We consider the equation $- \e^2 \D u + u= u^p$ in $Ω\subseteq \R^N$, where $Ω$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of $\partial Ø$, for $N \geq 3$ and for $k \in \{1, ..., N-2\}$. We impose Neumann boundary conditions, assuming $1<p <\frac{N-k+2}{N-k-2}$ and $\e \to 0^+$. This result settles in full generality a phenomenon previously considered only in the particular case $N = 3$ and $k = 1$.
Given a smooth Riemannian manifold (M,g)we investigate the existence of positive solutions to a singularly perturbed supercritical elliptic equation which concentrate at some submanifold of M. We obtain a posive answer for some manifolds, which include warped products.
In this paper we prove that a singularly perturbed Neumann problem with potentials admits the existence of interior spikes concentrating in maxima and minima of an auxiliary function depending only on the potentials.
We are interested in finding a family of solutions to a singularly perturbed biharmonic equation which has a concentration behavior. The proof is based on variational methods and it is used a weak version of the Ambrosetti-Rabinowitz condition.
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.
We show that the number of low energy solutions of a double singularly perturbed Schroedinger Maxwell system type on a smooth 3 dimensional manifold (M,g) depends on the topological properties of the manifold. The result is obtained via Lusternik Schnirelmann category theory.
We exhibit a singularly perturbed parabolic problems for which the asymptotic behavior can be described by an one-dimensional ordinary differential equation. We estimate the continuity of attractors in the Hausdorff metric by rate of convergence of resolvent operator.
A theorem is proved on the uniform estimation of the residual term of the asymptotic expansion with respect to a small parameter of the solution of the initial problem for a singularly perturbed differential operator weakly nonlinear transport equation in the critical case.
An asymptotic expansion with respect to a small parameter of a singularly perturbed system of hyperbolic equations, describing vibrations of two rigidly connected strings is constructed. Under certain conditions imposed on these problems, the principal term of the asymptotic expansion of the solution is described by the Korteweg-de Vries equation.
We exhibit a class of singularly perturbed parabolic problems which the asymptotic behavior can be described by a system of ordinary differential equation. We estimate the convergence of attractors in the Hausdorff metric by rate of convergence of resolvent operators. Application to spatial homogenization and large diffusion except in a neighborhood of a point will be considered.
For the singularly perturbed system \[Δu_{i,β}=βu_{i,β}\sum_{j\neq i}u_{j,β}^2, \quad 1\leq i\leq N,\] we prove that flat interfaces are uniformly Lipschitz. As a byproduct of the proof we also obtain the optimal lower bound near the flat interfaces, \[\sum_iu_{i,β}\geq cβ^{-1/4}.\]
We show that the number of solutions of a double singularly perturbed Schroedinger Maxwell system on a smooth bounded domain A depends on the topological properties of the domain. In particular if A is non contractible we obtain cat(A) + 1 positive solutions. The result is obtained via Lusternik Schnirelmann category theory
Given a symmetric Riemannian manifold (M, g), we show some results of genericity for non degenerate sign changing solutions of singularly perturbed nonlinear elliptic problems with respect to the parameters: the positive number ε and the symmetric metric g. Using these results we obtain a lower bound on the number of non degenerate solutions which change sign exactly once.
We consider a class of singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities. The dependence on the spatial coordinates comes from the presence of a potential and of a function representing a saturation effect. We investigate the existence of nontrivial nonnegative solutions concentrating around local minima of both the potential and of the saturation function. Necessary conditions to locate the possible concentration points are also given.
We consider a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions in a bounded domain $Ω\subset\R^{n}$ whose boundary has an $(n-2)$-dimensional singularity. Assuming $1<p<\frac{n+2}{n-2}$, we prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.
Asymptotic formulae for Green's functions for the operator $-\GD$ in domains with small holes are obtained. A new feature of these formulae is their uniformity with respect to the independent variables. The cases of multi-dimensional and planar domains are considered.
This paper is concerned with a Neumann type problem for singularly perturbed fractional nonlinear Schrödinger equations with subcritical exponent. For some smooth bounded domain $Ω\subset \mathbf R^n$, our boundary condition is given by \begin{equation*} \int_Ω\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy=0\quad\mbox{for }x\in \mathbf R^n\setminus\barΩ. \end{equation*} We establish existence of nonnegative small energy solutions, and also investigate the integrability of the solutions on $\mathbf R^n$.
