L. Mazet, M. M. Rodriguez, H. Rosenberg
In this paper, we study existence and uniqueness of solutions to Jenkins-Serrin type problems on domains in a Riemannian surface. In the case of unbounded domains, the study is focused on the hyperbolic plane.
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We derive the existence of $C^{1,1}$-solutions to the Dirichlet problem for degenerate fully nonlinear elliptic equations on Riemannian manifolds under appropriate assumptions.
Boundary determination of the Riemannian metric by the elastic Dirichlet-to-Neumann map
本文研究弹性 Dirichlet-to-Neumann 映射的边界逆问题，通过计算其全象征，证明它可唯一确定边界度量的任意阶偏导数。
Non-preservation of concavity properties by the Dirichlet heat flow on Riemannian manifolds
该研究刻画了 Riemann 流形全凸区域上 Dirichlet 热流保持凹性性质的必要条件，证明其仅在截面曲率处处为零时成立。
In this short paper we show a sufficient condition for the solvability of the Dirichlet problem at infinity in Riemannian cones (as defined below).This condition is related to a celebrated result of Milnor that classifies parabolic surfaces. When applied tosmooth Riemannian manifolds with a special type of metrics (which generalise rotational symmetry) we obtain generalisations of classical criteria for the solvability of the Dirichlet problem at infinity. Our proof is short and elementary: it uses separation of variables and comparison arguments for ODE's.
This article studies the Dirichlet problem for a class of degenerate fully nonlinear elliptic equations on Riemannian manifolds with \textit{mean concave} boundary in the sense that the mean curvature of the boundary is \textit{nonpositive}. The proof is primarily based on a quantitative boundary estimate. Also, we obtain analogous results in complex variables. In Appendix, the subsolutions are also constructed on certain topologically product manifolds.
Torsional Rigidity and Spherical Deficit for a Dirichlet Problem on Riemannian Manifolds
研究Ricci曲率下有界的黎曼流形上Dirichlet问题的扭转刚度不等式和球面亏损积分恒等式，并应用于Einstein流形。
We solve the Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds under essentially optimal structure conditions, especially with no restrictions to the curvature of the underlying manifold and the second fundamental form of its boundary. The main result (Theorem 1.1) includes a new (and optimal) result in the Euclidean case. We introduce some new ideas and methods in deriving a priori estimates, which can be used to treat other types of fully nonlinear elliptic and parabolic equations on real or complex manifolds.
We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in $M\times \mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish existence of solutions satisfying prescribed conditions at infinity, depending on the direction along which infinity is approached. Moreover, the large-time behavior of such solutions is studied. We consider also elliptic equations on $M$ with similar conditions at infinity.
In this article, we first show that for all compact Riemannian manifolds with non-empty smooth boundary and dimension at least 3, there exists a metric, pointwise conformal to the original metric, with constant scalar curvature in the interior, and constant scalar curvature on the boundary by considering the boundary as a manifold of its own with dimension at least 2. We then show a series of prescribed scalar curvature results in the interior and on the boundary, with pointwise conformal deformation. These type of results is both an analogy and an extension of Kazdan and Warner's "Trichotomy …
We prove the existence of minimal hypersurfaces for the Dirichlet that extends a similar result of Jenkins and Serrin in Euclidean Space to Riemannian ambient manifolds
Universal inequalities for eigenvalues of the Dirichlet Laplacian on conformally flat Riemannian manifolds
本文研究了共形平坦黎曼流形上Dirichlet Laplacian的特征值问题，并在双曲空间$\mathbb{H}^n(-1)$上建立了特征值的普遍不等式。
In this paper, on Riemannian manifolds with boundary, we establish a Yau type gradient estimate and Liouville theorem for harmonic functions under Dirichlet boundary condition. Under a similar setting, we also formulate a Souplet-Zhang type gradient estimate and Liouville theorem for ancient solutions to the heat equation.
Given a C2-domain with compact boundary in an arbitrary complete Riemannian manifold, we search for smallness conditions on the boundary data for which the Dirichlet problem for the minimal hypersurface equation is solvable. We obtain an extension to Riemannian manifolds of an existence result of G. H. Williams ( J. Reine Angew. Math. 354:123-140, 1984).
It is well known that the Serrin condition is a necessary condition for the solvability of the Dirichlet problem for the prescribed mean curvature equation in bounded domains of $\mathbb{R}^n$ with certain regularity. In this paper we investigate the sharpness of the Serrin condition for the vertical mean curvature equation in the product $ M^n \times \mathbb{R} $. Precisely, given a $\mathscr{C}^2$ bounded domain $Ω$ in $M$ and a function $ H = H (x, z) $ continuous in $\overlineΩ\times\mathbb{R}$ and non-decreasing in the variable $z$, we prove that the strong Serrin condition $(n-1)\mathcal…
We study constant mean curvature graphs in the Riemannian 3-dimensional Heisenberg spaces ${\cal H}={\cal H}(τ)$. Each such ${\cal H}$ is the total space of a Riemannian submersion onto the Euclidean plane $\mathbb{R}^2$ with geodesic fibers the orbits of a Killing field. We prove the existence and uniqueness of CMC graphs in ${\cal H}$ with respect to the Riemannian submersion over certain domains $Ω\subset\mathbb{R}^2$ taking on prescribed boundary values.
