussed for fields (i.e. field variations, space variations, time variations, spacetime variations), resulting in the Noether condition. Quasisymmetries and spontaneous symmetry breaking are discussed, as well as local symmetry and global symmetry. Following these definitions, Noether’s first theorem and Noether’s second theorem are developed. The classical Schrödinger field is investigated and the key equations of classical mechanics are summarised into a single Lagrangian. Symmetry properties of the field action and equations of motion are then compared. The chapter discusses the energy–momentum tensor, the Klein–Utiyama theorem, the Liouville equation and the Hamilton–Jacobi equation. It also discusses material science, special orthogonal groups and complex scalar fields.
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Mann, Peter. Autonomous Geometrical Mechanics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0022.

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This chapter examines the structure of the phase space of an integrable system as being constructed from invariant tori using the Arnold–Liouville integrability theorem, and periodic flow and ergodic flow are investigated using action-angle theory. Time-dependent mechanics is formulated by extending the symplectic structure to a contact structure in an extended phase space before it is shown that mechanics has a natural setting on a jet bundle. The chapter then describes phase space of integrable systems and how tori behave when time-dependent dynamics occurs. Adiabatic invariance is discussed, as well as slow and fast Hamiltonian systems, the Hannay angle and counter adiabatic terms. In addition, the chapter discusses foliation, resonant tori, non-resonant tori, contact structures, Pfaffian forms, jet manifolds and Stokes’s theorem.
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Mann, Peter. Canonical & Gauge Transformations. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0018.

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In this chapter, the Hamilton–Jacobi formulation is discussed in two parts: from a generating function perspective and as a variational principle. The Poincaré–Cartan 1-form is derived and solutions to the Hamilton–Jacobi equations are discussed. The canonical action is examined in a fashion similar to that used for analysis in previous chapters. The Hamilton–Jacobi equation is then shown to parallel the eikonal equation of wave mechanics. The chapter discusses Hamilton’s principal function, the time-independent Hamilton–Jacobi equation, Hamilton’s characteristic function, the rectification theorem, the Maupertius action principle and the Hamilton–Jacobi variational problem. The chapter also discusses integral surfaces, complete integral hypersurfaces, completely separable solutions, the Arnold–Liouville integrability theorem, general integrals, the Cauchy problem and de Broglie–Bohm mechanics. In addition, an interdisciplinary example of medical imaging is detailed.
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Book chapters on the topic "Liouville theorems"

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Wu, H. "Liouville theorems." In Lecture Notes in Mathematics, 331–49. Berlin, Heidelberg: Springer Berlin Heidelberg, 1987. http://dx.doi.org/10.1007/bfb0078254.

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Farina, Alberto. "I.6. Liouville Theorems." In James Serrin. Selected Papers, 363–428. Basel: Springer Basel, 2014. http://dx.doi.org/10.1007/978-3-0348-0845-3_6.

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Kha, Minh, and Peter Kuchment. "Specific Examples of Liouville-Riemann-Roch Theorems." In Liouville-Riemann-Roch Theorems on Abelian Coverings, 55–66. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-67428-1_4.

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Kha, Minh, and Peter Kuchment. "The Main Results." In Liouville-Riemann-Roch Theorems on Abelian Coverings, 23–33. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-67428-1_2.

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Kha, Minh, and Peter Kuchment. "Proofs of the Main Results." In Liouville-Riemann-Roch Theorems on Abelian Coverings, 35–53. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-67428-1_3.

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Kha, Minh, and Peter Kuchment. "Auxiliary Statements and Proofs of Technical Lemmas." In Liouville-Riemann-Roch Theorems on Abelian Coverings, 67–84. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-67428-1_5.

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Kha, Minh, and Peter Kuchment. "Preliminaries." In Liouville-Riemann-Roch Theorems on Abelian Coverings, 1–21. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-67428-1_1.

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Xin, Y. L. "Liouville type theorems and regularity of harmonic maps." In Lecture Notes in Mathematics, 198–208. Berlin, Heidelberg: Springer Berlin Heidelberg, 1987. http://dx.doi.org/10.1007/bfb0077691.

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Hua, Bobo, Matthias Keller, Daniel Lenz, and Marcel Schmidt. "On $$L^p$$ Liouville Theorems for Dirichlet Forms." In Springer Proceedings in Mathematics & Statistics, 201–21. Singapore: Springer Nature Singapore, 2022. http://dx.doi.org/10.1007/978-981-19-4672-1_12.

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Souplet, Philippe. "Liouville-Type Theorems for Nonlinear Elliptic and Parabolic Problems." In 2018 MATRIX Annals, 303–25. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-38230-8_21.

