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Berezin and Berezin-Toeplitz quantizations for general function spaces.

Miroslav Englis (2006)

Revista Matemática Complutense

The standard Berezin and Berezin-Toeplitz quantizations on a Kähler manifold are based on operator symbols and on Toeplitz operators, respectively, on weighted L2-spaces of holomorphic functions (weighted Bergman spaces). In both cases, the construction basically uses only the fact that these spaces have a reproducing kernel. We explore the possibilities of using other function spaces with reproducing kernels instead, such as L2-spaces of harmonic functions, Sobolev spaces, Sobolev spaces of holomorphic...

Bernstein and De Giorgi type problems: new results via a geometric approach

Alberto Farina, Berardino Sciunzi, Enrico Valdinoci (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We use a Poincaré type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the form div a ( | u ( x ) | ) u ( x ) + f ( u ( x ) ) = 0 . Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in  2 and  3 and of the Bernstein problem on the flatness of minimal area graphs in  3 . A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our analysis. Our approach...

Bernstein type properties of two-sided hypersurfaces immersed in a Killing warped product

Antonio W. Cunha, Eudes L. de Lima, Henrique F. de Lima, Eraldo A. Lima Jr., Adriano A. Medeiros (2016)

Studia Mathematica

Our purpose is to apply suitable maximum principles in order to obtain Bernstein type properties for two-sided hypersurfaces immersed with constant mean curvature in a Killing warped product M × ρ , whose curvature of the base Mⁿ satisfies certain constraints and whose warping function ρ is concave on Mⁿ. For this, we study situations in which these hypersurfaces are supposed to be either parabolic, stochastically complete or, in a more general setting, L¹-Liouville. Rigidity results related to entire...

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