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Rainer Mandel

Publications and source records attributed to Rainer Mandel.

At least 19 recordsLinked to original sources
arXiv2025

Ground state solutions to generalized nonlinear wave equations with infinite-dimensional kernel

The present paper is devoted to existence results for time-periodic solutions of generalized nonlinear wave equations in a closed Riemannian manifold M. Our main focus lies on the doubly degenerate setting where the associated generalized wave operator has an infinite dimensional kernel and the nonlinearity may vanish on open subsets of M. To deal with this setting, we apply a direct variational approach based on a new variant of the nonlinear saddle point reduction to the associated Nehari-Pankov set. This allows us to find ground state solutions and to characterize the associated ground state energy by a fairly simple minimax principle.

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arXiv2025

Dual variational methods for time-harmonic nonlinear Maxwell's equations

We prove the existence of infinitely many nontrivial solutions for time-harmonic nonlinear Maxwell's equations on bounded domains and on $\mathbb{R}^3$ using dual variational methods. In the dual setting we apply a new version of the Symmetric Mountain Pass Theorem that does not require the Palais-Smale condition.

math.AP↗
arXiv2023

Block-radial symmetry breaking for ground states of biharmonic NLS

We prove that the biharmonic NLS equation $\Delta^2 u +2\Delta u+(1+\varepsilon)u=|u|^{p-2}u$ in $\mathbb R^d$ has at least $k+1$ different solutions if $\varepsilon>0$ is small enough and $2<p<2_\star^k$, where $2_\star^k$ is an explicit critical exponent arising from the Fourier restriction theory of $O(d-k)\times O(k)$-symmetric functions. This extends the recent symmetry breaking result of Lenzmann-Weth and relies on a chain of strict inequalities for the corresponding Rayleigh quotients associated with distinct values of $k$. We further prove that, as $\varepsilon\to 0^+$, the Fourier transform of each ground state concentrates near the unit sphere and becomes rough in the scale of Sobolev spaces.

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arXiv2023

The Restriction-Extension Operator on Lebesgue spaces with symmetries and applications to PDEs

We prove $L^p$-$L^q$-estimates for the Restriction-Extension operator acting on block-radial functions with the aid of new oscillatory integral estimates and interpolation results in mixed Lorentz spaces. Similar techniques apply to the Limiting Absorption Principles for certain elliptic (pseudo-)differential operators with constant coefficients. In this way we obtain a richer existence theory for Helmholtz-type problems on $\mathbb{R}^d$ with block-radial right hand sides.

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arXiv2022

Inverse medium scattering for a nonlinear Helmholtz equation

We discuss a time-harmonic inverse scattering problem for a nonlinear Helmholtz equation with compactly supported inhomogeneous scattering objects that are described by a nonlinear refractive index in unbounded free space. Assuming the knowledge of a nonlinear far field operator, which maps Herglotz incident waves to the far field patterns of corresponding solutions of the nonlinear scattering problem, we show that the nonlinear index of refraction is uniquely determined. We also generalize two reconstruction methods, a factorization method and a monotonicity method, to recover the support of such nonlinear scattering objects. Numerical results illustrate our theoretical findings.

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arXiv2021

Ground states for Maxwell's equations in nonlocal nonlinear media

In this paper we investigate the existence of ground states and dual ground states for Maxwell's Equations in $\mathbb{R}^3$ in nonlocal nonlinear metamaterials. We prove that several nonlocal models admit ground states in contrast to their local analogues.

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arXiv2021

Dual variational methods for static Nonlinear Maxwell's Equations

We prove the existence of a ground state and infinitely many geometrically distinct solutions for static nonlinear Maxwell's equations on $\mathbb{R}^3$. Our existence result relies on a variant of the Symmetric Mountain Pass Theorem that applies to periodic as well as vanishing nonlinearities. It is applied in a dual variational setting and thus provides an alternative approach with respect to the direct variational method introduced by Mederski.

