Optimal discrete Neumann energy in a ball and a circular ring

球关于第一Dirichlet和Neumann $\infty-$特征值的一致稳定性

Uniform stability of the ball with respect to the first Dirichlet and Neumann $\infty-$eigenvalues

在这份说明中，我们分析如何扰动的球$\mathfrak{B}_r \subset \mathbb{R}^n$的行为在其第一（非平凡）诺依曼和狄利克雷$\infty-$特征值时，体积约束$\\mathscr{L}^n(Ω) = \mathscr{L}^n(\mathfrak{B}_r)$施加。我们的主要结果指出，$Ω$是一致接近一个球时，它有第一Neumann和Dirichlet特征值接近的球相同体积的$\mathfrak{B}_r$。事实上，我们证明了，如果$$ |λ_{1,\infty}^D(Ω) - λ_{1,\infty}^D(\mathfrak{B}_r)| = δ_1 \quad \text{and} \quad |λ_{1,\infty}^N(Ω) - λ_{1,\infty}^N(\mathfrak{B}_r)| = δ_2, $$，则存在两个球使得$$\mathfrak{B}_{\frac{r}{δ_1 r+1}} \subset Ω\subset \mathfrak{B}_{\frac{r+δ_2 r}{1-δ_2 r}}.$$。此外，我们还得到了关于Dirichlet $\infty-$特征函数稳定性的一个结果。

Heisenberg群$\mathbb H_n$区域上的Neumann边值问题

Neumann boundary value problem in domains of the Heisenberg Group $\mathbb H_n$

讨论了Heisenberg群$\mathbb{H}_n$的Korányi球上Kohn-Laplacian算子的Neumann问题解的存在唯一性.给出了半空间中格林函数（Neumann函数）的显式表示和$\mathbb{H}_n$中Korányi球对圆函数的显式表示.然后将这些函数用于$\mathbb{H}_n$中的上述区域，以解决圆形数据的非齐次Neumann边值问题。

三维单位球中临界 Neumann 问题的变号解

Sign-changing solutions for the critical Neumann problem in the three-dimensional unit ball

本文证明三维单位球中带零Neumann边界条件的临界方程对任意正参数均存在无穷多个非径向变号解，通过内外粘合与有限维约化构造，并利用约化能量展开定位临界点。

Analyticity of Steklov eigenvalues of nearly-circular and nearly-spherical domains

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…

Space quasiconformal composition operators with applications to Neumann eigenvalues

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.

欧几里得球上诺伊曼拉普拉斯算子的波利亚猜想

Pólya's Conjecture for the Neumann Eigenvalues on Euclidean Balls

研究欧几里得球上诺伊曼拉普拉斯算子的波利亚猜想下界，通过严格罗宾比较、变分界等方法，在不同频率范围进行估计，最终证明该猜想，有限有理计算辅助证明。

Sharp inequalities for Neumann eigenvalues on the sphere

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…

一类临界拟线性Neumann问题正解的多重性

Multiplicity of positive solutions for a critical quasilinear Neumann problem

在边界条件下，建立了扩张球和扩张半球中具有临界指数的拟线性Neumann问题的多个正解。

欧氏球的Steklov刚性

Geometric rigidity from the Dirichlet-to-Neumann operator

该研究针对Steklov谱的刚性问题，证明了n≥3维空间中，若有界光滑区域的Steklov谱以足够快速率趋近于球的Steklov谱则该区域必为球，且欧氏球由其Steklov谱唯一确定。

Spectrum of the Neumann-Poincaré operator and optimal estimates for transmission problems in presence of two circular inclusions

We consider the field concentration for the transmission problems of the homogeneous and inhomogeneous conductivity equations in the presence of closely located circular inclusions. We revisit these well-studied problems by exploiting the spectral nature residing behind the phenomenon of the field concentration. The spectral approach enables us not only to recover the existing results with new insights but also to produce significant new results. We show that when relative conductivities of inclusions have different signs, then the gradient of the solution is bounded regardless of the distance…

From Neumann to Steklov and beyond, via Robin: the Weinberger way

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.