We study fully nonlinear singularly perturbed parabolic equations and their limits. We show that solutions are uniformly Lipschitz continuous in space and Hölder continuous in time. For the limiting free boundary problem, we analyse the behaviour of solutions near the free boundary. We show, in particular, that, at each time level, the free boundary is a porous set and, consequently, is of Lebesgue measure zero. For rotationally invariant operators, we also derive the limiting free boundary condition.
The paper deals with the Neumann spectral problem for a singularly perturbed second order elliptic operator with bounded lower order terms. The main goal is to provide a refined description of the limit behaviour of the principal eigenvalue and eigenfunction. Using the logarithmic transformation we reduce the studied problem to additive eigenvalue problem for a singularly perturbed Hamilton-Jacobi equation. Then assuming that the Aubry set of the Hamiltonian consists of a finite number of points or limit cycles situated in the domain or on its boundary, we find the limit of the eigenvalue and …
The paper deals with a Dirichlet spectral problem for a singularly perturbed second order elliptic operator with rapidly oscillating locally periodic coefficients. We study the limit behaviour of the first eigenpair (ground state) of this problem. The main tool in deriving the limit (effective) problem is the viscosity solutions technique for Hamilton-Jacobi equations. The effective problem need not have a unique solution. We study the non-uniqueness issue in a particular case of zero potential and construct the higher order term of the ground state asymptotics.
We study a $q-$analog of a singularly perturbed Cauchy problem with irregular singularity in the complex domain which generalizes a previous result by S. Malek in \cite{malek}. First, we construct solutions defined in open $q-$spirals to the origin. By means of a $q-$Gevrey version of Malgrange-Sibuya theorem we show the existence of a formal power series in the perturbation parameter which turns out to be the $q-$Gevrey asymptotic expansion (of certain type) of the actual solutions.
By means of a penalization scheme due to del Pino and Felmer, we prove the existence of single-peaked solutions for a class of singularly perturbed quasilinear elliptic equations associated with functionals which lack of smoothness. We don't require neither uniqueness assumptions on the limiting autonomous equation nor monotonicity conditions on the nonlinearity. Compared with the semilinear case some difficulties arise and the study of concentration of the solutions needs a somewhat involved analysis in which the Pucci-Serrin variational identity plays an important role.
Asymptotic formulae for Green's kernels $G_ε({\bf x}, {\bf y})$ of various boundary value problems for the Laplace operator are obtained in regularly perturbed domains and certain domains with small singular perturbations of the boundary, as $ε\to 0$. The main new feature of these asymptotic formulae is their uniformity with respect to the independent variables ${\bf x}$ and ${\bf y}$.
In this paper we build models for short-term, mean-term and long-term dynamics of dune and megariple morphodynamics. They are models that are degenerated parabolic equations which are, moreover, singularly perturbed. We, then give an existence and uniqueness result for the short-term and mean-term models. This result is based on a time-space periodic solution existence result for degenerated parabolic equation that we set out. Finally the short-term model is homogenized.
We consider the equation $-ε^{2}Δu + u = u^ {p}$ in a bounded domain $Ω\subset\R^{3}$ with edges. We impose Neumann boundary conditions, assuming $1<p<5$, and prove concentration of solutions at suitable points of $\partialΩ$ on the edges.
In this article we are interested in studying regularity up to the boundary for one-phase singularly perturbed fully nonlinear elliptic problems, associated to high energy activation potentials, namely $$ F(X, \nabla u^{\varepsilon}, D^2 u^{\varepsilon}) = ζ_{\varepsilon}(u^{\varepsilon}) \quad \mbox{in} \quad Ω\subset \R^n $$ where $ζ_{\varepsilon}$ behaves asymptotically as the Dirac measure $δ_{0}$ as $\varepsilon$ goes to zero. We shall establish global gradient bounds independent of the parameter $\varepsilon$.
In this paper we consider models for short-term, mean-term and long-term morphodynamics of dunes and megariples. We give an existence and uniqueness result for long term dynamics of dunes. This result is based on a time-space periodic solution existence result for degenerated parabolic equation that we set out. Finally the mean-term and long-term models are homogenized.
