We show that the solutions of the three-dimensional critical defocusing nonlinear wave equation with Neumann boundary conditions outside a ball and radial initial data scatter. This is to our knowledge the first result of scattering for a nonlinear wave equation with Neumann boundary conditions. Our proof uses the scheme of concentration-compactness/rigidity introduced by Kenig and Merle, extending it to our setup, together with the so-called channels of energy method to rule out compact-flow solutions. We also obtain, for the focusing equation, the same exact scattering/blow-up dichotomy below the energy of the ground-state as in $\mathbb R^3$.
This paper concerns spectral clusters of the Neumann Laplacian on compact Riemannian manifolds with strictly geodesically concave boundary. We prove an inequality which controls the $L^p$ norms of spectral clusters.
This note introduces a class of nonlinear Neumann problems on balls expanding with the radii tending towards infinity. Performing singular perturbation arguments, we establish the corresponding concentration phenomenon and refined asymptotic expansions with the precise first two order terms. In doing so, we obtain the nontrivial boundary structure of solutions with effects coming from the nonlinear Neumann boundary condition and the boundary mean curvature varied with expanding domains.
In this paper, we treat the inverse problem of determining two time-dependent coefficients appearing in a dissipative wave equation, from measured Neumann boundary observations. We establish in dimension $n\geq 2$, stability estimates with respect to the Dirichlet-to-Neumann map of these coefficients provided that are known outside a cloaking regions. Moreover, we prove that it can be stably recovered in larger subsets of the domain by enlarging the set of data
We provide an elementary derivation of the Green's function for Poisson's equation with Neumann boundary data on balls of arbitrary dimension, which was recently found in [Sadybekov et al., Eurasian Math. J. 7(2):100-105, 2016]. The underlying idea consists of first computing the Green's function for the electroencephalography (EEG) equation (Poisson's equation with dipole right-hand side) and then deriving the Green's function for Poisson's equation from that.
In this note we analyze how perturbations of a ball $\mathfrak{B}_r \subset \mathbb{R}^n$ behaves in terms of their first (non-trivial) Neumann and Dirichlet $\infty-$eigenvalues when a volume constraint $\\mathscr{L}^n(Ω) = \mathscr{L}^n(\mathfrak{B}_r)$ is imposed. Our main result states that $Ω$ is uniformly close to a ball when it has first Neumann and Dirichlet eigenvalues close to the ones for the ball of the same volume $\mathfrak{B}_r$. In fact, we show that, if $$ |λ_{1,\infty}^D(Ω) - λ_{1,\infty}^D(\mathfrak{B}_r)| = δ_1 \quad \text{and} \quad |λ_{1,\infty}^N(Ω) - λ_{1,\infty}^N(\mat…
The purpose of this note is to prove dispersive estimates for the wave equation outside a ball in R^d. If d = 3, we show that the linear flow satisfies the dispersive estimates as in R^3. In higher dimensions d $\ge$ 4 we show that losses in dispersion do appear and this happens at the Poisson spot.
We consider spherically symmetric supercritical focusing wave equations outside a ball. Using mixed analytical and numerical methods, we show that the threshold for blowup is given by a codimension-one stable manifold of the unique static solution with exactly one unstable direction. We analyze in detail the convergence to this critical solution for initial data fine-tuned to the threshold.
In this paper, we prove some exact estimates for the discrete Neumann energy of a ball and a circular ring in Euclidean space for points located on circles. The proofs are based on dissymmetrization and analysis of the asymptotic behavior of the Dirichlet integral of the potential function.
Existence and uniqueness of the solution of the Neumann problem for the Kohn-Laplacian on the Korányi ball of the Heisenberg group $\mathbb{H}_n$ are discussed. Explicit representations of Green's type function (Neumann function) for the half space and Korányi ball in $\mathbb{H}_n$ for circular functions have been obtained. These functions are then used on above regions in $\mathbb{H}_n$ to solve the inhomogeneous Neumann boundary value problem for circular data.
