We are interested in the structure of the positive radial solutions of the supercritical Neumann problem $\varepsilon^2Δu-u+u^p=0$ on a unit ball in $\mathbb{R}^N$ , where $N$ is the spatial dimension and $p>p_S:=(N+2)/(N-2)$, $N\ge 3$. We show that there exists a sequence $\{\varepsilon_n^*\}_{n=1}^{\infty}$ ($\varepsilon_1^*>\varepsilon_2^*>\cdots\rightarrow 0$) such that this problem has infinitely many singular solutions $\{(\varepsilon_n^*,U_n^*)\}_{n=1}^{\infty}\subset\mathbb{R}\times (C^2(0,1)\cap C^1(0,1])$ and that the nonconstant regular solutions consist of infinitely many smooth curves in the $(\varepsilon,U(0))$-plane. It is shown that each curve blows up at $\varepsilon_n^*$ and if $p_{\rm{S}}<p<p_{\rm{JL}}$, then each curve has infinitely many turning points around $\varepsilon_n^*$. Here, $p_{\rm{JL}}$ stands for the Joseph-Lundgren exponent. In particular, the problem has infinitely many solutions if $\varepsilon\in\{\varepsilon_n^*\}_{n=1}^{\infty}$. We also show that there exists $\bar{\varepsilon}>0$ such that the problem has no nonconstant regular solution if $\varepsilon>\bar{\varepsilon}$. The main technical tool is the intersection number between the regular and singular solutions.
In this paper we study the Neumann problem\begin{equation*}\begin{cases}-Δu+u=u^p \& \text{ in }B\_1 \\u \textgreater{} 0, \& \text{ in }B\_1 \\\partial\_νu=0 \& \text{ on } \partial B\_1,\end{cases}\end{equation*}and we show the existence of multiple-layer radial solutions as $p\rightarrow+\infty$.
We show the existence and multiplicity of concentrating solutions to a pure Neumann slightly supercritical problem in a ball. This is the first existence result for this kind of problems in the supercritical regime. Since the solutions must satisfy a compatibility condition of zero average, all of them have to change sign. Our proofs are based on a Lyapunov-Schmidt reduction argument which incorporates the zero-average condition using suitable symmetries. Our approach also guarantees the existence and multiplicity of solutions to subcritical Neumann problems in annuli. More general symmetric d…
This paper deals with existence and multiplicity of positive solutions for a quasilinear problem with Neumann boundary conditions, set in a ball. The problem admits at least one constant non-zero solution and it involves a nonlinearity that can be supercritical in the sense of Sobolev embeddings. The main tools used are variational techniques and the shooting method for ODE's. These results are contained in [6,3].
For $p>2$, we consider the quasilinear equation $-Δ_p u+|u|^{p-2}u=g(u)$ in the unit ball $B$ of $\mathbb R^N$, with homogeneous Neumann boundary conditions. The assumptions on $g$ are very mild and allow the nonlinearity to be possibly supercritical in the sense of Sobolev embeddings. We prove the existence of a nonconstant, positive, radially nondecreasing solution via variational methods. In the case $g(u)=|u|^{q-2}u$, we detect the asymptotic behavior of these solutions as $q\to\infty$.
We establish existence of positive non-decreasing radial solutions for a nonlocal nonlinear Neumann problem both in the ball and in the annulus. The nonlinearity that we consider is rather general, allowing for supercritical growth (in the sense of Sobolev embedding). The consequent lack of compactness can be overcome, by working in the cone of non-negative and non-decreasing radial functions. Within this cone, we establish some a priori estimates which allow, via a truncation argument, to use variational methods for proving existence of solutions. As a side result, we prove a strong maximum p…
We look for nonconstant, positive, radially nondecreasing solutions of the quasilinear equation $-Δ_p u+u^{p-1}=f(u)$ with $p>2$, in the unit ball $B$ of $\mathbb R^N$, subject to homogeneous Neumann boundary conditions. The assumptions on the nonlinearity $f$ are very mild and allow it to be possibly supercritical in the sense of Sobolev embeddings. The main tools used are the truncation method and a mountain pass-type argument. In the pure power case, i.e., $f(u)=u^{q-1}$, we detect the limit profile of the solutions of the problems as $q\to\infty$.
In this paper we consider the slightly $L^2$-supercritical gKdV equations $\partial_t u+(u_{xx}+u|u|^{p-1})_x=0$, with the nonlinearity $5<p<5+\varepsilon$ and $0<\varepsilon\ll 1$ . We will prove the existence and stability of a blow-up dynamic with self-similar blow-up rate in the energy space $H^1$ and give a specific description of the formation of the singularity near the blow-up time.
