Homogenization and norm resolvent convergence for elliptic operators in a strip perforated along a curve

We consider an infinite planar straight strip perforated by small holes along a curve. In such domain, we consider a general second order elliptic operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic and satisfies rather weak assumptions, we describe all possible homogenized problems. Our main result is the norm resolvent convergence of the perturbed operator to a homogenized one in various operator norms and the estimates for the rate of convergence. On the basis of the norm resolvent convergence, we prove the convergence of the spectrum

Operator estimates for the crushed ice problem

Let $\Delta_{\Omega_\varepsilon}$ be the Dirichlet Laplacian in the domain $\Omega_\varepsilon:=\Omega\setminus\left(\cup_i D_{i \varepsilon}\right)$. Here $\Omega\subset\mathbb{R}^n$ and $\{D_{i \varepsilon}\}_{i}$ is a family of tiny identical holes ("ice pieces") distributed periodically in $\mathbb{R}^n$ with period $\varepsilon$. We denote by $\mathrm{cap}(D_{i \varepsilon})$ the capacity of a single hole. It was known for a long time that $-\Delta_{\Omega_\varepsilon}$ converges to the operator $-\Delta_{\Omega}+q$ in strong resolvent sense provided the limit $q:=\lim_{\varepsilon\to 0} \mathrm{cap}(D_{i\varepsilon}) \varepsilon^{-n}$ exists and is finite. In the current contribution we improve this result deriving estimates for the rate of convergence in terms of operator norms. As an application, we establish the uniform convergence of the corresponding semi-groups and (for bounded $\Omega$) an estimate for the difference of the $k$-th eigenvalue of $-\Delta_{\Omega_\varepsilon}$ and $-\Delta_{\Omega_\varepsilon}+q$. Our proofs relies on an abstract scheme for studying the convergence of operators in varying Hilbert spaces developed previously by the second author.Comment: now 24 pages, 3 figures; some typos fixed and references adde

Asymptotic analysis of a planar waveguide perturbed by a non periodic perforation

We consider a general second order elliptic operator in a planar waveguide perforated by small holes distributed along a curve and subject to classical boundary conditions on the holes. Under weak assumptions on the perforation, we describe all possible homogenized problems

Norm-Resolvent Estimates and Perforated Domains

In this thesis we are concerned with norm-resolvent estimates for unbounded linear operators. The text is structured into four parts. The first two parts contain mathematical preliminaries, reviews of previous work and an introduction into the two results which constitute parts three and four. In the third part we are concerned with the non-normal Schrödinger operator H = −∆+V on L²(Rᵈ), where Re(V(x))≥c|x|²−b for some c,b>0. The spectrum of this operator is discrete and its real part is bounded below by −b. In general, the ε-pseudospectrum of H will have an unbounded component for any ε > 0 and thus will not approximate the spectrum in a global sense. By exploiting the fact that the semigroup exp(−tH) is immediately compact, we show a complementary result, namely that for every δ > 0, R > 0 there exists an ε > 0 such that the ε-pseudospectrum is contained in the union of the half plane {Re(z)>R} and disks of radius δ around the eigenvalues. In particular, the unbounded component of the pseudospectrum escapes towards +∞ as ε decreases. Additionally, we give two examples of non-selfadjoint Schrödinger operators outside of our class and study their pseudospectra in more detail. In Part IV, we prove norm-resolvent convergence, as ε→0, for the operator −∆ in domain perforated ε-periodically, to the limit operator −∆+μ on L²(Ω), where μ∈C is a constant depending on the choice of boundary conditions on the holes (we consider Dirichlet, Neumann and Robin boundary conditions). This is an improvement of previous results by [Cioranescu-Murat(1997)], [Kaizu(1985)], which show strong resolvent convergence. In particular, our result implies Hausdorff convergence of the spectrum of the resolvent for the perforated domain problem

Uniform resolvent convergence for strip with fast oscillating boundary

In a planar infinite strip with a fast oscillating boundary we consider an elliptic operator assuming that both the period and the amplitude of the oscillations are small. On the oscillating boundary we impose Dirichlet, Neumann or Robin boundary condition. In all cases we describe the homogenized operator, establish the uniform resolvent convergence of the perturbed resolvent to the homogenized one, and prove the estimates for the rate of convergence. These results are obtained as the order of the amplitude of the oscillations is less, equal or greater than that of the period. It is shown that under the homogenization the type of the boundary condition can change

Norm-Resolvent Estimates and Perforated Domains

In this thesis we are concerned with norm-resolvent estimates for unbounded linear operators. The text is structured into four parts. The first two parts contain mathematical preliminaries, reviews of previous work and an introduction into the two results which constitute parts three and four. In the third part we are concerned with the non-normal Schrödinger operator H = −∆+V on L²(Rᵈ), where Re(V(x))≥c|x|²−b for some c,b>0. The spectrum of this operator is discrete and its real part is bounded below by −b. In general, the ε-pseudospectrum of H will have an unbounded component for any ε > 0 and thus will not approximate the spectrum in a global sense. By exploiting the fact that the semigroup exp(−tH) is immediately compact, we show a complementary result, namely that for every δ > 0, R > 0 there exists an ε > 0 such that the ε-pseudospectrum is contained in the union of the half plane {Re(z)>R} and disks of radius δ around the eigenvalues. In particular, the unbounded component of the pseudospectrum escapes towards +∞ as ε decreases. Additionally, we give two examples of non-selfadjoint Schrödinger operators outside of our class and study their pseudospectra in more detail. In Part IV, we prove norm-resolvent convergence, as ε→0, for the operator −∆ in domain perforated ε-periodically, to the limit operator −∆+μ on L²(Ω), where μ∈C is a constant depending on the choice of boundary conditions on the holes (we consider Dirichlet, Neumann and Robin boundary conditions). This is an improvement of previous results by [Cioranescu-Murat(1997)], [Kaizu(1985)], which show strong resolvent convergence. In particular, our result implies Hausdorff convergence of the spectrum of the resolvent for the perforated domain problem