We consider the following singularly perturbed elliptic problem \[ - {\varepsilon ^2}Δu + u = f(u){\text{ in }}Ω,{\text{ }}u > 0{\text{ in }}Ω,{\text{ }}u = 0{\text{ on }}\partial Ω, \] where $Ω$ is a domain in ${\mathbb{R}^N}(N \ge 3)$, not necessarily bounded, with boundary $\partial Ω\in {C^2}$ and the nonlinearity $f$ is of critical growth. In this paper, we construct a family of multi-peak solutions to the equation given above which concentrate around any prescribed finite sets of local maxima of the distance function from the boundary $\partial Ω$.
We consider the Laplace equation in a domain of $\mathbb{R}^n$, $n\ge 3$, with a small inclusion of size $ε$. On the boundary of the inclusion we define a nonlinear nonautonomous transmission condition. For $ε$ small enough one can prove that the problem has solutions. In this paper, we study the local uniqueness of such solutions.
We consider a two-dimensional singularly perturbed transmission problem with two different diffusion coefficients, in a domain with smooth (analytic) boundary. The solution will contain boundary layers only in the part of the domain where the diffusion coefficient is high and interface layers along the interface. Utilizing existing and newly derived regularity results for the exact solution, we design a robust $hp$ finite element method for its approximation. Under the assumption of analytic input data, we show that the method converges at an e{xponential} rate, provided the mesh and polynomia…
We perform a convergence analysis of a discrete-in-time minimization scheme approximating a finite dimensional singularly perturbed gradient flow. We allow for different scalings between the viscosity parameter $\varepsilon$ and the time scale $τ$. When the ratio $\frac{\varepsilon}τ$ diverges, we rigorously prove the convergence of this scheme to a (discontinuous) Balanced Viscosity solution of the quasistatic evolution problem obtained as formal limit, when $\varepsilon\to 0$, of the gradient flow. We also characterize the limit evolution corresponding to an asymptotically finite ratio betwe…
We deal with a singularly perturbed optimal control problem with slow and fast variable depending on a parameter ε. We study the asymptotic, as ε goes to 0, of the corresponding value functions, and show convergence, in the sense of weak semilimits, to sub and supersolution of a suitable limit equation containing the effective Hamiltonian. The novelty of our contribution is that no compactness condition are assumed on the fast variable. This generalization requires, in order to perform the asymptotic proce- dure, an accurate qualitative analysis of some auxiliary equations posed on the space o…
By means of a variational identity of Pohožaev-Pucci-Serrin type for solutions of class $C^1$ recently obtained, we give some necessary conditions for locating the concentration points for a class of quasi-linear elliptic problems in divergence form. More precisely we show that the points where the concentration occurs must be critical, either in a generalized or in the classical sense, for a suitable ground state function.
In this paper, we are concerned with the existence and concentration phenomena of solutions for the following singularly perturbed fractional Schrödinger problem \begin{align*} \varepsilon^{2s}(-Δ)^su+V(x)u=f(u) \ \ \ \mbox{in} \ \ \ \mathbb{R}^N, \end{align*} where $N>2s$ and the nonlinearity $f$ has critical growth. By using the variational approach, we construct a localized bound-state solution concentrating around an isolated component of the positive minimum point of $V$ as $\varepsilon\rightarrow 0$. Our result improves the study made in X. He and W. Zou ({\it Calc. Var. Partial Differen…
In this paper we generalize to arbitrary dimensions a one-dimensional equicoerciveness and $Γ$-convergence result for a second derivative perturbation of Perona-Malik type functionals. Our proof relies on a new density result in the space of special functions of bounded variation with vanishing diffuse gradient part. This provides a direction of investigation to derive approximation for functionals with discontinuities penalized with a "cohesive" energy, that is, whose cost depends on the actual opening of the discontinuity.
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.
In this paper we study the semiclassical limit for the singularly perturbed Choquard equation $$ -\vr^2Δu +V(x)u =\vr^{μ-3}\Big(\int_{\R^3} \frac{Q(y)G(u(y))}{|x-y|^μ}dy\Big)Q(x)g(u) \quad \mbox{in $\R^3$}, $$ where $0<μ<3$, $\vr$ is a positive parameter, $V,Q$ are two continuous real function on $\R^3$ and $G$ is the primitive of $g$ which is of critical growth due to the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on the nonlinearity $g$, we first establish the existence of ground states for the critical Choquard equation with constant coefficients in $\R^3$. Next we esta…
This work is devoted to study of a class of elliptic singular perturbed systems and their singular limit to a phase segregating system. We prove existence and uniqueness and study the asymptotic behaviour with convergence to a limiting problem as the interaction rate tends to infinity. The limiting problem is a free boundary problem such that at each point in the domain at least one of the components is zero which implies simultaneously all components can not coexist. We present a novel method, which provides an explicit solution of limiting problem for special choice of parameters. Moreover, …
Let (M,g) be a smooth connected compact Riemannian manifold of finite dimension n \geq 2 with a smooth boundary \partial M. We consider the problem -ε^2Δ_gu+u=|u|^{p-2}u, u>0 on M, \partial u/ \partialν=0 on \partial M where ν is an exterior normal to \partial M. The number of solutions of this problem depends on the topological properties of the manifold. In particular we consider the Lusternik Schnirelmann category of the boundary.