We are concerned with Dirichlet problems of the form $${\mathop{\rm div}\nolimits} (|D u|^{p-2}Du)+f(u)=0\ \mbox{ in }Ω,\qquad u=0\ \mbox{ on }\partialΩ, $$ where $Ω$ is a bounded domain of $\mathbb{R}^n$, $n\ge 2$, $1<p<n$ and $f$ is a continuous function with supercritical growth from the viewpoint of the Sobolev embedding. In particular, if $n=2$ and $γ:[a,b]\to\mathbb{R}^2$ is a smooth curve such that $γ(t_1)\neqγ(t_2)$ for $t_1\neq t_2$, we prove that, for $\varepsilon>0$ small enough, there exists a unique solution of the Dirichlet problem in the domain $Ω=Ω^Γ_\varepsilon=\{(x_1,x_2)\in\…
In this paper we consider the slightly $L^2$-supercritical gKdV equations $\partial_t u+(u_{xx}+u|u|^{p-1})_x=0$, with the nonlinearity $5<p<5+\varepsilon$ and $0<\varepsilon\ll 1$ . In the previous work of the author we know that there exists an stable self-similar blow-up dynamics for slightly $L^2$-supercritical gKdV equations. Such solution can be viewed as solutions with single blow-up point. In this paper we will prove the existence of solutions with multiple blow-up points, and give a description of the formation of the singularity near the blow-up time.
We deal with Dirichlet problems of the form $$ Δu+f(u)=0 \mbox{ in }Ω,\qquad u=0\ \mbox{ on }\partial Ω$$ where $Ω$ is a bounded domain of $\mathbb{R}^n$, $n\ge 3$, and $f$ has supercritical growth from the viewpoint of Sobolev embedding. In particular, we consider the case where $Ω$ is a tubular domain $T_\varepsilon(Γ_k)$ with thickness $\varepsilon>0$ and centre $Γ_k$, a $k$-dimensional, smooth, compact submanifold of $\mathbb{R}^n$. Our main result concerns the case where $k=1$ and $Γ_k$ is contractible in itself. In this case we prove that the problem does not have nontrivial solutions fo…
We consider radial solutions of the slightly subcritical problem $-Δu_\varepsilon = |u_\varepsilon|^{\frac{4}{n-2}-\varepsilon}u_\varepsilon$ either on $\mathbb R^n$ ($n\geq 3$) or in a ball $B$ satisfying Dirichlet or Neumann boundary conditions. In particular, we provide sharp rates and constants describing the asymptotic behavior (as $\varepsilon\to 0$) of all local minima and maxima of $u_\varepsilon$ as well as its derivative at roots. Our proof is done by induction and uses energy estimates, blow-up/normalization techniques, a radial pointwise Pohozaev identity, and some ODE arguments. A…
We provide a sufficient condition for the existence of a positive solution to $-Δu+V(|x|) u=u^p$ in $B_1$, when p is large enough. Here $B_1$ is the unit ball of $R^n$, n greater or equal to 2, and we deal both with Neumann and Dirichlet homogeneous boundary conditions. The solution turns to be a constrained minimum of the associated energy functional. As an application we show that, in case $V(|x|)$ is smooth, nonnegative and not identically zero, and p is sufficiently large, the Neumann problem always admits a solution.
The main purpose of this work is to study uniform regularity estimates for a family of elliptic operators $\{\mathcal{L}_\varepsilon, \varepsilon>0\}$, arising in the theory of homogenization, with rapidly oscillating periodic coefficients. We establish sharp $W^{1,p}$ estimates, Lipschitz estimates, and nontangential maximal function estimates, which are uniform in the parameter $\varepsilon$, on solutions with Neumann boundary conditions in $C^{1,α}$ domains.