球内超临界问题的正约束极小解

Positive constrained minimizers for supercritical problems in the ball

当p足够大时，我们给出了$-Δu+V(|x|) u=u^p$在$B_1$中存在正解的充分条件.这里$B_1$是$R^n$的单位球，n大于或等于2，我们处理Neumann和Dirichlet齐次边界条件。该解是相关能量泛函的约束最小值。作为一个应用，我们表明，在情况下$V(|x|)$是光滑的，非负的和不恒零，和p是足够大的，诺依曼问题总是承认一个解决方案。

Structure of the positive radial solutions for the supercritical Neumann problem $\varepsilon^2Δu-u+u^p=0$ in a ball

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…

Explicit inversion formulas for the two-dimensional wave equation from Neumann traces

In this article we study the problem of recovering the initial data of the two-dimensional wave equation from Neumann measurements on a convex domain with smooth boundary in the plane. We derive an explicit inversion formula of a so-called back-projection type and deduce exact inversion formulas for circular and elliptical domains. In addition, for circular domains, we show that the initial data can also be recovered from any linear combination of its solution and its normal derivative on the boundary. Numerical results of our implementation of the derived inversion formulas are presented demo…

Ashbaugh–Benguria 关于Neumann特征值倒数和的猜想的一个证明

A proof of the Ashbaugh--Benguria conjecture for reciprocal sums of Neumann eigenvalues

证明了在固定体积的光滑有界区域中，球体最小化前m个非零Neumann特征值的倒数和，解决了Ashbaugh–Benguria猜想。

Anisotropic Isoperimetric Inequality outside Euclidean Ball

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.

黎曼曲面上Neumann特征值的锐利调和平均界及高维球面上测地圆盘最优性的反例

Sharp Harmonic-Mean Bounds for Neumann Eigenvalues on Riemannian Surfaces and Counterexamples to Geodesic Ball Optimality on Higher-Dimensional Spheres

本文证明在满足曲率上界和面积条件的共形度量下，Neumann特征值调和平均下界由测地圆盘达到，且等号仅当等距时成立，从而完成Langford--Laugesen猜想。

无增长限制的Neumann问题的径向增长解

Increasing radial solutions for Neumann problems without growth restrictions

研究了球上超线性Neumann问题正增长径向解的存在性。我们不施加任何增长条件的非线性在无穷大，我们的假设允许与光谱的相互作用。在我们的方法中，我们同时使用了拓扑和变分两种方法，并且通过考虑H^1的非负、非减径向函数的锥来克服紧性的不足。

Maximization of the second non-trivial Neumann eigenvalue

In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of R N with prescribed measure m attains its maximum on the union of two disjoint balls of measure m 2. As a consequence, the P{ó}lya conjecture for the Neumann eigenvalues holds for the second eigenvalue and for arbitrary domains. We moreover prove that a relaxed form of the same inequality holds in the context of non-smooth domains and densities.

A priori bounds and multiplicity of positive solutions for $p$-Laplacian Neumann problems with sub-critical growth

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…

Dirichlet--Neumann 括号估计用于非局部算子

Dirichlet--Neumann bracketing for nonlocal operators

本文在有界Lipschitz域上为完全Bernstein函数的非局部算子的Dirichlet特征值建立Dirichlet--Neumann括号估计，证明特征值介于拉普拉斯算子的Neumann与Dirichlet特征值之间，并推广了分数幂情形的已知结果。

一般波数下$n$维球面上亥姆霍兹问题的Dirichlet-to-Neumann算子

Dirichlet-to-Neumann operator for the Helmholtz problem with general wavenumbers on the $n$-sphere

研究外部球域上带Dirichlet边界条件和辐射条件的亥姆霍兹问题，证明DtN算子的波数显式连续性估计，并导出波数无关的迹和Friedrichs型不等式。

Hermite微分方程第一类Neumann eigenvalue的等周估计

Isoperimetric estimates for the first Neumann eigenvalue of Hermite differential equations

在Gauss空间中给出了第一非平凡Neumann特征值$μ_{1}(Ω)$的等周Szegö-Weinberger型不等式，其中$Ω$是$\mathbb{R}^{N}$的可能无界区域.我们的主要结果包括在显示所有的$\mathbb{R}^{N}$对称的起源，有规定的高斯措施，$μ_{1}(Ω)$是最大的当且仅当$Ω$是欧几里得球中心在原点。