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arXiv2021

Nonlinear Helmholtz equations with sign-changing diffusion coefficient

In this paper we study nonlinear Helmholtz equations with sign-changing diffusion coefficients on bounded domains. The existence of an orthonormal basis of eigenfunctions is established making use of weak T-coercivity theory. All eigenvalues are proved to be bifurcation points and the bifurcating branches are investigated both theoretically and numerically. In a one-dimensional model example we obtain the existence of infinitely many bifurcating branches that are mutually disjoint, unbounded, and consist of solutions with a fixed nodal pattern.

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arXiv2021

The Stein-Tomas inequality under the effect of symmetries

We prove new Fourier restriction estimates to the unit sphere $S^{d-1}$ on the class of $O(d-k)\times O(k)$-symmetric functions, for every $d\geq 4$ and $2\leq k\leq d-2$. As an application, we establish the existence of maximizers for the endpoint Stein--Tomas inequality within that class. Moreover, we construct examples showing that the range of Lebesgue exponents in our estimates is sharp.

math.FA↗
arXiv2021

Time-harmonic solutions for Maxwell's equations in anisotropic media and Bochner-Riesz estimates with negative index for non-elliptic surfaces

We solve time-harmonic Maxwell's equations in anisotropic, spatially homogeneous media in intersections of $L^p$-spaces. The material laws are time-independent. The analysis requires Fourier restriction-extension estimates for perturbations of Fresnel's wave surface. This surface can be decomposed into finitely many components of the following three types: smooth surfaces with non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvature vanishing along one-dimensional submanifolds, but without flat points, and surfaces with conical singularities. Our estimates are based on new Bochner-Riesz estimates with negative index for non-elliptic surfaces.

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arXiv2020

Dual variational methods for a nonlinear Helmholtz equation with sign-changing nonlinearity

We prove new existence results for a Nonlinear Helmholtz equation with sign-changing nonlinearity of the form $$ - \Delta u - k^{2}u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}(\mathbb{R}^{N}) $$ with $k>0,$ $N \geq 3$, $p \in \left[\left.\frac{2(N+1)}{N-1},\frac{2N}{N-2}\right)\right.$ and $Q \in L^{\infty}(\mathbb{R}^{N})$. Due to the sign-changes of $Q$, our solutions have infinite Morse-Index in the corresponding dual variational formulation.

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arXiv2020

Some non-homogeneous Gagliardo-Nirenberg inequalities and application to a biharmonic non-linear Schr\"odinger equation

We study the standing waves for a fourth-order Schr\"odinger equation with mixed dispersion that minimize the associated energy when the $L^2-$norm (the \textit{mass}) } is kept fixed. We need some non-homogeneous Gagliardo-Nirenberg-type inequalities and we develop a method to prove such estimates that should be useful elsewhere. We prove optimal results on the existence of minimizers in the {\it mass-subcritical } and {\it mass-critical } cases. In the { \it mass supercritical} case we show that global minimizers do not exist, and we investigate the existence of local minimizers. If the mass does not exceed some threshold $ \mu_0 \in (0,+\infty)$, our results on "best" local minimizers are also optimal.

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arXiv2020

A limiting absorption principle for Helmholtz systems and time-harmonic isotropic Maxwell's equations

In this work we investigate the L^p-L^q-mapping properties of the resolvent associated with the time-harmonic isotropic Maxwell operator. As spectral parameters close to the spectrum are also covered by our analysis, we obtain an L^p-L^q-type Limiting Absorption Principle for this operator. Our analysis relies on new results for Helmholtz systems with zero order non-Hermitian perturbations. Moreover, we provide an improved version of the Limiting Absorption Principle for Hermitian (self-adjoint) Helmholtz systems.

math.AP↗
arXiv2020

Variational methods for breather solutions of Nonlinear Wave Equations

We construct infinitely many real-valued, time-periodic breather solutions of power-type nonlinear wave equations. These solutions are obtained from critical points of a dual functional and they are weakly localized in space. Our abstract framework allows to find similar existence results for the Klein-Gordon equation or biharmonic wave equations.

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