Existence of solutions for a higher order non-local equation appearing in crack dynamics

In this paper, we prove the existence of non-negative solutions for a non-local higher order degenerate parabolic equation arising in the modeling of hydraulic fractures. The equation is similar to the well-known thin film equation, but the Laplace operator is replaced by a Dirichlet-to-Neumann operator, corresponding to the square root of the Laplace operator on a bounded domain with Neumann boundary conditions (which can also be defined using the periodic Hilbert transform). In our study, we have to deal with the usual difficulty associated to higher order equations (e.g. lack of maximum principle). However, there are important differences with, for instance, the thin film equation: First, our equation is nonlocal; Also the natural energy estimate is not as good as in the case of the thin film equation, and does not yields, for instance, boundedness and continuity of the solutions (our case is critical in dimension $1$ in that respect)

Физические основы низкотемпературной лазерной абляции биотканей: квазинепрерывный режим облучения

Applied to solving practical problems of extremely accurate laser surgery; the present study examines the conditions of deep, controlled, low-temperature dissection of biological tissues by quasi-continuous laser radiation of the infrared spectrum based on forming a directional wave of environmental enlightment.Применительно к решению практических задач особо точной лазерной хирургии в работе рассматриваются условия глубокого, регулируемого, низкотемпературного рассечения биологических тканей квазинепрерывным лазерным излучением инфракрасного диапазона спектра на основе формирования направленной волны просветления среды

OPTIMIZATION OF LASER SURGERY OF THE SECONDARY CATARACT

Background: This is a theoretical and experimental study of the thermomechanical effects of laser radiation with a wavelength of 1.06 microns on the eye tissues during a laser surgery on the secondary cataract (pupillary membrane). Its relevance is related to the rates of complications after laser surgery of the eye associated with the choice of energy and time parameters of the laser irradiation. These parameters are related to the occurrence of such factors as unstable fields of thermal stress and pressure that are difficult to take into account and indirectly lead to adverse events when removing the secondary cataract. Aim: To minimize side effects and to optimize the existing technology of laser removal of the secondary cataract. Materials and methods: Samples of a  normal lens capsule and of lens capsules with various types of opacities taken during a  cataract surgery, with various optic characteristics and thickness, were treated with an infrared laser (Nd:YAG laser, 1064 nm). We performed morphometric measurements and built up a  theoretical model of the processes in a continuous medium under the effects of impulse laser irradiation. Results: The results of numerical modelling with this newly developed theoretical model are in satisfactory agreement with the experimental data on development of deformities obtained with the autopsy materials (posterior capsule of the human lens with various optical characteristics and thickness, from thin transparent membranes to more thick opaque samples). Conclusion: This study would allow for optimization of the technology of laser treatment for secondary cataracts by changing the irradiation parameters during the procedure