We study the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain in the Euclidean plane. Under certain conditions on the initial curve the flow does not develop any singularity, and it subconverges smoothly to an arc of a circle sitting outside of the given fixed domain and enclosing the same area as the initial curve.
For $1\leq q \leq n-2$, we provide explicit examples to demonstrate non-compactness of the Neumann operator for the Kohn Laplacian acting on $L^2$ $(0,q)$-forms on the unit ball in $(2n+1)$-dimensional Heisenberg space.
In this article we obtain estimates of Neumann eigenvalues of $p$-Laplace operators in a large class of space domains satisfying quasihyperbolic boundary conditions. The suggested method is based on composition operators generated by quasiconformal mappings and their applications to Sobolev-Poincaré-inequalities. By using a sharp version of the inverse Hölder inequality we refine our estimates for quasi-balls, that is, images of balls under quasiconformal mappings of the whole space.
We show that the third eigenvalue of the Neumann Laplacian in hyperbolic space is maximal for the disjoint union of two geodesic balls, among domains of given volume. This extends a recent result by Bucur and Henrot in Euclidean space, while providing a new proof of a key step in their argument
We prove that the second nontrivial Neumann eigenvalue of the Laplace-Beltrami operator on the unit sphere $\mathbb{S}^n \subseteq \mathbb{R}^{n+1}$ is maximized by the union of two disjoint, equal, geodesic balls among all subsets of $\mathbb{S}^n$ of prescribed volume. In fact, the result holds in a stronger version, involving the harmonic mean of the eigenvalues of order $2$ to $n$, and extends to densities. A (surprising) consequence occurs on the maximality of a geodesic ball for the first nontrivial eigenvalue under the volume constraint: the hemisphere inclusion condition of the Ashbaug…
We establish the multiplicity of positive solutions to a quasilinear Neumann problem in expanding balls and hemispheres with critical exponent in the boundary condition.
We prove an sharp anisotropic isoperimetric inequality for a domain outside an Euclidean ball in $\mathbb{R}^n$ for $n\geq 2$. The proof applies the ABP method to a Neumann boundary value problem.
In this paper, we consider the initial-boundary value problems with several fundamental boundary conditions (the Dirichlet/Neumann/Robin boundary condition) for the multi-component system of semi-linear classical damped wave equations outside a ball. By applying a test function approach with a judicious choice of test functions, which approximates the harmonic functions being subject to these boundary conditions on $\partial Ω$, simultaneously we have succeeded in proving the blow-up result in a finite time as well as in catching the sharp upper bound of lifespan estimates for small solutions …
The aim of this article is to prove an "almost" global existence result for some semilinear wave equations in the plane outside a bounded convex obstacle with the Neumann boundary condition.
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 paper is devoted to investigating long time behavior of smooth small data solutions to 3-D quasilinear wave equations outside of compact convex obstacles with Neumann boundary conditions.
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 cu…
Sharp Harmonic-Mean Bounds for Neumann Eigenvalues on Riemannian Surfaces and Counterexamples to Geodesic Ball Optimality on Higher-Dimensional Spheres
We consider the Dirichlet-to-Neumann operator (DNO) on nearly-circular and nearly-spherical domains in two and three dimensions, respectively. Treating such domains as perturbations of the ball, we prove the analyticity of the DNO with respect to the domain perturbation parameter. Consequently, the Steklov eigenvalues are also shown to be analytic in the domain perturbation parameter. To obtain these results, we use the strategy of Nicholls and Nigam (2004); we transform the equation on the perturbed domain to a ball and geometrically bound the Neumann expansion of the transformed Dirichlet-to…
The second eigenvalue of the Robin Laplacian is shown to be maximal for the ball among domains of fixed volume, for negative values of the Robin parameter $α$ in the regime connecting the first nontrivial Neumann and Steklov eigenvalues, and even somewhat beyond the Steklov regime. The result is close to optimal, since the ball is not maximal when $α$ is sufficiently large negative, and the problem admits no maximiser when $α$ is positive.
