We set up a dual variational framework to detect real standing wave solutions of the nonlinear Helmholtz equation $$ -Δu-k^2 u =Q(x)|u|^{p-2}u,\qquad u \in W^{2,p}(\mathbb{R}^N) $$ with $N\geq 3$, $\frac{2(N+1)}{(N-1)}< p<\frac{2N}{N-2}$ and nonnegative $Q \in L^\infty(\mathbb{R}^N)$. We prove the existence of nontrivial solutions for periodic $Q$ as well as in the case where $Q(x)\to 0$ as $|x|\to\infty$. In the periodic case, a key ingredient of the approach is a new nonvanishing theorem related to an associated integral equation. The solutions we study are superpositions of outgoing and incoming waves and are characterized by a nonlinear far field relation.
We prove new existence results for a Nonlinear Helmholtz equation with sign-changing nonlinearity of the form $$ - Δu - k^{2}u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}(\mathbb{R}^{N}) $$ 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.
We establish weighted $L^p$-Fourier-extension estimates for $O(N-k) \times O(k)$-invariant functions defined on the unit sphere $\mathbb{S}^{N-1}$, allowing for exponents $p$ below the Stein-Tomas critical exponent $\frac{2(N+1)}{N-1}$. Moreover, in the more general setting of an arbitrary closed subgroup $G \subset O(N)$ and $G$-invariant functions, we study the implications of weighted Fourier-extension estimates with regard to boundedness and nonvanishing properties of the corresponding weighted Helmholtz resolvent operator. Finally, we use these properties to derive new existence results f…
We couple the mixed variational problem for the generalized Hodge-Helmholtz or Hodge-Laplace equation posed on a bounded three-dimensional Lipschitz domain with the first-kind boundary integral equation arising from the latter when constant coefficients are assumed in the unbounded complement. Recently developed Calderon projectors for the relevant boundary integral operators are used to perform a symmetric coupling. We prove stability of the coupled problem away from resonant frequencies by establishing a generalized Garding inequality (T-coercivity). The resulting system of equations describ…
This paper considers a pair of coupled nonlinear Helmholtz equations \begin{align*} -Δu - μu = a(x) \left( |u|^\frac{p}{2} + b(x) |v|^\frac{p}{2} \right)|u|^{\frac{p}{2} - 2}u, \end{align*} \begin{align*} -Δv - νv = a(x) \left( |v|^\frac{p}{2} + b(x) |u|^\frac{p}{2} \right)|v|^{\frac{p}{2} - 2}v \end{align*} on $\mathbb{R}^N$ where $\frac{2(N+1)}{N-1} < p < 2^\ast$. The existence of nontrivial strong solutions in $W^{2, p}(\mathbb{R}^N)$ is established using dual variational methods. The focus lies on necessary and sufficient conditions on the parameters deciding whether or not both components…
Stability results for the Helmholtz equations in both deterministic and random periodic structures are proved in this paper. Under the assumption of excluding resonances, by a variational method and Fourier analysis in the energy space, the stability estimate for the Helmholtz equation in a deterministic periodic structure is established. For the stochastic case, by introducing a variable transform, the variational formulation of the scattering problem in a random domain is reduced to that in a definite domain with random medium. Combining the stability result for the deteministic case with re…
We prove well-posedness results and a priori bounds on the solution of the Helmholtz equation $\nabla\cdot(A\nabla u) + k^2 n u = -f$, posed either in $\mathbb{R}^d$ or in the exterior of a star-shaped Lipschitz obstacle, for a class of random $A$ and $n,$ random data $f$, and for all $k>0$. The particular class of $A$ and $n$ and the conditions on the obstacle ensure that the problem is nontrapping almost surely. These are the first well-posedness results and a priori bounds for the stochastic Helmholtz equation for arbitrarily large $k$ and for $A$ and $n$ varying independently of $k$. These…
This paper is concerned with reconstruction issue of some typical inverse problems and consists of three parts. First a framework of the enclosure method for an inverse source problem governed by the Helmholtz equation at a fixed wave number in three dimensions is introduced. It is based on the nonvanishing of the coefficient of the leading profile of an oscillatory integral over a domain having a conical singularity. Second an explicit formula of the coefficient for a domain having a circular cone singularity and its implication under the framework are given. Third, an application under the f…
We study a stationary scattering problem related to the nonlinear Helmholtz equation $-Δu - k^2 u = f(x,u) \ \ \text{in $\mathbb{R}^N$,}$ where $N \ge 3$ and $k>0$. For a given incident free wave $φ\in L^\infty(\mathbb{R}^N)$, we prove the existence of complex-valued solutions of the form $u=φ+u_{\text{sc}}$, where $u_{\text{sc}}$ satisfies the Sommerfeld outgoing radiation condition. Since neither a variational framework nor maximum principles are available for this problem, we use topological fixed point theory and global bifurcation theory to solve an associated integral equation involving …
In this paper, we study nonlinear Helmholtz equations (NLH) $-Δ_{\mathbb{H}^N} u - \frac{(N-1)^2}{4} u -λ^2 u = Γ|u|^{p-2}u$ in $\mathbb{H}^N$, $N\geq 2$ where $Δ_{\mathbb{H}^N}$ denotes the Laplace-Beltrami operator in the hyperbolic space $\mathbb{H}^N$ and $Γ\in L^\infty(\mathbb{H}^N)$ is chosen suitably. Using fixed point and variational techniques, we find nontrivial solutions to (NLH) for all $λ>0$ and $p>2$. The oscillatory behaviour and decay rates of radial solutions is analyzed, with possible extensions to Cartan-Hadamard manifolds and Damek-Ricci spaces. Our results rely on a new Li…