In this paper, we aim to investigate the following class of singularly perturbed elliptic problem $$ \left\{ \begin{array}{ll} \displaystyle -\varepsilon^2\triangle {u}+|x|^ηu =|x|^ηf(u)& \mbox{in}\,\, A, u=0 & \mbox{on}\,\, \partial A, \end{array} \right. $$ where $\varepsilon>0$, $η\in\mathbb{R}$, $A=\{x\in\R^{2N}:\,\,0<a<|x|<b\}$, $N\ge2$ and $f$ is a nonlinearity of $C^1$ class with supercritical growth. By a reduction argument, we show that there exists a nodal solution $u_\e$ with exactly two positive and two negative peaks, which concentrate on two different orthogonal spheres of dimens…
We prove uniform parabolic Hölder estimates of De Giorgi-Nash-Moser type for sequences of minimizers of the functionals \[ \mathcal{E}_\varepsilon(W) = \int_0^\infty \frac{e^{- t/\varepsilon}}{\varepsilon} \bigg\{ \int_{\mathbb{R}_+^{N+1}} y^a \left( \varepsilon|\partial_t W|^2 + |\nabla W|^2 \right) dX + \int_{\mathbb{R}^N \times\{0\}} Φ(w) dx \bigg\}dt, \qquad \varepsilon \in (0,1) \] where $a \in (-1,1)$ is a fixed parameter, $\mathbb{R}_+^{N+1}$ is the upper half-space and $dX = dxdy$. As a consequence, we deduce the existence and Hölder regularity of weak solutions to a class of weighted …
Utilizing a new variational principle that allows dealing with problems beyond the usual locally compactness structure, we study problems with a supercritical nonlinearity of the type $ -Δu + u= a(x) f(u)$ in $ Ω$ with $\partial_νu=0$ on $ \partial Ω$. Here $Ω$ is a bounded domain with certain symmetry assumptions. We find positive nontrivial solutions in the case of suitable supercritical nonlinearities $f$ by finding critical points of $I$ where \[ I(u)=\int_Ω\left\{ a(x) F^* \left( \frac{-Δu + u}{a(x)} \right) - a(x) F(u) \right\} dx, \] over the closed convex cone $K_m$ of nonnegative, sym…
A mixed Dirichlet-Neumann problem is regularized with a family of singularly perturbed Neumann-Robin boundary problems, parametrized by $\varepsilon > 0$. Using an asymptotic development by Gamma-convergence, the asymptotic behavior of the solutions to the perturbed problems is studied as $\varepsilon \to 0^+$, recovering classical results in the literature.
Let ${\mathcal O} \subset {\mathbb R}^d$ be a bounded domain with the boundary of class $C^{1,1}$. In $L_2({\mathcal O};{\mathbb C}^n)$, a matrix elliptic second order differential operator ${\mathcal A}_{N,\varepsilon}$ with the Neumann boundary condition is considered. Here $\varepsilon>0$ is a small parameter, the coefficients of ${\mathcal A}_{N,\varepsilon}$ are periodic and depend on ${\mathbf x} /\varepsilon$. There are no regularity assumptions on the coefficients. It is shown that the resolvent $({\mathcal A}_{N,\varepsilon}+λI)^{-1}$ converges in the $L_2({\mathcal O};{\mathbb C}^n)$…
In this paper we study a Neumann problem for the fractional Laplacian, namely \begin{equation}\left\{ \begin{array}{rcll} \varepsilon^{2s}(- Δ)^{s}u + u &=& f(u) \ \ &\mbox{in} \ \ Ω\\ \mathcal{N}_{s}u &=& 0 , \,\, &\text{in} \,\, \mathbb{R}^{N}\backslash Ω\end{array}\right. \end{equation} where $Ω\subset \mathbb{R}^{N}$ is a smooth bounded domain, $N>2s$, $s \in (0,1)$, $\varepsilon > 0$ is a parameter and $\mathcal{N}_{s}$ is the nonlocal normal derivative introduced by Dipierro, Ros-Oton, and Valdinoci. We establish the existence of a nonnegative, non-constant small energy solution $u_{\var…
We consider the following singularly perturbed elliptic problem $$ \varepsilon^2\triangle\tilde{u}-\tilde{u}+\tilde{u}^p=0, \ \tilde{u}>0\quad \mbox{in} \ Ω,\ \ \ \frac{\partial\tilde{u}}{\partial \mathbf{n}}=0 \quad \mbox{on}\ \partialΩ, $$ where $Ω$ is a bounded domain in $\mathbb{R}^3$ with smooth boundary, $\varepsilon$ is a small parameter, $\mathbf{n}$ denotes the inward normal of $ \partialΩ$ and the exponent $p>1$. Let $Γ$ be a hypersurface intersecting $\partialΩ$ in the right angle along its boundary $\partialΓ$ and satisfying a {\em non-degenerate condition}. We establish the existe…