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Conference papers on the topic "Liouville theorems"

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BREZIS, H., M. CHIPOT, and Y. XIE. "SOME REMARKS ON LIOUVILLE TYPE THEOREMS." In Proceedings of the International Conference on Nonlinear Analysis. WORLD SCIENTIFIC, 2008. http://dx.doi.org/10.1142/9789812709257_0003.

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YU, CHII-HUEI. "APPLICATIONS OF TWO IMPORTANT THEOREMS OF FRACTIONAL CALCULUS." In 2021 INTERNATIONAL CONFERENCE ON ADVANCED EDUCATION AND INFORMATION MANAGEMENT (AEIM 2021). Destech Publications, Inc., 2021. http://dx.doi.org/10.12783/dtssehs/aeim2021/36005.

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Abstract. In this article, we study the fundamental theorem of fractional calculus and integration by parts for fractional calculus, regarding the Jumarie type of modified Riemann-Liouville fractional derivatives. On the other hand, some examples are proposed to illustrate the applications of these two important theorems.
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Winston, Roland. "Beating the optical Liouville theorem." In Optics and Photonics for Advanced Energy Technology. Washington, D.C.: OSA, 2009. http://dx.doi.org/10.1364/energy.2009.wd8.

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Frederico, Gastao S. F., and Delfim F. M. Torres. "Fractional Noether's theorem with classical and Riemann-Liouville derivatives." In 2012 IEEE 51st Annual Conference on Decision and Control (CDC). IEEE, 2012. http://dx.doi.org/10.1109/cdc.2012.6426162.

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YAMANO, TAKUYA, and OSAMU IGUCHI. "ON CLASSICAL NO-CLONING THEOREM UNDER LIOUVILLE DYNAMICS AND DISTANCES." In Summer School on Mathematical Aspects of Quantum Computing. WORLD SCIENTIFIC, 2008. http://dx.doi.org/10.1142/9789812814487_0011.

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Jannson, Tomasz, and Joel Ng. "Nonimaging optics and the Liouville theorem in the XUV region." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1987. http://dx.doi.org/10.1364/oam.1987.thpo24.

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In general, Liouville's theorem holds for any time-reversible Hamiltonian system including the geometrical optics (both imaging and nonimaging1,2) model of monochromatic rays, where, in Hamiltonian formulation,2 the z-coordinate is treated as a parameter equivalent to the time variable in statistical classical mechanics. Thus the so-called ideal nonimaging optical elements, well known in the IR/VIS/UV region,1,3 can also be introduced to the collimation and concentration of XUV electromagnetic radiation (1–100 nm). Contrary to the IR/VIS/UV region, where the nonimagng elements are based on either metallic reflection or dielectric total internal reflection, the XUV concentrators/collimators are based on grazing incidence total external reflection, since the refractive index of all optical materials is smaller than 1 in the XUV region.
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OKAYASU, TAKASHI. "A LIOUVILLE THEOREM FOR HARMONIC MAPS WITH FINITE TOTAL N-ENERGY." In Proceedings in Honor of Professor K Sekigawa's 60th Birthday. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812701701_0016.

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Klimek, Malgorzata. "Fractional Sturm-Liouville Problem and 1D Space-Time Fractional Diffusion With Mixed Boundary Conditions." In ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/detc2015-46808.

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In the paper, we show a connection between a regular fractional Sturm-Liouville problem with left and right Caputo derivatives of order in the range (1/2, 1) and a 1D space-time fractional diffusion problem in a bounded domain. Both problems include mixed boundary conditions in a finite space interval. We prove that in the case of vanishing mixed boundary conditions, the Sturm-Liouville problem can be rewritten in terms of Riesz derivatives. Then, we apply earlier results on its eigenvalues and eigenfunctions to construct a weak solution of the 1D fractional diffusion equation with variable diffusivity. Adding an assumption on the summability of the eigenvalues’ inverses series, we formulate a theorem on a strong solution of the 1D fractional diffusion problem.
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Winston, Roland, Chunhua Wang, and Weiya Zhang. "Beating the optical Liouville theorem: How does geometrical optics know the second law of thermodynamics?" In SPIE Optical Engineering + Applications, edited by Roland Winston and Jeffrey M. Gordon. SPIE, 2009. http://dx.doi.org/10.1117/12.836029.

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González, Diego, and Sergio Davis. "Liouville’s theorem from the principle of maximum caliber in phase space." In TECHNOLOGIES AND MATERIALS FOR RENEWABLE ENERGY, ENVIRONMENT AND SUSTAINABILITY: TMREES. Author(s), 2016. http://dx.doi.org/10.1063/1.4959044.

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Reports on the topic "Liouville theorems"

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Moses, Joel, and Jamil Baddoura. A Dilogarithmic Extension of Liouville's Theorem on Integration in Finite Terms. Fort Belvoir, VA: Defense Technical Information Center, February 1988. http://dx.doi.org/10.21236/ada206681.

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