58 research outputs found
Separation of Boundary Singularities for Holomorphic Generators
We prove a theorem on separation of boundary null points for generators of
continuous semigroups of holomorphic self-mappings of the unit disk in the
complex plane. Our construction demonstrates the existence and importance of a
particular role of the binary operation given by on generators
Autoresonance in a Dissipative System
We study the autoresonant solution of Duffing's equation in the presence of
dissipation. This solution is proved to be an attracting set. We evaluate the
maximal amplitude of the autoresonant solution and the time of transition from
autoresonant growth of the amplitude to the mode of fast oscillations.
Analytical results are illustrated by numerical simulations.Comment: 22 pages, 3 figure
Boundary Value Problems with Non-Local Conditions
We describe a new algebra of boundary value problems which contains Lopatinskii elliptic as well as Toeplitz type conditions. These latter are necessary, if an analogue of the Atiyah-Bott obstruction does not vanish. Every elliptic operator is proved to admit up to a stabilisation elliptic conditions of such a kind. Corresponding boundary value problems are then Fredholm in adequate scales of spaces. The crucial novelty consists of the new type of weighted Sobolev spaces which fit well to the nature of pseudodifferential operators
An Explicit Carleman Formula for the Dolbeault Cohomology
ΠΠ·ΡΡΠ°ΡΡΡΡ ΡΠΎΡΠΌΡΠ»Ρ, ΠΊΠΎΡΠΎΡΡΠ΅ Π²ΠΎΡΡΡΠ°Π½Π°Π²Π»ΠΈΠ²Π°ΡΡ ΠΊΠ»Π°ΡΡ ΠΊΠΎΠ³ΠΎΠΌΠΎΠ»ΠΎΠ³ΠΈΠΉ ΠΠΎΠ»ΡΠ±ΠΎ Π² ΠΎΠ±Π»Π°ΡΡΡΡ
ΠΈΠ· Cn ΠΏΠΎ
ΠΈΡ
Π·Π½Π°ΡΠ΅Π½ΠΈΡΠΌ Π½Π° ΠΎΡΠΊΡΡΡΠΎΠΉ ΡΠ°ΡΡΠΈ Π³ΡΠ°Π½ΠΈΡΡ. ΠΠ½ΠΈ Π½Π°Π·ΡΠ²Π°ΡΡΡΡ ΡΠΎΡΠΌΡΠ»Π°ΠΌΠΈ ΠΠ°ΡΠ»Π΅ΠΌΠ°Π½Π° ΠΏΠΎ ΠΈΠΌΠ΅Π½ΠΈ
ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΠΊΠ°, ΠΊΠΎΡΠΎΡΡΠΉ Π½Π°ΡΠ΅Π» ΠΈΡ
ΠΏΠ΅ΡΠ²ΡΠΌ Π² ΠΏΡΠΎΡΡΠ΅ΠΉΡΠ΅ΠΌ ΡΠ»ΡΡΠ°Π΅ Π΄Π»Ρ ΠΏΡΠΎΠ±Π»Π΅ΠΌΡ Π°Π½Π°Π»ΠΈΡΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ
ΠΏΡΠΎΠ΄ΠΎΠ»ΠΆΠ΅Π½ΠΈΡ. ΠΠ»Ρ ΡΡΠ½ΠΊΡΠΈΠΉ ΠΌΠ½ΠΎΠ³ΠΈΡ
ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ½ΡΡ
ΠΏΠ΅ΡΠ΅ΠΌΠ΅Π½Π½ΡΡ
Π½Π°Ρ ΠΏΠΎΠ΄Ρ
ΠΎΠ΄ Π΄Π°Π΅Ρ ΠΏΡΠΎΡΡΠ΅ΠΉΡΡΡ ΡΠΎΡΠΌΡΠ»Ρ Π΄Π»Ρ Π°Π½Π°Π»ΠΈΡΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ ΠΏΡΠΎΠ΄ΠΎΠ»ΠΆΠ΅Π½ΠΈΡ Ρ ΡΠ°ΡΡΠΈ Π³ΡΠ°Π½ΠΈΡΡ. ΠΡΠΎΠ±Π»Π΅ΠΌΠ° ΠΏΡΠΎΠ΄ΠΎΠ»ΠΆΠ΅Π½ΠΈΡ Π΄Π»Ρ ΠΊΠΎΠ³ΠΎΠΌΠΎΠ»ΠΎΠ³ΠΈΠΉ
ΠΠΎΠ»ΡΠ±ΠΎ Π½Π΅ΠΎΠΆΠΈΠ΄Π°Π½Π½ΠΎ ΡΡΡΠΎΠΉΡΠΈΠ²Π° Π² ΠΏΠΎΠ»ΠΎΠΆΠΈΡΠ΅Π»ΡΠ½ΡΡ
ΠΊΠ»Π°ΡΡΠ°Ρ
, Π΅ΡΠ»ΠΈ Π½Π°ΡΠ°Π»ΡΠ½ΡΠ΅ Π΄Π°Π½Π½ΡΠ΅ Π΄Π°ΡΡΡΡ Π½Π° Π²ΠΎΠ³Π½ΡΡΠΎΠΉ ΡΠ°ΡΡΠΈ Π³ΡΠ°Π½ΠΈΡΡ. Π ΡΡΠΎΠΌ ΡΠ»ΡΡΠ°Π΅ Π΄Π°Π΅ΡΡΡ ΡΠΎΡΠ½Π°Ρ ΡΠΎΡΠΌΡΠ»Π° ΠΏΡΠΎΠ΄ΠΎΠ»ΠΆΠ΅Π½ΠΈΡ.We study formulas which recover a Dolbeault cohomology class in a domain of Cn through its values on
an open part of the boundary. These are called Carleman formulas after the mathematician who first
used such a formula for a simple problem of analytic continuation. For functions of several complex
variables our approach gives the simplest formula of analytic continuation from a part of the boundary.
The extension problem for the Dolbeault cohomology proves surprisingly to be stable at positive steps if
the data are given on a concave piece of the boundary. In this case we construct an explicit extension
formula
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