Massimiliano Berti, Alberto Maspero, Paolo Ventura
We prove an analyticity result for the Dirichlet-Neumann operator under space periodic boundary conditions in any dimension in an unbounded domain with infinite depth. We derive an analytic bifurcation result of analytic Stokes waves -- i.e. space periodic traveling solutions -- of the water waves equations in deep water.
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We consider the inhomogeneous heat equation on the half-space $\mathbb R_{+}^{d}$ with Neumann boundary conditions. We prove a space-time Gevrey regularity of the solution, with a radius of analyticity uniform up to the boundary of the half-space. We also address the case of homogeneous Robin boundary conditions. Our results generalize the case of homogeneous Dirichlet boundary conditions established by Kukavica and Vicol in [10].
Let $Ω$ be a bounded open subset with $C^{1+κ}$-boundary for some $κ> 0$. Consider the Dirichlet-to-Neumann operator associated to the elliptic operator $- \sum \partial_l ( c_{kl} \, \partial_k ) + V$, where the $c_{kl} = c_{lk}$ are Hölder continuous and $V \in L_\infty(Ω)$ are real valued. We prove that the Dirichlet-to-Neumann operator generates a $C_0$-semigroup on the space $C(\partial Ω)$ which is in addition holomorphic with angle $\fracπ{2}$. We also show that the kernel of the semigroup has Poisson bounds on the complex right half-plane. As a consequence we obtain an optimal holomorp…
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…
We study the Cauchy problem for the quasi-geostrophic equations with the critical dissipation in the two dimensional half space under the homogeneous Dirichlet boundary condition. We show the global existence, the uniqueness and the analyticity of solutions, and the real analyticity up to the boundary is obtained. We will show one of natural ways to estimate the nonlinear term for functions satisfying the Dirichlet boundary condition.
Analyticity of Exponential Dirichlet Series and Applications to the Approximate Controllability of Parabolic Equations
本文研究一类指数 Dirichlet 级数的解析性、幂级数系数及余项估计，并将其用于矩量法，证明线性抛物方程在局部分布或集中控制下的近似能控性。
We prove the time analyticity for weak solutions of inhomogeneous parabolic equations with measurable coefficients in the half space with either the Dirichlet boundary condition or the conormal boundary condition under the assumption that the solution and the source term have the exponential growth of order $2$ with respect to the space variables. We also obtain the time analyticity for bounded mild solutions of the incompressible Navier-Stokes equations in the half space with the Dirichlet boundary condition. Our work is an extension of the recent work by Zhang [Proc. Amer. Math. Soc. 148 (20…
Dirichlet to Neumann Maps for Infinite Quantum Graphs
研究了大量无限量子图上椭圆型方程的狄利克雷问题和Dirichlet到Neumann映射。对于图边界上的连续函数的稠密集，Dirichlet到Neumann映射的值在图边界上的Radon测度中。
According to the Schwarz symmetry principle, every harmonic function vanishing on a real analytic curve has an odd continuation, while a harmonic function satisfying homogeneous Neumann condition has the even continuation. There are different generalizations of the Schwarz symmetry principle. Most of them are dealing with homogeneous conditions and the Dirichlet case. Using a technique of Dirichlet to Neumann and Robin to Neumann operators, we derive reflection formulae for nonhomogeneous Neumann and Robin conditions from a reflection formula subject to a nonhomogeneous Dirichlet condition.
We consider elliptic equations in planar domains with mixed boundary conditions of Dirichlet-Neumann type. Sharp asymptotic expansions of the solutions and unique continuation properties from the Dirichlet-Neumann junction are proved.
In this chapter of the book entitled, "Extending the Theory of Composites to Other Areas of Science" [edited by Graeme W. Milton, 2016] we derive the analyticity properties of the electromagnetic Dirichlet-to-Neumann map for the time-harmonic Maxwell's equations for passive linear multicomponent media. Moreover, we discuss the connection of this map to Herglotz functions for isotropic and anisotropic multicomponent composites.
Lower bounds for nodal sets of Dirichlet and Neumann eigenfunctions
设λ是紧致有界黎曼流形上Laplace-Beltrami算子的Dirichlet或Neumann本征函数.我们证明了节点集{n =0}的大小的下界。
In this paper we consider the inverse problem of determining on a compact Riemannian manifold the metric tensor in the wave equation with Dirichlet data from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the wave equation. We prove in dimension $n\geq 2$ that the knowledge of the Dirichlet-to-Neumann map for the wave equation uniquely determines the metric tensor and we establish logarithm-type stability.
In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the Spectral Fractional Laplacian.