We study the existence of positive increasing radial solutions for superlinear Neumann problems in the ball. We do not impose any growth condition on the nonlinearity at infinity and our assumptions allow for interactions with the spectrum. In our approach we use both topological and variational arguments, and we overcome the lack of compactness by considering the cone of nonnegative, nondecreasing radial functions of H^1.
We study the polyharmonic Neumann and mixed boundary value problems on the Korányi ball in the Heisenberg group $\H_n$. Necessary and sufficient solvability conditions are obtained for the nonhomogeneous polyharmonic Neumann problem and Neumann-Dirichlet boundary value problems.
In this note we consider a semilinear elliptic equation in $B_R$ with the nonlinear boundary condition, where $B_R$ is a ball of radius $R$. Under certain conditions, we establish a sufficient condition on the non-existence of solutions provided that $R$ is sufficiently large. The main argument is based on applying the asymptotic analysis to the equation with respect to $R\gg1$.
Let $1<p<+\infty$ and let $Ω\subset\mathbb R^N$ be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the type \[ -Δ_p u = f(u), \quad u>0 \mbox{ in } Ω, \quad \partial_νu = 0 \mbox{ on } \partialΩ. \] We suppose that $f(0)=f(1)=0$ and that $f$ is negative between the two zeros and positive after. In case $Ω$ is a ball, we also require that $f$ grows less than the Sobolev-critical power at infinity. We prove a priori bounds of radial solutions, focusing in particular…
We study a defocusing semilinear wave equation, with a power nonlinearity $|u|^{p-1}u$, defined outside the unit ball of $\mathbb{R}^{n}$, $n\ge3$, with Dirichlet boundary conditions. We prove that if $p>n+4$ and the initial data are nonradial perturbations of large radial data, there exists a global smooth solution. The solution is unique among energy class solutions satisfying an energy inequality. The main tools used are the Penrose transform and a pointwise decay estimate for the exterior linear wave equation perturbed with a large, time dependent potential.
We analyze existence, multiplicity and oscillatory behavior of positive radial solutions to a class of quasilinear equations governed by the Lorentz-Minkowski mean curvature operator. The equation is set in a ball or an annulus of $\mathbb R^N$, is subject to homogeneous Neumann boundary conditions, and involves a nonlinear term on which we do not impose any growth condition at infinity. The main tool that we use is the shooting method for ODEs.
The present work devoted to the finding explicit solution of a boundary problem with the Dirichlet-Neumann condition for elliptic equation with singular coefficients in a quarter of ball. For this aim the method of Green's function have been used. Since, found Green's function contains a hypergeometric function of Appell, we had to deal with decomposition formulas, formulas of differentiation and some adjacent relations for this hypergeometric function in order to get explicit solution of the formulated problem.
We consider the focusing wave equation outside a ball of $\R^3$, with Dirichlet boundary condition and a superquintic power nonlinearity. We classify all radial stationary solutions, and prove that all radial global solutions are asymptotically the sum of a stationary solution and a radiation term.
This paper is devoted to the study of stable radial solutions of $-Δu=f(u) \mbox{ in } \mathbb{R}^N\setminus B_1=\{ x\in \mathbb{R}^N : \vert x\vert\geq 1\}$, where $f\in C^1(\mathbb{R})$ and $N\geq 2$. We prove that such solutions are either large [in the sense that $\vert u(r)\vert \geq M r^{-N/2+\sqrt{N-1}+2}\ $, if $2\leq N\leq 9$; $\vert u(r)\vert \geq M \log (r)\ $, if $N=10$; $\vert u(r)-u_\infty \vert \geq M r^{\-N/2+\sqrt{N-1}+2}\ $, if $N\geq 11$; $\forall r\geq r_0$, for some $M>0$, $r_0\geq 1$] or small [in the sense that $\vert u(r)\vert \leq M\log (r)\ $, if $N=2$; $\vert u(r)-u_…
The purpose of this paper is to study weak solutions of a nonlinear Neumann problem considered on a ball. Assuming that the potential is invariant, we consider an orbit of critical points, i.e. we do not assume that critical points are isolated. We apply techniques of the equivariant analysis to examine bifurcations from the orbits of trivial solutions. We formulate sufficient conditions for local and global bifurcations, in terms of the right-hand side of the system and eigenvalues of the Laplace operator. Moreover, we characterise orbits at which the global symmetry-breaking phenomenon occur…
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 $q\to\infty$.
