On the existence of Feller semigroups with boundary conditions

On Feller Semigroups Generated by Elliptic Operators with Integro-differential Boundary Conditions

“…However later the results of [4] allowed to obtain different explicit forms of sufficient conditions for transversal case [5][6][7][8][9][10]. The most complete theory in this case was developed by K. Taira [7][8][9]. He considered arbitrary transversal boundary operators up to the second order with nonlocal terms having the form of pseudo-differential operators.…”

“…The existence of a Feller semigroup in general case is unsolved problem (see [8,14]). For the first time, non-transversal case was studied in [15] and then in [16][17][18][19].…”

“…Moreover, unlike [15][16][17][18][19] an elliptic operator and nonlocal conditions can contain an integro-differential operator with a singular measure. In transversal case in contrast to [8,9] we restrict ourselves to the Neumann problem. However our approach allows to consider more general integral term, which is not necessarily a pseudo-differential operator.…”

Нелокальные Эллиптические Задачи С Нелинейными Преобразованиями Переменных Вблизи Точек Сопряжения

“…известно [17]- [19], всякому диффузионному процессу соответствует некоторая по лугруппа Феллера, исследование которой сводится в силу теоремы Хилле-Иосиды к изучению эллиптического оператора с краевыми условиями, содержащими ин теграл по G относительно некоторой неотрицательной борелевской меры [20]. В наиболее сложном случае, когда мера атомарна, нелокальные условия принима ют вид (0.2).…”

Modeling of the diffusion phenomenon in media with membranes by generalized diffusion processes

“…Moreover, the method of solving the parabolic conjugation problems, mentioned above, was applied in [10] in the problem of high-energy astrophysics to construct the classical solution of the special case of a non-stationary kinetic equation describing the acceleration of charged particles in the vicinity of strong shock waves. We also mention works [11,12], where the one-parameter Feller semigroups associated with diffusion processes in a domain with different variants of the Wentzell boundary condition were constructed by methods of functional analysis. Here we can also include the work [9] in which the problem of pasting together two homogeneous diffusion processes on a line was studied.…”