Commutator Estimates for the Dirichlet-to-Neumann Map in Lipschitz Domains
建立了Lipschitz域上一类散度型二阶椭圆方程组Dirichlet-to-Neumann映射的交换子估计.我们的方法是基于Dahlberg的双线性估计。
Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map
我们证明了从Laplace-Beltrami算子波动方程相关的Dirichlet到Neumann映射恢复此类度量的逆问题的一般简单Riemannian度量附近的Hölder型稳定性估计。
A mixed Dirichlet-Neumann problem is regularized with a family of singularly perturbed Neumann-Robin boundary problems, parametrized by $\varepsilon > 0$. Using an asymptotic development by Gamma-convergence, the asymptotic behavior of the solutions to the perturbed problems is studied as $\varepsilon \to 0^+$, recovering classical results in the literature.
Using a distributed representation formula of the Gateaux derivative of the Dirichlet to Neumann map with respect to movements of a polygonal conductivity inclusion, [11], we extend the results obtained in [8] proving global Lipschitz stability for the determination of a polygonal conductivity inclusion embedded in a layered medium from knowledge of the Dirichlet to Neumann map.
We study the inverse problem of identifying a periodic potential perturbation of the Dirichlet Laplacian acting in an infinite cylindrical domain, whose cross section is assumed to be bounded. We prove log-log stable determination of the potential with respect to the partial Dirichlet-to-Neumann map, where the Neumann data is taken on slightly more than half of the boundary of the domain.
In the framework of Hilbert spaces we shall give necessary and sufficient conditions to define a Dirichlet-to-Neumann operator via Dirichlet principle. For singular Dirichlet-to-Neumann operators we will establish Laurent expansion near singularities as well as Mittag--Leffler expansion for the related quadratic form. The established results will be exploited to solve definitively the problem of positivity of the related semigroup in the $L^2$ setting. The obtained results are supported by some examples on Lipschitz domains. Among other results, we shall demonstrate that regularity of the boun…
Some linear integro-differential operators have old and classical representations as the Dirichlet-to-Neumann operators for linear elliptic equations, such as the 1/2-Laplacian or the generator of the boundary process of a reflected diffusion. In this work, we make some extensions of this theory to the case of a \emph{nonlinear} Dirichlet-to-Neumann mapping that is constructed using a solution to a \emph{fully nonlinear} elliptic equation in a given domain, mapping Dirichlet data to its normal derivative of the resulting solution. Here we begin the process of giving detailed information about …
We give formulae that yield an information about the location of an unknown polygonal inclusion having unknown constant conductivity inside a known conductive material having known constant conductivity from a partial knowledge of the Neumann -to-Dirichlet operator.
In this paper we study the dependence of the Hölder estimates on the geometry of a domain with holes for the Neumann problem. For this, we study the Hölder regularity of the solutions to the Dirichlet and Neumann problems in the disk (and in the exterior of the disk), from which we get a relation between harmonic extensions and harmonic functions with prescribed Neumann condition on the boundary of the disk (for both interior and exterior problems).
We study a nonlinear elliptic boundary value problem defined on a smooth bounded domain involving the fractional Laplace operator, a concave-convex powers term together with mixed Dirichlet-Neumann boundary conditions.
Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map
在这篇文章中，我们寻求稳定性估计的反问题，确定在各向异性介质中的波动方程的电位或速度从测量的诺依曼边界观测。这些信息包含在与波动方程相关的动力学狄利克雷-诺依曼映射中。我们证明在尺寸$n\geq 2$的知识的Dirichlet-诺伊曼映射的波动方程唯一地确定了电势，我们证明了霍尔德型稳定性确定的潜力。
In the very influential paper \cite{CS07} Caffarelli and Silvestre studied regularity of $(-Δ)^s$, $0<s<1$, by identifying fractional powers with a certain Dirichlet-to-Neumann operator. Stinga and Torrea \cite{ST10} and Galé, Miana and Stinga \cite{GMS13} gave several more abstract versions of this extension procedure. The purpose of this paper is to study precise regularity properties of the Dirichlet and the Neumann problem in Hilbert spaces. Then the Dirichlet-to-Neumann operator becomes an isomorphism between interpolation spaces and its part in the underlying Hilbert space is exactly the…
In this paper we derive rigorously the derivative of the Dirichlet to Neumann map and of the Neumann to Dirichlet map of the conductivity equation with respect to movements of vertices of triangular conductivity inclusions. We apply this result to formulate an optimization problem based on a shape derivative approach.
