Liouville s theorem for a fractional elliptic system. Lets first think further about paths in phase space. Suppose that the isoperimetric inequality with 2 is satisfied in. Dynamics on the space of harmonic functions and the foliated liouville problem r. The eigenfunctions of a sturm liouville problem can be chosen to be real. Liouvilletype theorems for subharmonic functions on complete. The result is trivially true if is a simple eigenvalue. We prove liouville type theorems for p harmonic functions on exterior domains of rd, where 1 harmonic function satisfying zero dirichlet, neumann. The classical theorem of liouville states that any bounded harmonic function on rn is constant. On a liouville type theorem for harmonic maps to convex. Datar recall that an entire function is a function that is holomorphic on the entire complex plane c. The liouville property has been extended to harmonic functions associated with markov processes. Pdf a liouville theorem for pharmonic functions on. The holomorphy of h is clear except at points in g.
If f is a harmonic function defined on all of r n which is bounded above or bounded below, then f is constant. Harmonic functions definitions and examples harmonic functions, for us, live on open subsets of real euclidean spaces. A beautiful consequence of this is a proof of the fundamental theorem. A discrete harmonic function bounded on a large portion of z2 is constant lev buhovsky, alexander logunov, eugenia malinnikova, and mikhail sodin to fedya nazarov with admiration abstract. Applying the generalized maximum principle, a liouvilletype theorem of subharmonic functions on complete riemannian manifolds is shown and a liouville type differential inequality on properly immersed complete submanifolds is given. Rigolisetti 55 also gave a liouville type theorem for harmonic maps from a certain riemannian manifold with pole. It provides a complete description of the system both at equilibrium and also away from equilibrium.
It asserts that the phasespace distribution function is constant along the trajectories of the systemthat is that the density of system points in the vicinity of a given system point traveling through phasespace is constant with time. An improvement of the liouville theorem for discrete harmonic functions on z2 is obtained. Liouvilles theorem on integration in terms of elementary. These are called liouville theorems for harmonic functions, regarded as the general. We prove liouville type theorems for equation harmonic functions on exterior domains of equation, where equation and equation. Dynamics on the space of harmonic functions and the. Then the subspace spanned by the eigenfunctions corresponding to admits an orthonormal basis of realvalued functions. Liouvilles theorem on integration in terms of elementary functions r. More precisely, we prove that there exists a positive constant such. Pdf a strong liouville theorem for pharmonic functions on. The classical liouville theorem asserts that every harmonic function on the whole space rd is constant if it is bounded from below or from above. We can obtain additional examples of harmonic functions by differentiation, noting that for smooth functions the laplacian commutes with any partial derivative. Bounded entire functions liouville s theorem suppose fz is an entire function. A proof of liouville s theorem edward nelson consider a bounded harmonic function on euclidean space.
The main reference is chapter 2 of gilbarg and trudingers elliptic partial di erential equations of second order. Introduction to liouvilles theorem paths in simple phase spaces. On the liouville theorem for harmonic maps hyeong in choi abstract. Liouville s theorem for harmonic functions as follows. Liouville s theorem and the fundamental theorem of algebra 45 6. Given two points, choose two balls with the given points as. It is proved in 6 that there is no nonconstant quasi harmonic function with polynomial growth. Compare liouvilles theorem for functions of a complex variable.
Hallenbeck received october 5, 1999 liouville s classical theorem assures that every harmonic function on the whole. A liouville type theorem for harmonic functions on exterior domains f. In complex analysis, see liouville s theorem complex analysis. The mean value property mathematics and statistics. R on the space of harmonic functions on the unit disc bounded by a common constant, as well as the relationship this action has with the foliated liouville prob. Consequently, every positive pharmonic function on such a graph must be constant. It is easy to see that there are no nonconstant quasi harmonic functions with finite energy, from 14. Liouville theorem for fractional henonlaneemden systems on a half space phuong le. Moreover, if a harmonic function is negative positive on the boundary of an open uit will be negative positive throughout u. Bounded entire functions liouvilles theorem suppose fz is an entire function. Integration of probability density function why is the most common hydraulic fluid so poisonous. Cauchy, liouville, and the fundamental theorem of algebra these notes supplement the lectures of wed. This equation describes the evolution of phase space distribution function for the conservative hamiltonian system.