关于具有常数Neumann边界数据的Laplacian特征值方程

On Laplacian eigenvalue equation with constant Neumann boundary data

本文研究了具有常数Neumann边界数据的Laplacian特征值方程解的边界行为，利用Bessel函数与初等对称多项式证明了特定区域上的边界极限不等式，并推广了对称破缺结果。

一个p-Laplacian超临界Neumann问题

A p-Laplacian supercritical Neumann problem

对于$p>2$，我们考虑拟线性方程$-Δ_p u+|u|^{p-2}u=g(u)$在$\mathbb R^N$的单位球$B$中，具有齐次Neumann边界条件.对$g$的假设是非常温和的，并允许非线性在Sobolev嵌入的意义上可能是超临界的。通过变分方法，我们证明了一个非常数的，正的，径向非减解的存在性。在$g(u)=|u|^{q-2}u$的情况下，我们检测这些解的渐近行为为$q\to\infty$。

球中Minkowski曲率边值问题的节点解对

Pairs of nodal solutions for a Minkowski-curvature boundary value problem in a ball

利用打靶技术证明了拟线性边值问题$$ \textrm{div} \, \left( \frac{\nabla u}{\sqrt{1-| \nabla u |^2}}\right) + λq(| x |) | u |^{p-1} u = 0, \qquad u|_{\partial \mathcal{B}} = 0,$$，其中$\mathcal{B} \subset \mathbb{R}^N$是一个球，而$p > 1$随着参数$λ> 0$的增长而具有越来越多的节点解对.径向Neumann问题和相应的一维方程的周期问题，以及考虑。

Sharp concentration estimates near criticality for radial sign-changing solutions of Dirichlet and Neumann problems

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…

Dirichlet-to-Neumann算子交换性性质的稳定性估计

Stability Estimates for Commutativity Properties of the Dirichlet-to-Neumann Operator

本文针对Dirichlet-to-Neumann算子与球面拉普拉斯算子交换子的小性，证明了区域接近球的稳定性估计，并推广到共形流形，给出Gohberg引理的现代阐述。

The Third Boundary Value Problem of Potential Theory for the Exterior Ball and the Approximation behaviour of the solution; a Novel Open Problem

The paper is concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet and Neumann problems of harmonic analysis for the three-dimensional unit ball with the corresponding behaviour of the associated ergodic inverse problems for the punched unlimited space. The basis is the theory of semigroups of linear operators mapping a Banach space $X$ into itself. The rates of approximation play a basic role. Another tool is a Drazin-like inverse operator $B$ for the infinitesimal generator $A$ of a semigroup that arises naturally in ergodic theory. This oper…

可能具有超临界非线性项的$p$ -Laplacian Neumann问题

A $p$-Laplacian Neumann problem with a possibly supercritical nonlinearity

我们寻找非常数，积极的，径向非递减的解决方案的拟线性方程$-Δ_p u+u^{p-1}=f(u)$与$p>2$，在单位球$B$的$\mathbb R^N$，受齐次诺依曼边界条件。非线性$f$的假设是非常温和的，并允许它可能是超临界的意义上的Sobolev嵌入。所使用的主要工具是截断法和山路式论证。在纯功率情况下，即，$f(u)=u^{q-1}$，我们检测的极限轮廓的解决方案的问题作为$q\to\infty$。

秩一对称空间中Neumann特征值的耦合互和不等式

Coupled reciprocal-sum inequalities for Neumann eigenvalues in rank-one symmetric spaces

该研究针对秩一对称空间，通过耦合广义特征值问题与正投影Jensen不等式，证明了非紧致型及满足特定条件的紧致型空间中Neumann特征值的完整m项互和尖锐不等式，明确了等号成立条件与相关障碍。

关于Neumann问题振动径向解的Bonheure-Noris-Weth猜想的证明

Proof of the Bonheure-Noris-Weth conjecture on oscillatory radial solutions of Neumann problems

