Section 4 is devoted to the connection between the relative size of the linear term. We then study the boundary behaviour of sets of zeros theorem 3 and 4. Harmonic functions definitions and examples harmonic functions, for us, live on open subsets of real euclidean spaces. Moreover, by a theorem on complex variables, the real part of an analytic function on an open set. By the maximum principle, its zero set z does not contain any simple closed curve. Alexander logunov, eugenia malinnikova submitted on 26 jun 2015 v1, last revised 26 apr 2016 this version, v3. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext.
Pdf suppose that u is a nonconstant harmonic function on the plane. First we show that for any polynomial, not necessarily harmonic, the relative size of the linear term controls local. On zero sets of harmonic and real analytic functions. Thus u has a zero on the boundary of every sufficiently small ball. In a broader context, this paper also complements the recent investigations by cheeger, naber, and valtorta cnv15 and naber and valtorta nv14 into volume estimates for the critical sets of. Thus a function such as u rncosn is a harmonic function on r2 since u is the real part of zn. It was proved in 7 that if f is the ratio of two harmonic functions in. Set of zeros of harmonic functions of two variables. Finally, we prove that a harmonic function on r2 with finitely many maximal curves of. We study the ratio of harmonic functions u, v which have the same zero set. A basic fact is that zeroes of holomorphic functions. Datar a complex number ais called a zero of a holomorphic function f. We also prove that, for a certain category of sets \ e \subset \ mathbb r n \ containing the finely open sets, each function f defined on e is the restriction of a real analytic respectively harmonic function on an open neighbourhood of e if and only if f is analytic respectively harmonic at each point of e. Pdf zero subsets for spaces of holomorphic functions.
To complete the tight connection between analytic and harmonic functions we show that any harmonic function is the real part of an analytic function. The next lemma captures a weak rigidity property of realvalued harmonic functions. Ratios of harmonic functions with the same zero set. We obtain various general conditions in terms of the balayage and greens functions under which the sequence of points is the zero set for weighted spaces of holomorphic functions in a domain on. Pdf geometric remarks on the level curves of harmonic functions. In this section, we will show how greens theorem is closely connected with solutions to laplaces partial di. Structure of sets which are well approximated by zero sets of. Ratios of harmonic functions with the same zero set authors.814 672 1322 1052 1093 377 951 762 392 572 1507 430 1240 911 343 1048 1154 1000 1496 1056 573 704 18 30 1372 436 1401 504 43 1273 936 179