Using a dual variational approach we obtain nontrivial real-valued solutions of the critical nonlinear Helmholtz equation $$ - Δu - k^{2}u = Q(x)|u|^{2^{\ast} - 2}u, \quad u \in W^{2,2^{\ast}}(\mathbb{R}^{N}) $$ for $N\geq 4$, where $2^{\ast} := \frac{2N}{N-2}$. The coefficient $Q \in L^{\infty}(\mathbb{R}^{N})\setminus\{0\}$ is assumed to be nonnegative, asymptotically periodic and to satisfy a flatness condition at one of its maximum points. The solutions obtained are so-called dual ground states, i.e., solutions arising from critical points of the dual functional with the property of having…
In this paper, we study real solutions of the nonlinear Helmholtz equation $$ - Δu - k^2 u = f(x,u),\qquad x\in \R^N $$ satisfying the asymptotic conditions $$ u(x)=O(|x|^{\frac{1-N}{2}}) \quad \text{and} \quad \frac{\partial^2 u}{\partial r^2}(x)+k^2 u(x)) =o(|x|^{\frac{1-N}{2}}) \qquad \text{as $r=|x| \to \infty$.} $$ We develop the variational framework to prove the existence of nontrivial solutions for compactly supported nonlinearities without any symmetry assumptions. In addition, we consider the radial case in which, for a larger class of nonlinearities, infinitely many solutions are sh…
The Cauchy problem for the inhomogeneous Helmholtz equation with non-uniform refraction index is considered. The ill-posedness of this problem is tackled by means of the variational form of mollification. This approach is proved to be consistent, and the proposed numerical simulations are quite promising.
In this paper, we investigate the limiting absorption principle associated to and the well-posedness of the Helmholtz equations with sign changing coefficients which are used to model negative index materials. Using the reflecting technique introduced in \cite{Ng-Complementary}, we first derive Cauchy problems from these equations. The limiting absorption principle and the well-posedness are then obtained via various a priori estimates for these Cauchy problems. There approaches are proposed to obtain the a priori estimates. The first one follows from a priori estimates of elliptic systems equ…
For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior Dirichlet-to-Neumann map on the truncation boundary). Our motivation for proving this result is that a) the finite-element method for computing approximations to solutions of the Helmholtz…
We use the Floquet-Bloch transform to reduce variational formulations of surface scattering problems for the Helmholtz equation from periodic and locally perturbed periodic surfaces to equivalent variational problems formulated on bounded domains. To this end, we establish various mapping properties of that transform between suitable weighted Sobolev spaces on periodic strip-like domains and coupled families of quasiperiodic Sobolev spaces. Our analysis shows in particular that the decay of solutions to surface scattering problems from locally perturbed periodic surfaces is precisely character…
Systems of ordinary differential equations (or dynamical forms in Lagrangian mechanics), induced by embeddings of smooth fibered manifolds over one-dimensional basis, are considered in the class of variational equations. For a given non-variational system, conditions assuring variationality (the Helmholtz conditions) of the induced system with respect to a submanifold of a Euclidean space are studied, and the problem of existence of these "variational submanifolds" is formulated in general and solved for second-order systems. The variational sequence theory on sheaves of differential forms is …
For the problem of diffraction of harmonic scalar waves by a lossless periodic slab scatterer, we analyze field sensitivity with respect to the material coefficients of the slab. The governing equation is the Helmholtz equation, which describes acoustic or electromagnetic fields. The main theorem establishes the variational Frechet derivative of the scattered field measured in the H^1 (root-mean-square-gradient) norm as a function of the material coefficients measured in an L^p (p-power integral) norm, with 2<p<infinity, as long as these coefficients are bounded above and below by positive con…
We study the stability of traveling waves of nonlinear Schrödinger equation with nonzero condition at infinity obtained via a constrained variational approach. Two important physical models are Gross-Pitaevskii (GP) equation and cubic-quintic equation. First, under a non-degeneracy condition we prove a sharp instability criterion for 3D traveling waves of (GP), which had been conjectured in the physical literature. This result is also extended for general nonlinearity and higher dimensions, including 4D (GP) and 3D cubic-quintic equations. Second, for cubic-quintic type sub-critical or critica…
We consider the scattering problem on locally perturbed periodic penetrable dielectric layers, which is formulated in terms of the full vector-valued time-harmonic Maxwell's equations. The right-hand side is not assumed to be periodic. At first, we derive a variational formulation for the electromagnetic scattering problem in a suitable Sobolev space on an unbounded domain and reformulate the problem into a family of bounded domain problems using the Bloch-Floquet transform. For this family we can show the unique existence of the solution by applying a carefully designed Helmholtz decompositio…