We consider the Calderòn problem in an infinite cylindrical domain, whose cross section is a bounded domain of the plane. We prove log-log stability in the determination of the isotropic periodic conductivity coefficient from partial Dirichlet data and partial Neumann boundary observations of the solution.
We study the Dirichlet-to-Neumann map for the stationary linear equation of elasticity in a bounded domain in R d , d $\ge$ 2, with smooth boundary. We show that it can be approximated by a pseudodifferential operator on the boundary with a matrix-valued symbol and we compute the principal symbol modulo conjugation by unitary matrices.
Asymptotic Approximation of the Dirichlet to Neumann Map of High Contrast Conductive Media
我们提出了一个渐近研究的狄利克雷诺依曼映射的高对比度的复合介质与完美的导电夹杂物是接近接触。其结果是一个明确的表征的地图在渐近极限的粒子之间的距离趋于零。
It is well known that both the heat equation with Dirichlet or Neumann boundary conditions are null controlable as soon as the control acts in a non trivial domain (i.e. a set of positive measure, see [10, 11, 12, 1, 6]. In this article, we show that for any couple of initial data (u0, v0) we can achieve the null control for both equations (Dirichlet and Neumann boundary conditions respectively) simultaneously with the same control function for both equations.
Analysis of the heat kernel of the Dirichlet-to-Neumann operator
证明了Dirichlet-to-Neumann算子生成的半群的核$K$的Poisson上界，如果定义域有界且有$C^\infty$ -边界.我们还证明了Poisson界的$K_z$的所有$z$在右半平面和其所有的衍生物。
In this article, we study the boundary inverse problem of determining the aligned magnetic fiaeld appearing in the magnetic Schrödinger equation in a periodic quantum cylindrical waveguide. Provided that the Dirichlet-to-Neumann map of the magnetic Schrödinger equation. We prove a Holder stability estimate with respect to the Dirichlet-to-Neumann map, by means of the geometrical optics solutions of the magnetic Schrodinger equation.
A large class of initial-boundary value problems of linear evolution partial differential equations formulated on the half-line is analyzed via the unified transform method. In particular, explicit formulae are presented for the generalized Dirichlet to Neumann maps. Namely, the determination of the unknown boundary values in terms of an essential set of given data.
Stable determination of coefficients in the dynamical anisotropic Schr{ö}dinger equation from the Dirichlet-to-Neumann map
在本文中，我们感兴趣的是建立稳定性估计的反问题，确定一个紧凑的黎曼流形上的电势或共形因子的薛定谔方程与Dirichlet数据从测量的诺依曼边界观测。这些信息包含在与薛定谔方程相关的动力学狄利克雷-诺依曼映射中。我们证明了在n维大于2的情况下，薛定谔方程的Dirichlet-Neumann映射的知识唯一地决定了电势，并建立了确定电势的Hölder型稳定性估计.我们证明了类似的结果为确定的共形因子接近1。
G-Convergence, Dirichlet to Neumann maps and invisibility
建立了线性椭圆算子G-收敛蕴涵相应Dirichlet到Neumann映射收敛的最优条件。作为一个应用，我们表明，近似隐身各向同性材料[ Greenleaf，A.; Kurylev，Y.; Lassas，M.; Uhlmann，G.近似量子和声学隐身。J. Spectr. Theory（2011），no. 1，27- 80.]独立于来源。
We consider the Dirichlet-to-Neumann operator in strip-like and half-space domains with Lipschitz boundary. It is shown that the quadratic form generated by the Dirichlet-to-Neumann operator controls some sharp homogeneous fractional Sobolev norm. As an application, we prove that the global Lipschitz solutions constructed in \cite{DGN} for the one-phase Muskat problem decays exponentially in time in any Hölder norm $C^α$, $α\in (0, 1)$.
We study the bilinear estimates in the Sobolev spaces with the Dirichlet and the Neumann boundary condition. The optimal regularity is revealed to get such estimates in the half space case, which is related to not only smoothness of functions and but also boundary behavior. The crucial point for the proof is how to handle boundary values of functions and their derivatives.
We deal with the sharp asymptotic behaviour of eigenvalues of elliptic operators with varying mixed Dirichlet-Neumann boundary conditions. In case of simple eigenvalues, we compute explicitly the constant appearing in front of the expansion's leading term. This allows inferring some remarkable consequences for Aharonov-Bohm eigenvalues when the singular part of the operator has two coalescing poles.
We show that the fractional wave operator, which is usually studied in the context of hypersingular integrals but had not yet appeared in mathematical physics, can be constructed as the Dirichlet-to-Neumann map associated with the Klein-Gordon equation in anti-de Sitter spacetimes. Several generalizations of this relation will be discussed too.
