Non-Algebraic Quadrature Domains
نویسندگان
چکیده
منابع مشابه
Non-algebraic quadrature domains
It is well known that, in the plane, the boundary of any quadrature domain (in the classical sense) coincides with the zero set of a polynomial. We show, by explicitly constructing some four-dimensional examples, that this is not always the case. This confirms, in dimension 4, a conjecture of the second author. Our method is based on the Schwarz potential and involves elliptic integrals of the ...
متن کاملTopology of Quadrature Domains
Formulas like (0.1) and (0.2) are called quadrature identities, and the corresponding domains of integration are called (classical) quadrature domains. Various classes of quadrature domains have been known for quite some time, see e.g. Neumann’s papers [39, 40] from the beginning of the last century, but the systematic study began only with the work of Davis [13], and Aharonov and Shapiro [2]. ...
متن کاملThe Dirichlet and Neumann and Dirichlet-to-neumann Problems in Quadrature, Double Quadrature, and Non-quadrature Domains
We demonstrate that solving the classical problems mentioned in the title on quadrature domains when the given boundary data is rational is as simple as the method of partial fractions. A by-product of our considerations will be a simple proof that the Dirichlet-to-Neumann map on a double quadrature domain sends rational functions on the boundary to rational functions on the boundary. The resul...
متن کاملSelected topics on quadrature domains ?
This is a selection of facts, old and new, about quadrature domains. The text, written in the form of a survey, is addressed to non-experts and covers a variety of phenomena related to quadrature domains. Such as: the difference between quadrature domains for subharmonic, harmonic and respectively complex analytic functions, geometric properties of the boundary, instability in the reverse Hele-...
متن کاملNon-Laplacian growth, algebraic domains and finite reflection groups
Dynamics of planar domains with multiply connected moving boundaries driven by the gradient of a scalar field that satisfies an elliptic PDE is studied. We consider the question: For which kind of PDEs the domains are algebraic, provided the field has singularities at a finite number of fixed points? The construction reveals a direct connection with the theory of the Calogero-Moser systems rela...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Potential Analysis
سال: 2012
ISSN: 0926-2601,1572-929X
DOI: 10.1007/s11118-012-9297-6