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…
In this paper we analyse a boundary value problem for the Laplace equation with a nonlinear non-autonomous transmission conditions on the boundary of a small inclusion of size $ε$. We show that the problem has solutions for $ε$ small enough and we investigate the dependence of a specific family of solutions upon $ε$. By adopting a functional analytic approach we prove that the map which takes $ε$ to (suitable restrictions of) the corresponding solution can be represented in terms of real analytic functions.
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 investigate the effective thermal conductivity of a two-phase composite with thermal resistance at the interface. The composite is obtained by introducing into an infinite homogeneous matrix a periodic set of inclusions of a different material. The diameter of each inclusion is assumed to be proportional to a positive real parameter ε. Under suitable assumptions, we show that the effective conductivity can be continued real analytically in the parameter εaround the degenerate value ε=0, in correspondence of which the inclusions collapse to points.
We consider a singularly perturbed fourth-order problem with third-order terms on the unit square. With a formal power series approach, we decompose the solution into solutions of reduced (third-order) problems and various layer parts. The existence of unique solutions for the problem itself and for the reduced third-order problems is also addressed.
Analytic solutions and their formal asymptotic expansions for a family of the singularly perturbed $q-$difference-differential equations in the complex domain are constructed. They stand for a $q-$analog of the singularly perturbed partial differential equations considered in our recent work [A. Lastra, S. Malek, Boundary layer expansions for initial value problems with two complex time variables, submitted 2019]. In the present work, we construct outer and inner analytic solutions of the main equation, each of them showing asymptotic expansions of essentially different nature with respect to …
We study the asymptotic behavior of the solutions related to a family of singularly perturbed partial differential equations in the complex domain. The analytic solutions are asymptotically represented by a formal power series in the perturbation parameter. The geometry of the problem and the nature of the elements involved in it give rise to different asymptotic levels related to the so-called strongly regular sequences. The result leans on a novel version of a multi-level Ramis-Sibuya theorem.
We study the asymptotic behavior of the solutions related to a family of singularly perturbed linear partial differential equations in the complex domain. The analytic solutions obtained by means of a Borel-Laplace summation procedure are represented by a formal power series in the perturbation parameter. Indeed, the geometry of the problem gives rise to a decomposition of the formal and analytic solutions so that a multi-level Gevrey order phenomenon appears. This result leans on a Malgrange-Sibuya theorem in several Gevrey levels.
Singularly perturbed vector nonlinear Schroedinger equations (PVNLS) are investigated. Emphasis is placed upon the relation with their restriction: The singularly perturbed scalar nonlinear Schroedinger equation (PNLS) studied earlier by Li. It turns out that the persistent homoclinic orbit for the PNLS is the only one for the PVNLS, asymptotic to the same saddle.
This work is concerned with a singularly perturbed stochastic nonlinear wave equation with a random dynamical boundary condition. A splitting skill is used to derive the approximating equation of the system in the sense of probability distribution, when the singular perturbation parameter is sufficiently small. The approximating equation is a stochastic parabolic equation when the power exponent of singular perturbation parameter is in $[1/2, 1)$, but a deterministic hyperbolic (wave) equation when the power exponent is in $(1, +\infty)$.
We study a family of singularly perturbed $q-$difference-differential equations in the complex domain. We provide sectorial holomorphic solutions in the perturbation parameter $ε$. Moreover, we achieve the existence of a common formal power series in $ε$ which represents each actual solution, and establish $q-$Gevrey estimates involved in this representation. The proof of the main result rests on a new version of the so-called Malgrange-Sibuya Theorem regarding $q-$Gevrey asymptotics. A particular Dirichlet like series is studied on the way.