Dynamics on the space of harmonic functions and the foliated. Liouvilles theorem for a fractional elliptic system. The liouville type theorem about the p harmonic map with finite l q energy from complete submanifold in a partially nonnegatively curved manifold to nonpositively. A liouvilletype theorem and bochner formula for harmonic. Throughout, we call a p harmonic function v positive if v. Since it is harmonic, its value at any point is its average over any sphere, and hence over any ball, with the point as center.
The classical liouville theorem was generalised to p harmonic functions on the whole. Harmonic functions on graphs and on other discrete structures. C2rn is bounded and harmonic in rn, then u is constant. Nelson proof of the liouville theorem for harmonic functions will be useful here, see references 1 and 2. As a corollary, any bounded harmonic function is constant. In fact, it was cauchy who proved liouville s theorem. Properties of sturmliouville eigenfunctions and eigenvalues. Unless other wise specified, repeated latin letters are summed from 1 to m, repeated greek letters are summed from 1 to n. The theorem provides some estimates that place limitations on how slowfast harmonic functions.
Harmonic function theory second edition sheldon axler paul bourdon wade ramey 26 december 2000 this ed pdf. In physics, liouville s theorem, named after the french mathematician joseph liouville, is a key theorem in classical statistical and hamiltonian mechanics. Below is a brief introduction to properties of harmonic functions. Phragmenlindelof theorem for infinity harmonic functions. To prove liouville s theorem, it is enough to show that the derivative of any entire function vanishes. A liouville theorem for p harmonic functions on exterior domains. Removable singularity theorem and liouville s theorem for harmonic functions are proven by maximum principle and poisson integral formula.
Cauchy, liouville, and the fundamental theorem of algebra. A sharp l q liouville theorem for p harmonic functions. Oct 02, 20 liouvilles theorem applied to 1d harmonic oscillator wolframmathematica. A strong liouville theorem for pharmonic functions on graphs 209 caccioppolitype inequality theorem 2. Our second main result is a liouville theorem for p harmonic functions on exterior domains with zero neumann boundary conditions, that is, t he case. Liouville s theorem has various meanings, all mathematical results named after joseph liouville. It is easy to see that there are no nonconstant quasi harmonic functions with finite energy, from. A liouville theorem for pharmonic functions on exterior domains e. Ltcc course on potential theory, spring 2011, qmul. Yau 10 generalized the liouville theorem to harmonic functions on riemannian manifolds of nonnegative ricci curvature. Liouvilles theorem applied to 1d harmonic oscillator youtube. How to convert pdf to word without software duration. A classical liouville theorem for harmonic functions may be stated as follows.
Underlined contents were not included in the tutorial because of time constraint, but. It is enough to prove that h can be extended to an entire function, in which case the result follows by liouville s theorem. This ed pdf file is available without charge only to individuals. In this section we will study the harmonic functions, which play a crucial role in the field theory. Liouville type theorem about p harmonic function and p. Some liouvilletype theorems for harmonic functions on. We proved that a lipschitz entire infinity harmonic function on rn must asymp totically tend to a plane at infinity and it must be a plane. All the eigenvalues of a sturm liouville problem are real. Hallenbeck received october 5, 1999 liouvilles classical theorem assures that every harmonic function on the whole.
Thus h can be extended to an entire bounded function which by liouville s theorem. Adapting the common terminology we say that a graph. Liouvilles theorem applied to 1d harmonic oscillator. We prove an lp liouville type theorem which is a quantitative integral lp estimate of harmonic functions analogous to karps theorem for riemannian manifolds. Introduction to liouville s theorem paths in simple phase spaces. Harmonic functions are real analytic1 on this very short note we prove that harmonic functions are real analytic functions. Cheng 1 proved our theorem for harmonic maps assuming n is simply connected with nonpositive sectional curvature. Proposition 4 let be an eigenvalue of a regular or periodic sturm liouville problem. Liouville theorems for quasiharmonic functions sciencedirect. Zeghib july 12, 2002 abstract we study here the action of subgroups of psl2. A liouville theorem for \alphaharmonic functions in. A liouviletype theorem for harmonic functions on exterior. These functions arise in the theory of gravity, magnetic, and electromagnetic fields.
A liouville theorem for infinity harmonic functions. We show that every positive equation harmonic function. The theorem below asserts that harmonic functions do not have local maxima or minima on open sets u unless they are constant. Pdf a discrete harmonic function bounded on a large portion. A strong liouville theorem for pharmonic functions on graphs. If jfzj m for all z 2c, for some m, then f is constant. The generalized liouvilles theorems obtained in this paper will help us to further study the properties of entire functions and harmonic functions. Since the 1970s, various liouville theorems for harmonic subharmonic functions have been extensively studied on complete riemannian manifolds. Keywords liouvilles theorem entire function extension harmonic function harnacks inequality. Liouville theorem for fractional henonlaneemden systems on. Let w be an exterior domain with no regularity assumption on w.