In the paper, we establish commutator estimates for the Dirichlet-to-Neumann map of Stokes systems in Lipschitz domains. The approach is based on Dahlberg's bilinear estimates, and the results may be regarded as an extension of [Dahlberg, Poisson semigroups and singular integrals, Proc. Amer. Math. Soc., 97(1986), no.1, 41-48.] and [Shen, Commutator estimates for the Dirichlet-to-Neumann map in Lipschitz domains, Some topics in harmonic analysis and applications, 369C384, Adv. Lect. Math. (ALM)34, Int. Press, Somerville, MA, 2016.] to Stokes systems.
We revisit the stability issue of determining the conductivity at the boundary from the corresponding Dirichlet-to-Neumann map. We discuss both the method based on singular solutions and the one built on the localized oscillating solutions. Our primary objective is not establishing new results on this subject even if the present work contains some new ones. We mainly clarify some points in the existing proofs and make some comments. We also derive some consequences of the stability inequality of the determination of the conductivity at the boundary from the Dirichlet-to-Neumann map.
Based on the $H^2$ existence of the solution, we investigate weighted estimates for a mixed boundary elliptic system in a two-dimensional corner domain, when the contact angle $\om\in(0,π/2)$. This system is closely related to the Dirichlet-Neumann operator in the water-waves problem, and the weight we choose is decided by singularities of the mixed boundary system. Meanwhile, we also prove similar weighted estimates with a different weight for the Dirichlet boundary problem as well as the Neumann boundary problem when $\om\in(0,π)$.
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 consider the Schrödinger operator with a potential q on a disk and the map that associates to q the corresponding Dirichlet to Neumann (DtN) map. We give some numerical and analytical results on the range of this map and its stability, for the particular class of one-step radial potentials.
We examine the stability issue in the inverse problem of determining a scalar potential appearing in the stationary Schr{ö}dinger equation in a bounded domain, from a partial elliptic Dirichlet-to-Neumann map. Namely, the Dirichlet data is imposed on the shadowed face of the boundary of the domain and the Neumann data is measured on its illuminated face. We establish a log log stability estimate for the L2-norm (resp. the H minus 1-norm) of bounded (resp. L2) potentials whose difference is lying in any Sobolev space of order positive order.
Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map
当能级/频率增长时，我们导出了一些边界，这些边界可以被看作是从Dirichlet-Neumann映射中恢复薛定谔方程中的势系数的问题的稳定性增加的证据.这些界限保持在某些先验的规则性约束下的未知系数。证明使用复杂和有界的复杂几何光学解决方案。
We consider smooth solutions to the Monge-Amp`ere equation subject to mixed boundary conditions on annular domains. We establish global $C^2$ estimates when the boundary of the domain consists of two smooth strictly convex closed hypersurfaces.
We study the Neumann and Dirichlet problems for the total variation flow in metric measure spaces. We prove existence and uniqueness of weak solutions and study their asymptotic behaviour. Furthermore, in the Neumann problem we provide a notion of solutions which is valid for $L^1$ initial data, as well as prove their existence and uniqueness. Our main tools are the first-order linear differential structure due to Gigli and a version of the Gauss-Green formula.
On Poisson operators and Dirichlet-Neumann maps in H^s for divergence form elliptic operators with Lipschitz coefficients
考虑$\R^{d+1}$中系数与一个变量无关的散度型二阶一致椭圆算子.在利普希茨条件下，我们刻画了Poisson算子和Dirichlet-Neumann映射在索伯列夫空间$s\in [0,1]$中的定义域.此外，我们还证明了一个因子分解公式的椭圆算子的泊松算子。
Homogenization of Green and Neumann Functions
对于一类具有快速振荡周期系数的二阶椭圆算子族，利用Dirichlet和Neumann校正，研究了Green函数和Neumann函数的渐近性态.作为结果，我们得到的泊松核和Dirichlet-到-Neumann映射的渐近展开式以及在$W^{1,p}$的Dirichlet或Neumann边界条件的解决方案的近最优收敛速度。
We consider the semilinear wave equation $\Box_g u+a u^4=0$, $a\neq 0$, on a Lorentzian manifold $(M,g)$ with timelike boundary. We show that from the knowledge of the Dirichlet-to-Neumann map one can recover the metric $g$ and the coefficient $a$ up to natural obstructions. Our proof rests on the analysis of the interaction of distorted plane waves together with a scattering control argument, as well as Gaussian beam solutions.
