arXiv subjects

Lutz Recke

Publications and source records attributed to Lutz Recke.

16 recordsLinked to original sources

arXiv2026

Higher Regularity of Time-Periodic Solutions to Nonautonomous Hyperbolic Problems: Away from Resonances

We study higher regularity and its relation to nonresonant behavior for time-periodic solutions of boundary value problems for one-dimensional linear and nonlinear nonautonomous first-order integro-differential strictly hyperbolic systems. The boundary conditions include integral operators and various types of boundary reflections. We prove that continuous and classical solutions have $C^k$-regularity, provided the coefficients are sufficiently smooth and a suitable number of nonresonance conditions is satisfied. In the linear case, these conditions involve the principal coefficients, the diagonal lower-order coefficients, and the boundary reflection coefficients. In the nonlinear case, they also depend on the nonlinearities and on the solution itself. For nonautonomous hyperbolic systems, higher regularity generally requires additional nonresonance conditions, whose number depends on the desired order of differentiability. These conditions are not only sufficient but, in general, also necessary, revealing a distinctive feature of nonautonomous hyperbolic PDEs. By contrast, in the autonomous case, a single nonresonance condition (if one is needed at all) suffices to obtain arbitrarily high regularity. We also identify a class of nonautonomous hyperbolic problems for which no nonresonance conditions are required. In this case, the higher regularity of solutions is determined solely by the regularity of the data. The main technical tool underlying the proofs is an abstract regularity principle formulated in the setting of vector spaces.

math.AP↗

arXiv2026

An H-convergence-based implicit function theorem for homogenization of nonlinear non-smooth elliptic systems

We consider homogenization of Dirichlet problems for semilinear elliptic systems with non-smooth data. We suppose that the diffusion tensors H-converge if the homogenization parameter tends to zero. Our result is of implicit function theorem type: For small homogenization parameter there exists exactly one weak solution close to a given non-degenerate weak solution to the homogenized problem. For the proofs we use gradient estimates of Meyers (if the space dimension is two) or Morrey (if the diffusion tensors are triangular) type for solutions to linear elliptic systems.

math.AP↗

arXiv2025

A common approach to singular perturbation and homogenization III: Nonlinear periodic homogenization with localized defects

We consider periodic homogenization with localized defects for semilinear elliptic equations and systems of the type $$ \nabla\cdot\Big(\Big(A(x/\varepsilon)+B(x/\varepsilon)\Big)\nabla u(x)+c(x,u(x)\Big)=d(x,u(x)) \mbox{ in } \Omega $$ with Dirichlet boundary conditions. For small $\varepsilon>0$ we show existence of weak solutions $u=u_\varepsilon$ as well as their local uniqueness for $\|u-u_0\|_\infty \approx 0$, where $u_0$ is a given non-degenerate weak solution to the homogenized problem. Moreover, we prove that $\|u_\varepsilon-u_0\|_\infty\to 0$ for $\varepsilon \to 0$, and we estimate the corresponding rate of convergence. Our assumptions are, roughly speaking, as follows: $\Omega$ is a bounded Lipschitz domain, $A$, $B$, $c(\cdot,u)$ and $d(\cdot,u)$ are bounded and measurable, $c(x,\cdot)$ and $d(x,\cdot)$ are $C^1$-smooth, $A$ is periodic, and $B$ is a localized defect. Neither global uniqueness is supposed nor growth restriction for $c(x,\cdot)$ or $d(x,\cdot)$. The main tool of the proofs is an abstract result of implicit function theorem type which permits a common approach to nonlinear singular perturbation and homogenization.

math.AP↗

arXiv2024

Nonlinear non-periodic homogenization: Existence, local uniqueness and estimates

We consider periodic homogenization with localized defects of boundary value problems for semilinear ODE systems of the type $$ \Big((A(x/\varepsilon)+B(x/\varepsilon))u'(x)+c(x,u(x))\Big)'= d(x,u(x)) \mbox{ for } x \in (0,1),\; u(0)=u(1)=0. $$ For small $\varepsilon>0$ we show existence of weak solutions $u=u_\varepsilon$ as well as their local uniqueness for $\|u-u_0\|_\infty \approx 0$, where $u=u_0$ is a given solution to the homogenized problem $$ \Big(A_0u'+c(x,u(x))\Big)'= d(x,u(x)) \mbox{ for } x \in (0,1),\; u(0)=u(1)=0,\; A_0:=\left(\int_0^1A(y)^{-1}dy\right)^{-1} $$ such that the linearized problem $$ \Big(A_0u'+\partial_uc(x,u_0(x))u(x)\Big)'= \partial_ud(x,u_0(x))u(x) \mbox{ for } x \in (0,1),\; u(0)=u(1)=0 $$ does not have weak solutions $u\not=0$. Further, we prove that $\|u_\varepsilon-u_0\|_\infty\to 0$ and, if $c(\cdot,u)\in W^{1,\infty}((0,1);\mathbb{R}^n)$, that $\|u_\varepsilon-u_0\|_\infty=O(\varepsilon)$ for $\varepsilon \to 0$. Moreover, all these statements are true, roughly speaking, uniformly with respect to the localized defects $B$. We assume that $A \in L^\infty(\mathbb{R};\mathbb{M}_n)$ is 1-periodic, $B \in L^\infty(\mathbb{R};\mathbb{M}_n)\cap L^1(\mathbb{R};\mathbb{M}_n)$, $A(y)$ and $A(y)+B(y)$ are positive definite uniformly with respect to $y$, $c(x,\cdot),d(x,\cdot)\in C^1(\mathbb{R}^n;\mathbb{R}^n)$ and $c(\cdot,u),d(\cdot,u) \in L^\infty((0,1);\mathbb{R}^n)$. The main tool of the proofs is an abstract result of implicit function theorem type which has been tailored for applications to nonlinear singular perturbation and homogenization problems.