We study a two dimensional collision problem for a rigid solid immersed in a cavity filled with a perfect fluid. We are led to investigate the asymptotic behavior of the Dirichlet energy associated to the solution of a Laplace Neumann problem as the distance $\varepsilon>0$ between the solid and the cavity's bottom tends to zero. Denoting by $α>0$ the tangency exponent at the contact point, we prove that the solid always reaches the cavity in finite time, but with a non zero velocity for $α<2$ (real shock case), and with null velocity for $α\geqslant 2$ (smooth landing case). Our proof is base…
Let $Ω\subset\mathbb{R}^n$ be a bounded domain. We perturb it to a domain $Ω^\varepsilon$ attaching a family of small protuberances with "room-and-passage"-like geometry ($\varepsilon>0$ is a small parameter). Peculiar spectral properties of Neumann problems in so perturbed domains were observed for the first time by R. Courant and D. Hilbert. We study the case, when the number of protuberances tends to infinity as $\varepsilon\to 0$ and they are $\varepsilon$-periodically distributed along a part of $\partialΩ$. Our goal is to describe the behaviour of the spectrum of the operator $\mathcal{A…
In this article, we investigate the spectrum of the Neumann-Poincaré operator associated to a periodic distribution of small inclusions with size $\varepsilon$, and its asymptotic behavior as the parameter $\varepsilon$ vanishes. Combining techniques pertaining to the fields of homogenization and potential theory, we prove that the limit spectrum is composed of the `trivial' eigenvalues $0$ and $1$, and of a subset which stays bounded away from $0$ and $1$ uniformly with respect to $\varepsilon$. This non trivial part is the reunion of the \textit{Bloch spectrum}, accounting for the collective…
Let $\mathcal{O}\subset\mathbb{R}^d$ be a bounded domain of class $C^{2p}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, we study a selfadjoint strongly elliptic operator $A_{N,\varepsilon}$ of order $2p$ given by the expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon) b({\mathbf D})$, $\varepsilon >0$, with the Neumann boundary conditions. Here $g({\mathbf x})$ is a bounded and positive definite $(m\times m)$-matrix-valued function in ${\mathbb R}^d$, periodic with respect to some lattice; $b({\mathbf D})=\sum_{|α|=p} b_α{\mathbf D}^α$ is a differential operator of order $p$ with constant coeffici…
We study the stiff spectral Neumann problem for the Laplace operator in a smooth bounded domain $Ω\subset\mathbb{R}^d$ which is divided into two subdomains: an annulus $Ω_1$ and a core $Ω_0$. The density and the stiffness constants are of order $\varepsilon^{-2m}$ and $\varepsilon^{-1}$ in $Ω_0$, while they are of order $1$ in $Ω_1$. Here $m\in\mathbb{R}$ is fixed and $\varepsilon>0$ is small. We provide asymptotics for the eigenvalues and the corresponding eigenfunctions as $\varepsilon \to 0$ for any $m$. In dimension $2$ the case when $Ω_0$ touches the exterior boudary $\partialΩ$ and $Ω_1$…
We propose and study a concept of renormalized solution to the problem $Δ_p u=0$ in $\mathbb{R}^N_+$, $|\nabla u|^{p-2}u_ν + g(u) = μ$ on $\partial\mathbb{R}^N_+$, where $1<p\leq N$, $N\geq 2$, $\mathbb{R}^N_+=\left\lbrace(x',x_N):x'\in\mathbb{R}^{N-1}, x_N>0\right\rbrace $, $u_ν$ is the normal derivative of $u$, $μ$ is a bounded Radon measure, and $g:\mathbb{R}\rightarrow\mathbb{R}$ is a nonlinear term. We develop stability results and, using the symmetry of the domain, apriori estimates on hyperplanes, and potential methods, we obtain several existence results. In particular, we show existen…
In this paper we study the number of the boundary single peak solutions of the problem {align*} {cases} -\varepsilon^2 Δu + u = u^p, &\text{in}Ωu > 0, &\text{in}Ω\frac{\partial u}{\partial ν} = 0,& \text{on}\partial Ω{cases} {align*} for $\varepsilon$ small and $p$ subcritical. Under some suitable assumptions on the shape of the boundary near a critical point of the mean curvature, we are able to prove exact multiplicity results. Note that the degeneracy of the critical point is allowed.
