We construct the asymptotics of the sharp constant in the Friedrich-type inequality for functions, which vanish on the small part of the boundary Γ1ɛ. It is assumed that Γ1ɛ consists of (1/δ)n-1 pieces with diameter of order O(ɛδ). In addition, δ=δ(ɛ) and δ→0 as ɛ→0.

This paper is devoted to the asymptotic analysis of functions depending on a small parameter characterizing the microinhomogeneous structure of the domain on which the functions are defined. We derive the Friedrichs inequality for these functions and prove the convergence of solutions to corresponding problems posed in a domain perforated aperiodically along the boundary. Moreover, we use numerical simulation to illustrate the results.

We derive a new three-dimensional Hardy-type inequality for a cube for the class of functions from the Sobolev space $H^1$ having zero trace on small holes distributed periodically along the boundary. The proof is based on a careful analysis of the asymptotic expansion of the first eigenvalue of a related spectral problem and the best constant of the corresponding Friedrichs-type inequality.

In this paper we construct and verify the asymptotic expansion for the spectrum of a boundary-value problem in a unit circle periodically perforated along the boundary. It is assumed that the size of perforation and the distance to the boundary of the circle are of the same smallness. As an application of the obtained results the asymptotic behavior of the best constant in a Friedrichs-type inequality is investigated.

We consider a problem that models fluid flow in a thin domain bounded by two surfaces. One of the surfaces is rough and moving, whereas the other is flat and stationary. The problem involves two small parameters ε and μ that describe film thickness and roughness wavelength, respectively. Depending on the ratio λ = ε/μ, three different flow regimes are obtained in the limit as both of them tend to zero. Time-dependent equations of Reynolds type are obtained in all three cases (Stokes roughness, Reynolds roughness and high-frequency roughness regime). The derivations of the limiting equations are based on formal expansions in the parameters ε and μ.

We study the asymptotic behavior of solutions of the evolution Stokes equation in a thin three-dimensional domain bounded by two moving surfaces in the limit as the distance between the surfaces approaches zero. Using only a priori estimates and compactness it is rigorously verified that the limit velocity field and pressure are governed by the time-dependent Reynolds equation.

We consider a boundary value problem for the Laplace operator in a model domain periodically perforated along the boundary. We assume that the homogeneous Neumann condition is posed on the exterior boundary and the homogeneous Dirichlet condition is posed on the boundary of the cavities. We construct and justify the asymptotic expansions of eigenelements of the boundary value problem.

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.

This paper is devoted to the application of asymptotic analysisfor functions depending on small parameter which characterizes themicroinhomogeneous structure of the domain where the functions aredefined. We derive the Friedrichs inequality for functions set inthe three dimensional domain perforated periodically along apart of the boundary and prove the convergence of solutions of theoriginal problems to the solution of the respective homogenizedproblem in this domain.

We derive and discuss a new two-dimensional weightedHardy-type inequality in a rectangle for the class of functions fromthe Sobolev space $H^1$ vanishing on small alternating pieces of theboundary

In present paper we derive three-dimensional weighted Hardy-typeinequality in a cube for the class of functions from Sobolev space$H^1$ having zero trace on small holes distributed periodicallyalong the boundary

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 are periodically 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.

We consider the Poission problem in a model domain periodically perforatedalong the boundary. It is assumed that on the external boundary the homogenizedNeumann condition is imposed while on the boundary of the cavities the Dirichletcondition is supposed. We construct and justify the asymptotic expansion of thesolution to this problem.

This paper is devoted to the Friedrichs inequality, where the domain isperiodically perforated along the boundary. It is assumed that the functionssatisfy homogeneous Neumann boundary conditions on the outer boundary andthat they vanish on the perforation. In particular, it is proved that thebest constant in the inequality converges to the best constant in aFriedrichs-type inequality as the size of the perforation goes to zero muchfaster than the period of perforation. The limit Friedrichs-type inequalityis valid for functions in the Sobolev space $H^{1}$.