math.CA↗

arXiv2023

A common approach to singular perturbation and homogenization I: Quasilinear ODE systems

We consider periodic homogenization of boundary value problems for quasilinear second-order ODE systems in divergence form of the type $a(x,x/\varepsilon,u(x),u'(x))'= f(x,x/\varepsilon,u(x),u'(x))$ for $x \in [0,1]$. For small $\varepsilon>0$ we show existence of weak solutions $u=u_\varepsilon$ as well as their local uniqueness for $\|u-u_0\|_\infty \approx 0$, where $u_0$ is a given non-degenerate solution to the homogenized boundary value problem, and we describe the rate of convergence to zero for $\varepsilon \to 0$ of the homogenization error $\|u_\varepsilon-u_0\|_\infty$. In particular, we show that this rate depends on the smoothness of the maps $a(\cdot,y,u,u')$ and $f(\cdot,y,u,u')$. Our assumptions are, roughly speaking, as follows: The maps $a,f:[0,1]\times\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n$ are continuous, the maps $a(x,y,\cdot,\cdot)$ and $f(x,y,\cdot,\cdot)$ are $C^1$-smooth, the maps $a(x,\cdot,u,u')$ and $f(x,\cdot,u,u')$ are 1-periodic, and the maps $a(x,y,u,\cdot)$ are strongly monotone and Lipschitz continuous uniformly with respect to $x$, $y$ and bounded $u$. No global solution uniqueness is supposed. Because $x$ is one-dimensional, no correctors and no cell problems are needed. But, because the problem is nonlinear, we have to care about commutability of homogenization and linearization. The main tool of the proofs is an abstract result of implicit function theorem type which in the past has been applied to singularly perturbed nonlinear ODEs and elliptic and parabolic PDEs and, hence, which permits a common approach to existence and local uniqueness results for singularly perturbed problems and and for homogenization problems.

math.CA↗

arXiv2023

A Common Approach to Singular Perturbation and Homogenization II: Semilinear Elliptic Systems

We consider periodic homogenization of boundary value problems for second-order semilinear elliptic systems in 2D of the type $$ \partial_{x_i}\left(a_{ij}^{\alpha \beta}(x/\varepsilon)\partial_{x_j}u(x)+b_i^\alpha(x,u(x))\right)=b^\alpha(x,u(x)) \mbox{ for } x \in \Omega. $$ For small $\varepsilon>0$ we prove existence of weak solutions $u=u_\varepsilon$ as well as their local uniqueness for $\|u-u_0\|_\infty \approx 0$, where $u_0$ is a given non-degenerate weak solution to the homogenized boundary value problem, and we estimate the rate of convergence to zero of $\|u_\varepsilon-u_0\|_\infty$ for $\varepsilon \to 0$. Our assumptions are, roughly speaking, as follows: The functions $a_{ij}^{\alpha \beta}$ are bounded, measurable and $\mathbb{Z}^2$-periodic, the functions $b_i^\alpha(\cdot,u)$ and $b^\alpha(\cdot,u)$ are bounded and measurable, the functions $b_i^\alpha(x,\cdot)$ and $b^\alpha(x,\cdot)$ are $C^1$-smooth, and $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^2$. Neither global solution uniqueness is supposed nor growth restrictions of $b_i^\alpha(x,\cdot)$ or $b^\alpha(x,\cdot)$ nor higher regularity of $u_0$, and cross-diffusion is allowed. The main tool of the proofs is an abstract result of implicit function theorem type which in the past has been applied to singularly perturbed nonlinear ODEs and elliptic and parabolic PDEs and, hence, which permits a common approach to existence, local uniqueness and error estimates for singularly perturbed problems and and for homogenization problems.