Let $(M,g)$ and $(K,κ)$ be two Riemannian manifolds of dimensions $m$ and $k ,$ respectively. Let $ω\in C^2(N),$ $ω> 0.$ The warped product $ M\times _ωK$ is the $ (m+k)$-dimensional product manifold $M\times K$ furnished with metric $ g+ω^2 κ.$ We prove that the supercritical problem $$-Δ_{g+ω^2 κ}u+h u=u^{ {m+2\over m-2} \pm\varepsilon},\ u>0,\ \hbox{in}\ (M\times _ωK,g+ω^2 κ)$$ has a solution which concentrate along a $k$-dimensional minimal submanifold $Γ$ of $M\times _ωN$ as the real parameter $\varepsilon$ goes to zero, provided the function $h$ and the sectional curvatures along $Γ$ sat…
In this note we investigate spectral properties of a periodic waveguide $Ω^\varepsilon$ ($\varepsilon$ is a small parameter) obtained from a straight strip by attaching an array of $\varepsilon$-periodically distributed identical protuberances having "room-and-passage" geometry. In the current work we consider the operator $\mathcal{A}^\varepsilon=-ρ^\varepsilonΔ_{Ω^\varepsilon}$, where $Δ_{Ω^\varepsilon}$ is the Neumann Laplacian in $Ω^\varepsilon$, the weight $ρ^\varepsilon$ is equal to $1$ everywhere except the union of the "rooms". We will prove that the spectrum of $\mathcal{A}^\varepsilo…
Let $Ω$ be a domain in $\mathbb{R}^n$, $Γ$ be a hyperplane intersecting $Ω$, $\varepsilon>0$ be a small parameter, and $D_{k,\varepsilon}$, $k=1,2,3\dots$ be a family of small "holes" in $Γ\capΩ$; when $\varepsilon \to 0$, the number of holes tends to infinity, while their diameters tends to zero. Let $\mathscr{A}_\varepsilon$ be the Neumann Laplacian in the perforated domain $Ω_\varepsilon=Ω\setminusΓ_\varepsilon$, where $Γ_\varepsilon=Γ\setminus (\cup_k D_{k,\varepsilon})$ ("sieve"). It is well-known that if the sizes of holes are carefully chosen, $\mathscr{A}_\varepsilon$ converges in the …
We generalize pressure boundary conditions of an $\varepsilon$-Stokes problem. Our $\varepsilon$-Stokes problem connects the classical Stokes problem and the corresponding pressure-Poisson equation using one parameter $\varepsilon>0$. For the Dirichlet boundary condition, it is proven in K. Matsui and A. Muntean (2018) that the solution for the $\varepsilon$-Stokes problem converges to the one for the Stokes problem as $\varepsilon$ tends to 0, and to the one for the pressure-Poisson problem as $\varepsilon$ tends to $\infty$. Here, we extend these results to the Neumann and mixed boundary con…
We consider the generalized Stokes resolvent problem in an infinite layer with Neumann boundary conditions. This problem arises from a free boundary problem describing the motion of incompressible viscous one-phase fluid flow without surface tension in an infinite layer bounded both from above and from below by free surfaces. We derive a new exact solution formula to the generalized Stokes resolvent problem and prove the $\mathscr{R}$-boundedness of the solution operator families with resolvent parameter $λ$ varying in a sector $Σ_{\varepsilon,γ_0}$ for any $γ_0>0$ and $0<\varepsilon<π/2$, whe…
We consider the following fractional semilinear Neumann problem on a smooth bounded domain $Ω\subset\mathbb{R}^n$, $n\geq2$, $$\begin{cases} (-\varepsilonΔ)^{1/2}u+u=u^{p},&\hbox{in}~Ω,\\ \partial_νu=0,&\hbox{on}~\partialΩ,\\ u>0,&\hbox{in}~Ω, \end{cases}$$ where $\varepsilon>0$ and $1<p<(n+1)/(n-1)$. This is the fractional version of the semilinear Neumann problem studied by Lin--Ni--Takagi in the late 80's. The problem arises by considering steady states of the Keller--Segel model with nonlocal chemical concentration diffusion. Using the semigroup language for the extension method and variat…
The article deals with a convergence of the spectrum of the Neumann Laplacian in a periodic unbounded domain $Ω^\varepsilon$ depending on a small parameter $\varepsilon>0$. The domain has the form $Ω^\varepsilon=\mathbb{R}^n\setminus S^\varepsilon$, where $S^\varepsilon$ is an $\varepsilon\mathbb{Z}^n$-periodic family of trap-like screens. We prove that for an arbitrarily large $L$ the spectrum has just one gap in $[0,L]$ when $\varepsilon$ small enough, moreover when $\varepsilon\to 0$ this gap converges to some interval whose edges can be controlled by a suitable choice of geometry of the sc…