Boundary value problems for second-order elliptic equations in divergence form, whose nonlinearity is governed by a convex function of non-necessarily power type, are considered. The global boundedness of their solutions is established under boundary conditions of Dirichlet, or Neumann, or Robin type. A decisive role in the results is played by optimal forms of Orlicz-Sobolev embeddings and boundary trace embeddings, which allow for critical growths of the coefficients.
On the space-time analyticity of the Keller-Segel-Navier-Stokes system
本文研究周期条带上耦合Keller-Segel-Navier-Stokes系统，通过能量方法证明了解直到边界的同步时空解析性。
We study the semi-classical microlocal structure of the Dirichlet-to-Neumann map for an arbitrary compact Riemannian manifold with a non-empty smooth boundary. We build a new, improved parametrix in the glancing region compaired with that one built in [9], [12]. We also study the way in which the parametrix depends on the refraction index. As a consequence, we improve the transmission eigenvalue-free regions obtained in [12] in the isotropic case when the restrictions of the refraction indices on the boundary coincide.
Spatial boundary problem with the Dirichlet-Neumann condition for a singular elliptic equation
本文研究了四分之一球内具有奇异系数的椭圆型方程的Dirichlet-Neumann边值问题的显式解。为此目的，格林函数的方法已被使用。由于发现Green函数中含有Appell的一个超几何函数，为了得到问题的显式解，必须处理这个超几何函数的分解公式、微分公式和一些邻接关系。
Motivated by recent results regarding the equivalence of the Dirichlet and Neumann problems for the Laplace operator in the case of simply connected regions, the present paper takes a step further and provides a similar equivalence between the above mentioned problems in the case of planar doubly-connected regions. The equivalence means that solving any of these problems leads by an explicit formula to a solution of the other problem. In addition, suffcient conditions guaranteeing the uniform Holder continuity of the higher order partial derivatives of the solutions to the Neumann problem are …
We consider the elliptic estimates for Dirichlet-Neumann operator related to the water-wave problem on a two-dimensional corner domain in this paper. Due to the singularity of the boundary, there will be singular parts in the solution of the elliptic problem for D-N operator. To begin with, we study elliptic problems with mixed boundary condition to derive singularity decompositions and estimates. Based on the analysis, we present the estimates for both D-N operator and its shape derivative with the existence of singular parts.
Point measurements for a Neumann-to-Dirichlet map and the Calderón problem in the plane
这项工作认为性质的Neumann-Dirichlet映射的电导率方程的假设下，电导率是相同的一个接近检查光滑，有界和单连通域的边界。它表明，所谓的bisweep数据，即，当在完全相同的点处施加相反符号的δ电流时，两个边界点之间的（相对）电势差唯一地确定整个Neumann-to-Dirichlet映射。在二维中，双扫描数据作为两个变量的全纯函数延伸到产品边界的某个（内部）邻域。由此可见，整个Neumann-to-Dirichlet映射的特征在于bisweep数据在任意点的导数。在乘积边界的对角线上，这些导数可以借助于在某个不动点处的（相对）边界势的导数给出，这些导数是由在同一点处支持的分布电流密度引起的，因此这样的点测量唯一地定义了Neumann-to-Dirichlet映射。这一观察也导致了一个新的，真正的本地唯一性的结果，所谓的卡尔德龙逆电导率问题。
The Dirichlet-to-Neumann operator via hidden compactness
我们证明了对于有界Lipschitz域$Ω\subset \mathbb{R}^d$上的每个形式为\[ \mathcal{A} = - \sum \partial_k \, a_{kl} \, \partial_l + c \]的对称椭圆算子，可以关联$L_2(\partial Ω)$上的自伴Dirichlet-to-Neumann算子，如果0在$\mathcal{A}$的Dirichlet谱中，则该算子可以是多值的。为了克服在这种情况下，缺乏连续性，我们采用了一个新版本的拉克斯-米尔格拉姆引理的基础上间接的椭圆属性，我们称之为隐藏的紧凑性。当Dirichlet-to-Neumann算子序列的系数一致收敛且$L_2(Ω)$中的二阶极限算子具有唯一的连续性时，我们建立了Dirichlet-to-Neumann算子序列的一致预解式收敛.我们还考虑了半群收敛。
In this paper we consider the overdetermined boundary problem for a general second order semilinear elliptic equation on bounded domains of $\mathbf{R}^n$, where one prescribes both the Dirichlet and Neumann data of the solution. We are interested in the case where the data are not necessarily constant and where the coefficients of the equation can depend on the position, so that the overdetermined problem does not generally admit a radial solution. Our main result is that, nevertheless, under minor technical hypotheses nontrivial solutions to the overdetermined boundary problem always exist.
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