math.AP↗

arXiv2023

Regularity of Time-Periodic Solutions to Autonomous Semilinear Hyperbolic PDEs

This paper concerns autonomous boundary value problems for 1D semilinear hyperbolic PDEs. For time-periodic classical solutions, which satisfy a certain non-resonance condition, we show the following: If the PDEs are continuous with respect to the space variable $x$ and $C^\infty$-smooth with respect to the unknown function $u$, then the solution is $C^\infty$-smooth with respect to the time variable $t$, and if the PDEs are $C^\infty$-smooth with respect to $x$ and $u$, then the solution is $C^\infty$-smooth with respect to $t$ and $x$. The same is true for appropriate weak solutions. Moreover, we show examples of time-periodic functions, which do not satisfy the non-resonance condition, such that they are weak, but not classical solutions, and such that they are classical solutions, but not $C^\infty$-smooth, neither with respect to $t$ nor with respect to $x$, even if the PDEs are $C^\infty$-smooth with respect to $x$ and $u$. For the proofs we use Fredholm solvability properties of linear time-periodic hyperbolic PDEs and a result of E. N. Dancer about regularity of solutions to abstract equivariant equations.

math.AP↗

arXiv2022

Forced Frequency Locking for Semilinear Dissipative Hyperbolic PDEs

This paper concerns the behavior of time-periodic solutions to 1D dissipative autonomous semilinear hyperbolic PDEs under the influence of small time-periodic forcing. We show that the phenomenon of forced frequency locking happens similarly to the analogous phenomena known for ODEs or parabolic PDEs. However, the proofs are essentially more difficult than for ODEs or parabolic PDEs. In particular, non-resonance conditions are needed, which do not have counterparts in the cases of ODEs or parabolic PDEs. We derive a scalar equation which answers the main question of forced frequency locking: Which time shifts of the solution to the unforced equation do survive under which forcing?

math.AP↗

arXiv2021

Bounded and Almost Periodic Solvability of Nonautonomous Quasilinear Hyperbolic Systems

The paper concerns boundary value problems for general nonautonomous first order quasilinear hyperbolic systems in a strip. We construct small global classical solutions, assuming that the right hand sides are small. In the case that all data of the quasilinear problem are almost periodic, we prove that the bounded solution is also almost periodic. For the nonhomogeneous version of a linearized problem, we provide stable dissipativity conditions ensuring a unique bounded continuous solution for any smooth right-hand sides. In the autonomous case, this solution is two times continuously differentiable. In the nonautonomous case, the continuous solution is differentiable under additional dissipativity conditions, which are essential. A crucial ingredient of our approach is a perturbation theorem for general linear hyperbolic systems. One of the technical complications we overcome is the "loss of smoothness" property of hyperbolic PDEs.

math.AP↗

arXiv2020

Hopf Bifurcation for General 1D Semilinear Wave Equations with Delay

We consider boundary value problems for 1D autonomous damped and delayed semilinear wave equations of the type $$ \partial^2_t u(t,x)- a(x,\lambda)^2\partial_x^2u(t,x)= b(x,\lambda,u(t,x),u(t-\tau,x),\partial_tu(t,x),\partial_xu(t,x)), \; x \in (0,1) $$ with smooth coefficient functions $a$ and $b$ such that $a(x,\lambda)>0$ and $b(x,\lambda,0,0,0,0) = 0$ for all $x$ and $\lambda$. We state conditions ensuring Hopf bifurcation, i.e., existence, local uniqueness (up to time shifts), regularity (with respect to $t$ and $x$) and smooth dependence (on $\tau$ and $\lambda$) of small non-stationary time-periodic solutions, which bifurcate from the stationary solution $u=0$, and we derive a formula which determines the bifurcation direction with respect to the bifurcation parameter $\tau$. To this end, we transform the wave equation into a system of partial integral equations by means of integration along characteristics, and then we apply a Lyapunov-Schmidt procedure and a generalized implicit function theorem to this system. The main technical difficulties, which have to be managed, are typical for hyperbolic PDEs (with or without delay): small divisors and the "loss of derivatives" property. We do not use any properties of the corresponding initial-boundary value problem. In particular, our results are true also for negative delays $\tau$.

math.AP↗

arXiv2014

Time-Periodic Second-Order Hyperbolic Equations: Fredholmness, Regularity, and Smooth Dependence

The paper concerns the general linear one-dimensional second-order hyperbolic equation $$ \partial^2_tu - a^2(x,t)\partial^2_xu + a_1(x,t)\partial_tu + a_2(x,t)\partial_xu + a_3(x,t)u=f(x,t), \quad x\in(0,1) $$ with periodic conditions in time and Robin boundary conditions in space. Under a non-resonance condition (formulated in terms of the coefficients $a$, $a_1$, and $a_2$) ruling out the small divisors effect, we prove the Fredholm alternative. Moreover, we show that the solutions have higher regularity if the data have higher regularity and if additional non-resonance conditions are fulfilled. Finally, we state a result about smooth dependence on the data, where perturbations of the coefficient $a$ lead to the known loss of smoothness while perturbations of the coefficients $a_1$, $a_2$, and $a_3$ do not.

