Some new Friedrichs-type inequalities in domains with microinhomogeneous structure

Abstract: This Licentiate thesis is devoted to derive and discuss some new Friedrichs-type inequalities for functions, which belong to the Sobolev space $H^1$ in domains with microinhomogeneous structure and which vanish on a part of the boundary. The classical Friedrics inequality holds for functions from the space $mathop{H^{smash 1}}limits^{circ}$ with a constant depending only on the measure of the domain. It is well known that if the function has not zero trace on the whole boundary, but only on a subset of the boundary of a positive measure, then the Friedrichs inequality is still valid. Moreover, in such a case the constant in the inequality increases when the measure of the set where the function vanishes tends to zero. In particular, in this thesis we derive and discuss the corresponding behavior of the constant in our new Friedrichs-type inequalities. In paper A we prove a Friedrichs-type inequality for functions, having zero trace on small pieces of the boundary of a two-dimensional domain, which areperiodically alternating. The total measure of the set, where the function vanishes, tends to zero. It turns out that for this case the constant in the Friedrichs inequality is bounded. Moreover, we describe the precise asymptotics of the constant in the derived Friedrichs inequality as the small parameter, describing the microinhomogeneous structure of the boundary, tends to zero. Paper B is devoted to the asymptotic analysis of functions depending on a small parameter, which characterizes the microinhomogeneous structure of the domain, where the functions are defined. We consider a boundary-value problem in a two-dimensional domain perforated nonperiodically along the boundary in the case when the diameter of circles and the distance between them have the same order. In particular, we prove that the limit of the original problems is a Dirichlet problem. Moreover, we use numerical simulations to illustrate the results. We also derive a new version of the Friedrichs inequality for functions, vanishing on the boundary of the cavities, and prove that the constant in the obtained inequality is close to the constant in the corresponding inequality for functions from $mathop{H^{smash 1}}limits^{circ}$. In paper C we consider a boundary-value problem in a three-dimensional domain, which is periodically perforated along the boundary in the case when the diameter of the holes and the distance between them have the same order. We suppose that the Dirichlet boundary condition holds on the boundary of the cavities. In particular, we derive the limit (homogenized) problem for the original problem. Moreover, we establish strong convergence in $H^1$ for the solutions of the considered problems to the corresponding solution of the limit problem. Moreover, we prove that the eigenelements of the original spectral problems converge to the corresponding eigenelement of the limit spectral problem. We apply these results to obtain that the constant in the derived Friedrichs inequality tends to the constant of the classical Friedrichs inequality for functions from $mathop{H^{smash 1}}limits^{circ}$, when the small parameter describing the size of perforation tends to zero.