令$B_1$为具有$N \geq 2$的$\mathbb{R}^N$中的单位球。设$f\in C^1([0, \infty), \mathbb{R})$、$f(0)=0$、$f(β) = β, \ f(s)<s\ \text{for}\ s\in (0,β), \ f(s)>s\ \text{for}\ s\in (β, \infty)$和$f'(β)>λ^{r}_k$。D.邦赫尔湾Noris和T. Weth [Ann. Inst. H.庞加莱肛门。Non Linéaire 29（4）（2012）]证明了$k=2$的半线性Neumann问题$$ -Δu+u=f(u) \ \text{in}\ B_1,\ \ \ \ \partial_νu=0 \ \text{on}\ \partial B_1 $$的非减径向正解的存在性，并证明了存在$k$与$β$相交的径向解，条件是$k>2$的$f'(β) >λ^r_k$。在本文中，我们表明，答案是肯定的。

Blow-up and lifespan estimates for solutions to the weakly coupled system of nonlinear damped wave equations outside a ball

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 …

Neumann筛问题和降维：多尺度方法

The Neumann sieve problem and dimensional reduction: a multiscale approach

我们进行多尺度分析的弹性能量的$n$ -维双层薄膜的厚度$2δ$，其层连接通过$ε$ -周期性分布的接触区。将接触区描述为半径为$r\ll ε$（筛孔）的$(n-1)$ -维球的联合体，并假设$δ\ll ε$，我们证明了筛的渐近记忆（如$ε\to 0$）是由一个额外的界面能项的存在见证。此外，我们发现三个不同的极限行为（或制度）依赖于相互消失率的$δ$和$r$。我们还给出了一个明确的非线性电容型公式的界面能密度在每个政权。

次线性Neumann问题最小能量节点解的存在性、唯一延拓性和对称性

Existence, unique continuation and symmetry of least energy nodal solutions to sublinear Neumann problems

考虑次线性问题\begin {equation*} \left \{ \begin{array}{r c l c} -Δu & = &|u|^{q-2}u & \textrm{in }Ω, \\ u_n & = & 0 & \textrm{on }\partialΩ,\end{array} \right .其中\end是有界域，以及$1 \leq q < 2$。对于$q=1$，$|u|^{q-2}u$将用$\sgn(u)$标识。建立了最小能量节点解的变分原理，并研究了其定性性质。特别是，我们表明，他们满足一个独特的连续性（他们的零集是勒贝格忽略不计）。此外，如果$Ω$是径向的，那么最小能量节点解是叶状Schwarz对称的，而当$Ω$是球时，它们是非径向的。$q=1$的情况下，需要特殊的处理，因为正式相关的能量泛函是不可微的，许多参数必须进行调整。

Linear stability estimates for Serrin's problem via a modified implicit function theorem

We examine Serrin's classical overdetermined problem under a perturbation of the Neumann boundary condition. The solution of the problem for a constant Neumann boundary condition exists provided that the underlying domain is a ball. The question arises whether for a perturbation of the constant there still are domains admitting solutions to the problem. Furthermore, one may ask whether a domain that admits a solution for the perturbed problem is unique up to translation and whether it is close to the ball. We develop a new implicit function theorem for a pair of Banach triplets that is applica…

Maximization of Neumann eigenvalues

This paper is motivated by the maximization of the $k$-th eigenvalue of the Laplace operator with Neumann boundary conditions among domains of ${\mathbb R}^N$ with prescribed measure. We relax the problem to the class of (possibly degenerate) densities in ${\mathbb R}^N$ with prescribed mass and prove the existence of an optimal density. For $k=1,2$ the two problems are equivalent and the maximizers are known to be one and two equal balls, respectively. For $k \ge 3$ this question remains open, except in one dimension of the space where we prove that the maximal densities correspond to a union…

High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem

We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing $Γ$ for the boundary of the obstacle, the relevant integral operators map $L^2(Γ)$ to itself. We prove new frequency-explicit bounds on the norms of both the integral operator and its inverse. The bounds on the norm are valid for piecewise-smooth $Γ$ and are sharp up to factors of $\log k$ (where $k$ is the wavenumber), and the bounds on the norm of the inverse are valid for smooth $Γ$ and are observed to be sharp at least when $Γ$ is smooth with …

任意维度下Neumann-Poincaré算子的严格等周不等式

A sharp isoperimetric inequality for the Neumann--Poincaré operator in every dimension

该研究证明了任意维度下Neumann-Poincaré算子的严格等周不等式，在三维情形证实了Miyanishi和Suzuki的1/6猜想，揭示了等号成立的充要条件为区域是球。