math.AP↗

arXiv2014

Solution regularity and smooth dependence for abstract equations and applications to hyperbolic PDEs

In the first part we present a generalized implicit function theorem for abstract equations of the type $F(\lambda,u)=0$. We suppose that $u_0$ is a solution for $\lambda=0$ and that $F(\lambda,\cdot)$ is smooth for all $\lambda$, but, mainly, we do not suppose that $F(\cdot,u)$ is smooth for all $u$. Even so, we state conditions such that for all $\lambda \approx 0$ there exists exactly one solution $u \approx u_0$, that $u$ is smooth in a certain abstract sense, and that the data-to-solution map $\lambda \mapsto u$ is smooth. In the second part we apply the results of the first part to time-periodic solutions of first-order hyperbolic systems of the type $$ \partial_tu_j + a_j(x,\lambda)\partial_xu_j + b_j(t,x,\lambda,u) = 0, \; x\in(0,1), \;j=1,\dots,n $$ with reflection boundary conditions and of second-order hyperbolic equations of the type $$ \partial_t^2u-a(x,\lambda)^2\partial^2_xu+b(t,x,\lambda,u,\partial_tu,\partial_xu)=0, \; x\in(0,1) $$ with mixed boundary conditions (one Dirichlet and one Neumann). There are at least two distinguishing features of these results in comparison with the corresponding ones for parabolic PDEs: First, one has to prevent small divisors from coming up, and we present explicit sufficient conditions for that in terms of $u_0$ and of the data of the PDEs and of the boundary conditions. And second, in general smooth dependence of the coefficient functions $b_j$ and $b$ on $t$ is needed in order to get smooth dependence of the solution on $\lambda$, this is completely different to what is known for parabolic PDEs.

math.AP↗

arXiv2012

Frequency locking by external forcing in systems with rotational symmetry

We study locking of the modulation frequency of a relative periodic orbit in a general $S^1$-equivariant system of ordinary differential equations under an external forcing of modulated wave type. Our main result describes the shape of the locking region in the three-dimensional space of the forcing parameters: intensity, wave frequency, and modulation frequency. The difference of the wave frequencies of the relative periodic orbit and the forcing is assumed to be large and differences of modulation frequencies to be small. The intensity of the forcing is small in the generic case and can be large in the degenerate case, when the first order averaging vanishes. Applications are external electrical and/or optical forcing of selfpulsating states of lasers.

math.DS↗

arXiv2010

Frequency locking of modulated waves

We consider the behavior of a modulated wave solution to an $\mathbb{S}^1$-equivariant autonomous system of differential equations under an external forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the modulation frequency of the modulated wave solution, while the wave frequency of the forcing is supposed to be far from that of the modulated wave solution. We describe the domain in the three-dimensional control parameter space (of frequencies and amplitude of the forcing) where stable locking of the modulation frequencies of the forcing and the modulated wave solution occurs. Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external periodically modulated optical signals.

math.DS↗

arXiv2010

Fredholmness and Smooth Dependence for Linear Time-Periodic Hyperbolic System

This paper concerns $n\times n$ linear one-dimensional hyperbolic systems of the type $$ \partial_tu_j + a_j(x)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x)u_k = f_j(x,t),\; j=1,...,n, $$ with periodicity conditions in time and reflection boundary conditions in space. We state conditions on the data $a_j$ and $b_{jk}$ and the reflection coefficients such that the system is Fredholm solvable. Moreover, we state conditions on the data such that for any right hand side there exists exactly one solution, that the solution survives under small perturbations of the data, and that the corresponding data-to-solution-map is smooth with respect to appropriate function space norms. In particular, those conditions imply that no small denominator effects occur. We show that perturbations of the coefficients $a_j$ lead to essentially different results than perturbations of the coefficients $b_{jk}$, in general. Our results cover cases of non-strictly hyperbolic systems as well as systems with discontinuous coefficients $a_j$ and $b_{jk}$, but they are new even in the case of strict hyperbolicity and of smooth coefficients.

math.AP↗

arXiv2006

Fredholm Alternative for Periodic-Dirichlet Problems for Linear Hyperbolic Systems

This paper concerns hyperbolic systems of two linear first-order PDEs in one space dimension with periodicity conditions in time and reflection boundary conditions in space. The coefficients of the PDEs are supposed to be time independent, but allowed to be discontinuous with respect to the space variable. We construct two scales of Banach spaces (for the solutions and for the right hand sides of the equations, respectively) such that the problem can be modeled by means of Fredholm operators of index zero between corresponding spaces of the two scales.

math.AP↗