Integral inequalities of Hardy and Friedrichs types with applications to homogenization theory

Abstract: This PhD thesis deals with some new integral inequalities of Hardy and Friedrichs types in domains with microinhomogeneous structure in a neighborhood of the boundary. The thesis consists of six papers (Paper A -- Paper F) and an introduction, which put these papers into a more general frame and which also serves as an overview of this interesting field of mathematics.In papers A -- D we 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 or on boundaries of small sets (cavities). The classical Friedrichs 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 increases when the measure (the harmonic capacity) of the set where the function vanishes, tends to zero. In particular, in this thesis we study the corresponding behavior of the constant in our new Friedrichs-type inequalities. In papers E--F some corresponding Hardy-type inequalities are proved and discussed. More precisely:In paper A we prove a Friedrichs-type inequality for functions, having zero trace on the small pieces of the boundary of a two-dimensional domain, which are periodically alternating. %small pieces of the boundary of a %two-dimensional domain. The total measure of the set, where the function vanishes, tends to zero. It turns out that for this case the capacity is positive and hence 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 characterizing the microinhomogeneous structure of the boundary, tends to zero.Paper B is devoted to the asymptotic analysis of functions depending on the 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 Dirichlet problem is the limit for the original problem. Moreover, we use numerical simulations to illustrate the results. We also derive 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 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 is set 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 problem to the corresponding solutions of the limit problem. Moreover, we prove that the eigenelements of the original spectral problem converge to the eigenelements 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 inequality for functions from $\mathop{H^{\smash 1}}\limits^{\circ},$ when the small parameter describing the size of perforation tends to zero.Paper D deals with the construction of the asymptotic expansions for the first eigenvalue of the boundary-value problem in a perforated domain. This asymptotics gives an asymptotic expansion for the best constant in a corresponding Friedrichs inequality.In paper E we derive and discuss a new two-dimensional weighted Hardy-type inequality in a rectangle for the class of functions from the Sobolev space $H^1$ vanishing on small alternating pieces of the boundary. The dependence of the best constant in the derived inequality on a small parameter describing the size of microinhomogenity, is established.Paper F deals with a three-dimensional weighted Hardy-type inequality in the case when the domain $\Omega$ is bounded and has nontrivial microstructure. It is assumed that the small holes are distributed periodically along the boundary. We derive a weighted Hardy-type inequality for the class of functions from the Sobolev space $H^1$ having zero trace on the small holes under the assumption that a weight function decreases to zero in a neighborhood of the microinhomogenity on the boundary.