We consider the following Liouville-type equation with exponential Neumann boundary condition: $$ -Δ\tilde u = \varepsilon^2 K(x) e^{2\tilde u}, \quad x\in D, \qquad \frac{\partial \tilde u}{\partial n} + 1 = \varepsilon κ(x) e^{\tilde u}, \quad x\in\partial D, $$ where $D\subset \mathbb R^2$ is the unit disc, $\varepsilon^2 K(x)$ and $\varepsilon κ(x)$ stand for the prescribed Gaussian curvature and the prescribed geodesic curvature of the boundary, respectively. We prove the existence of concentration solutions if $κ(x) + \sqrt{K(x)+κ(x)^2}$ ($x\in\partial D$) has a strictly local extremum p…
Let $Ω$ be a Lipschitz domain in $\mathbb R^d$, and let $\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon)\nabla$ be a strongly elliptic operator on $Ω$. We suppose that $\varepsilon$ is small and the function $A$ is Lipschitz in the first variable and periodic in the second, so the coefficients of $\mathcal A^\varepsilon$ are locally periodic and rapidly oscillate. Given $μ$ in the resolvent set, we are interested in finding the rates of approximations, as $\varepsilon\to0$, for $(\mathcal A^\varepsilon-μ)^{-1}$ and $\nabla(\mathcal A^\varepsilon-μ)^{-1}$ in the operator topology o…
In this paper we deal with the equation \[-Δ_p u+|u|^{p-2}u=|u|^{q-2}u\] for $1<p<2$ and $q>p$, under Neumann boundary conditions in the unit ball of $\mathbb R^N$. We focus on the three positive, radial, and radially non-decreasing solutions, whose existence for $q$ large is proved in [13]. We detect the limit profile as $q\to\infty$ of the higher energy solution and show that, unlike the minimal energy one, it converges to the constant $1$. The proof requires several tools borrowed from the theory of minimization problems and accurate a priori estimates of the solutions, which are of indepen…
We consider the spectral Neumann problem for the Laplace operator in an acoustic waveguide $Π_{l}^{\varepsilon}$ obtained from a straight unit strip by a low box-shaped perturbation of size $2l\times\varepsilon,$ where $\varepsilon>0$ is a small parameter. We prove the existence of the length parameter $l_{k}^{\varepsilon}=πk+O\left( \varepsilon\right) $ with any $k=1,2,3,...$ such that the waveguide $Π_{l_{k}^{\varepsilon}}^{\varepsilon }$ supports a trapped mode with an eigenvalue $λ_{k}^{\varepsilon}% =π^{2}-4π^{4}l^{2}\varepsilon^{2}+O\left( \varepsilon^{3}\right) $ embedded into the conti…
In this article, we study the impact of a change in the type of boundary conditions of an elliptic boundary value problem. In the context of the conductivity equation we consider a reference problem with mixed homogeneous Dirichlet and Neumann boundary conditions. Two different perturbed versions of this ``background'' situation are investigated, when (i) The homogeneous Neumann boundary condition is replaced by a homogeneous Dirichlet boundary condition on a ``small'' subset $ω_\varepsilon$ of the Neumann boundary; and when (ii) The homogeneous Dirichlet boundary condition is replaced by a ho…
In this paper we use the method of layer potentials to study $L^2$ boundary value problems in a bounded Lipschitz domain $Ω$ for a family of second order elliptic systems with rapidly oscillating periodic coefficients, arising in the theory of homogenization. Let $\mathcal{L}_\varepsilon=-\text{div}\big(A(\varepsilon^{-1}X)\nabla \big)$. Under the assumption that $A(X)$ is elliptic, symmetric, periodic and Hölder continuous, we establish the solvability of the $L^2$ Dirichlet, regularity, and Neumann problems for $\mathcal{L}_\varepsilon (u_\varepsilon)=0$ in $Ω$ with optimal estimates uniform…
We examine the band-gap structure of the spectrum of the Neumann problem for the Laplace operator in a strip with periodic dense transversal perforation by identical holes of a small diameter $\varepsilon>0$. The periodicity cell itself contains a string of holes at a distance $O(\varepsilon)$ between them. Under assumptions on the symmetry of the holes, we derive and justify asymptotic formulas for the endpoints of the spectral bands in the low-frequency range of the spectrum as $\varepsilon \to 0$. We demonstrate that, for $\varepsilon$ small enough, some spectral gaps are open. The position…