In this paper, we are interested in the liouville theorems for the quasi harmonic functions. A liouvilletype theorem for harmonic functions on exterior domains f. A simple example of strong liouville graphs is the ndimensional grid zn. A liouville theorem for p harmonic functions on exterior domains daniel hauer school of mathematics and statistics university of sydney, australia joint work with prof.
Our rst goal is to prove a liouville type theorem for harmonic maps from riemannian polyhedra into cat1 spaces. In particular, we prove that everyp harmonic functionu. To complete the tight connection between analytic and harmonic functions we show that any harmonic function is the real part of an analytic function. Harmonic function an overview sciencedirect topics. There is also a related theorem on harmonic functions.
But since h is bounded and all the zeroes of g are isolated, any singularities must be removable. Lemma 1 is used on the inductive step of the proof of the main theorem. We study harmonic functions on general weighted graphs which allow for a compatible intrinsic metric. A liouville theorem for pharmonic functions on exterior. Yau 10 generalized the liouville theorem to harmonic functions on riemannian manifolds of. Below, always denotes a nonempty connected open set in r2 c. We prove liouville type theorems for pharmonic functions on exterior domains of rd, where 1. Compare liouville s theorem for functions of a complex variable. It is proved in that there is no nonconstant quasi harmonic function with polynomial growth. Noting that partial derivatives of harmonic functions are also harmonic, and by using the mean value property for the partial derivatives, we can bound the derivatives of harmonic functions by the size of the function itself. Consider a bounded harmonic function on euclidean space. Our second main result is a liouville theorem for pharmonic functions on exterior domains with zero neumann boundary conditions, that is, the case g 2 w and h 0. Thus any increase in the number of oscillators increases the phasespace density and also enhances the current flow through the boundary of the region.
A strong liouville theorem for pharmonic functions. Liouville theorems for harmonic maps 3 here and throughout the paper, the summation convention is used. Pdf a discrete harmonic function bounded on a large. M with ricci curvature bounded below by a, a 0, n with sectional curvature bounded above by a positive constant k. It is easy to verify that the following assertions are also true.
A liouville theorem for p harmonic functions on exterior domains e. A beautiful consequence of this is a proof of the fundamental theorem of algebra, that any polynomial is completely factorable over the complex numbers. We prove lemma 1 using the mean value property of harmonic functions, greens theorem. Harmonic functions of general graph laplacians bobo hua and matthias keller abstract. The classical liouville theorem states that every nonnegative or bounded harmonic function on r n must be constant. The moser iteration may be a too complicated method in the discrete setting.
Recently, a gradient estimate in the case of singular targets has been carried out in zzz, resulting in a liouville type theorem. Complex analysis questions october 2012 contents 1 basic complex analysis 1 2 entire functions 5 3 singularities 6 4 in nite products 7 5 analytic continuation 8 6 doubly periodic functions 9 7 maximum principles 9 8 harmonic functions 10 9 conformal mappings 11 10 riemann mapping theorem 12 11 riemann surfaces 1 basic complex analysis. Communications of the moscow mathematical society related. The fact that the domain of a nonconstant elliptic function f can not be is what liouville actually proved, in 1847, using the theory of elliptic functions. Liouville theorem for bounded harmonic functions on manifolds and.
A liouville theorem for equationharmonic functions on. We studyl q liouville properties of nonnegativepsuperharmonic and, respectively,psubharmonic functions on a complete riemannian manifoldm. Pdf a strong liouville theorem for pharmonic functions. The main analytic result generalizes a result of y2, where yau shows that on a complete manifold, any nonnegative subharmonic function in lpp1 must be constant. For example, the simple harmonic oscillator, with hamiltonian, describes circles in phase space parameterized with the variables. For an ensemble of harmonic oscillators, a region in phase space can be associated with the density of oscillators. A liouville theorem for pharmonic functions on exterior domains. Coldingminicozzi ii 8,9 gave related results to 3 on harmonic functions.
1245 1168 856 1132 311 766 254 1529 685 1176 1385 1501 319 1615 844 1606 44 891 1021 948 1286 429 1313 716 457 445 748 1459 984 528 923 783 187 109 67 17 462 532