

New Trends in Mathematical and Theoretical Physics
October 7, 2016 17:40, Moscow, MIAN, Gubkina, 8






Generalkind differentialdifference elliptic equations in halfplane: asymptotic properties of solutions
Andrey Muravnik^{ab} ^{a} "Sozvezdie"
^{b} Peoples Friendship University of Russia, Moscow

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Abstract:
The contemporary theory of nonlocal problems (actively developed by various researchers
nowadays) and the general theory of functionaldifferential
equations, i. e., equations containing other operators (apart from differential ones)
acting on the unknown function, are closely related between each
other. Actually, it is reasonably to talk about a unified research area
of mathematical physics, specified by the following circumstance:
unlike the classical theory of equations of mathematical physics, the equation
(boundaryvalue condition) links the values of derivatives of the unknown function
at different points. This qualitative difference generates
various novelties both in research results and in available research
methods. Also, it opens application areas that are not available
for the classical theory.
An important part of the above theory is the theory of partial differentialdifference
equations. To investigate them, one can apply operational methods because translation
operators are Fourier multipliers. For the parabolic case, it is implemented by the author in
J. Math. Sci. (New York), 216 (2016), No 3, 345–496.
The elliptic case (for unbounded domains) is still an almost open
research area.
In the present talk, the Dirichlet problem (with bounded continuous boundaryvalue
functions) in the halfplane
is investigated for the secondorder
differentialdifference equation containing, apart from
differential operators, translation operators with respect to the
variable parallel to the boundary line (the socalled
nonlocal variable). Those nonlocal terns are assumed to be
restricted by the only condition: the real part of the symbol of the operator
acting with respect to the nonlocal variable is bounded from below by a positive constant.
We investigate the solution
of the specified problem, which is smooth outside the boundary line,
and prove its asymptotical closeness to
the unique bounded solution of the same Dirichlet problem for
the partial differential equation obtained from the
original differentialdifference
one by the following way: all translations are assigned to be
equal to zero.
Language: English