In this paper, we consider the initial-boundary value problem to the compressible Navier-Stokes equations for ideal gases without heat conduction in the half space or outside a fixed ball in $\mathbb R^N$, with $N\geq1$. We prove that any classical solutions $(ρ, u, θ)$, in the class $C^1([0,T]; H^m(Ω))$, $m>[\frac N2]+2$, with bounded from below initial entropy and compactly supported initial density, which allows to touch the physical boundary, must blow-up in finite time, as long as the initial mass is positive. This paper extends the classical reault by Xin [CPAM, 1998], in which the Cauch…
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…
In this note, we find an equivalent boundary integral equation to the classical $\bar{\partial}$-Neumann problem. The new equation contains an equivalent regularity to the global regularity of the $\bar{\partial}$-Neumann problem. We also use the integral equations to observe a new-found rigidity property of the $\bar{\partial}$-Neumann problem on the unit ball in $\mathbb{C}^2$.
By using a shooting technique, we prove that the quasilinear boundary value problem $$ \textrm{div} \, \left( \frac{\nabla u}{\sqrt{1-| \nabla u |^2}}\right) + λq(| x |) | u |^{p-1} u = 0, \qquad u|_{\partial \mathcal{B}} = 0,$$ where $\mathcal{B} \subset \mathbb{R}^N$ is a ball and $p > 1$, has more and more pairs of nodal solutions on growing of the parameter $λ> 0$. The radial Neumann problem and the periodic problem for the corresponding one-dimensional equation are considered, as well.
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 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 study Neumann type boundary value problems for nonlocal equations related to Lévy processes. Since these equations are nonlocal, Neumann type problems can be obtained in many ways, depending on the kind of reflection we impose on the outside jumps. To focus on the new phenomenas and ideas, we consider different models of reflection and rather general non-symmetric Lévy measures, but only simple linear equations in half-space domains. We derive the Neumann/reflection problems through a truncation procedure on the Lévy measure, and then we develop a viscosity solution theory which includes co…
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 $q\to\infty$.
We provide isoperimetric Szegö-Weinberger type inequalities for the first nontrivial Neumann eigenvalue $μ_{1}(Ω)$ in Gauss space, where $Ω$ is a possibly unbounded domain of $\mathbb{R}^{N}$. Our main result consists in showing that among all sets of $\mathbb{R}^{N}$ symmetric about the origin, having prescribed Gaussian measure, $μ_{1}(Ω)$ is maximum if and only if $Ω$ is the euclidean ball centered at the origin.
Let $B_1$ be the unit ball in $\mathbb{R}^N$ with $N \geq 2$. Let $f\in C^1([0, \infty), \mathbb{R})$, $f(0)=0$, $f(β) = β, \ f(s)<s\ \text{for}\ s\in (0,β), \ f(s)>s\ \text{for}\ s\in (β, \infty)$ and $f'(β)>λ^{r}_k$. D. Bonheure, B. Noris and T. Weth [Ann. Inst. H. Poincaré Anal. Non Linéaire 29(4) (2012)] proved the existence of nondecreasing, radial positive solutions of the semilinear Neumann problem $$ -Δu+u=f(u) \ \text{in}\ B_1,\ \ \ \ \partial_νu=0 \ \text{on}\ \partial B_1 $$ for $k=2$, and they conjectured that there exists a radial solution with $k$ intersections with $β$ provided …