We establish existence and uniqueness of compact graphs of constant mean curvature in MxR over bounded multiply connected domains of Mx{0} with boundary lying in two parallel horizontal slices of MxR
Estimates for Eigenvalues of the Dirichlet Laplacian on Riemannian Manifolds
本文重新探讨了完备黎曼流形有界域上 Dirichlet Laplacian 的特征值问题，基于 Li-Yau 和 Yang 不等式等经典结果推导出特征值上下界，并对射影空间及其极小子流形给出了特征值下界的显式估计。
We prove the existence of classical solutions to the Dirichlet problem for a class of fully nonlinear elliptic equations of curvature type on Riemannian manifolds. We also derive new second derivative boundary estimates which allows us to extend some of the existence theorems of Caffarelli, Nirenberg and Spruck [4] and Ivochkina, Trundinger and Lin [19] to more general curvature functions and less convex domains.
Gauge freedoms in the anisotropic elastic Dirichlet-to-Neumann map
本文研究从 Dirichlet-to-Neumann 映射恢复弹性参数的反问题，通过分析坐标变换下方程的不变性，揭示了黎曼弹性波方程及其映射中的规范自由度。
In this paper we study the Dirichlet problem for fully nonlinear second-order equations on a riemannian manifold. As in a previous paper we define equations via closed subsets of the 2-jet bundle. Basic existence and uniqueness theorems are established in a wide variety of settings. However, the emphasis is on starting with a constant coefficient equation as a model, which then universally determines an equation on every riemannian manifold which is equipped with a topological reduction of the structure group to the invariance group of the model. For example, this covers all branches of the ho…
In this paper we prove a sub-Riemannian version of the classical Santaló formula: a result in integral geometry that describes the intrinsic Liouville measure on the unit cotangent bundle in terms of the geodesic flow. Our construction works under quite general assumptions, satisfied by any sub-Riemannian structure associated with a Riemannian foliation with totally geodesic leaves (e.g. CR and QC manifolds with symmetries), any Carnot group, and some non-equiregular structures such as the Martinet one. A key ingredient is a "reduction procedure" that allows to consider only a simple subset of…
Riemannian metrics with prescribed volume and finite parts of Dirichlet spectrum
本文证明了在带光滑边界的紧流形上，可以构造黎曼度量同时实现规定的体积和任意给定的有限Dirichlet特征值序列。
Yau-type gradient estimates for p-harmonic functions on Riemannian manifolds with boundary under Dirichlet boundary condition
本文在带边黎曼流形上，利用边界径向障碍与内部最大值原理，将Yau型梯度估计推广至p-调和函数，并导出Liouville定理。
We prove a Hardy inequality for uniformly elliptic operators subject to Dirichlet or mixed boundary conditions on domains $Ω$ with piecewiese smooth boundary in arbitrary Riemannian Manifolds (M, g). Employing an approach of E.B. Davies for the euclidean case, we show that it implies a sufficient geometric criterion under which the Laplace- Beltrami operator with Dirichlet boundary conditions $Δ^D$ has purely discrete spectrum on $Ω$. We proceed to classify all non-compact $Ω$ with discrete spectrum up to a boundary regularity condition and show that these include for example polygons with ide…
In this paper we consider determining a minimal surface embedded in a Riemannian manifold $Σ\times \mathbb{R}$. We show that if $Σ$ is a two dimensional Riemannian manifold with boundary, then the knowledge of the associated Dirichlet-to-Neumann map for the minimal surface equation determine $Σ$ up to an isometry.
This paper is devoted to investigate the heat trace asymptotic expansion corresponding to the magnetic Steklov eigenvalue problem on Riemannian manifolds with boundary. We establish an effective procedure, by which we can calculate all the coefficients $a_0$, $a_1$, $\dots$, $a_{n-1}$ of the heat trace asymptotic expansion. In particular, we explicitly give the expressions for the first four coefficients. These coefficients are spectral invariants which provide precise information concerning the volume and curvatures of the boundary of the manifold and some physical quantities by the magnetic …
On a compact Riemannian manifold with boundary, the absolute and relative cohomology groups appear as certain subspaces of harmonic forms. DeTurck and Gluck showed that these concrete realizations of the cohomology groups decompose into orthogonal subspaces corresponding to cohomology coming from the interior and boundary of the manifold. The principal angles between these interior subspaces are all acute and are called Poincare duality angles. This paper determines the Poincare duality angles of a collection of interesting manifolds with boundary derived from complex projective spaces and fro…
The main result of this note is the existence of martingale solutions to the stochastic heat equation (SHE) in a Riemannian manifold by using suitable Dirichlet forms on the corresponding path/loop space. Moreover, we present some characterizations of the lower bound of the Ricci curvature by functional inequalities of various associated Dirichlet forms.
We show that the degenerate special Lagrangian equation, recently introduced by Rubinstein-Solomon, induces a global equation on every Riemannian manifold, and that for certain associated geometries this equation governs, as it does in the Euclidean setting, geodesics in the space of positive Lagrangians. For example, geodesics in the space of positive Lagrangian sections of a smooth Calabi-Yau torus fibration are governed by the Riemannian DSL on the base manifold. We then develop their analytic techniques, specifically modifications of the Dirichlet duality theory of Harvey-Lawson, in the Ri…
An Inequality Comparing the Dirichlet Energy and the Bienergy of Maps Between Riemannian Manifolds
本文研究了Riemann流形间光滑映射的Dirichlet能量与Bi能量之间的不等式关系，证明了在特定曲率条件下，Bi能量下界由域的Ricci曲率决定，并探讨了等号成立时的刚性结果。
In these notes we study the Dirichlet problem for critical points of a convex functional of the form \[ F(u)=\int_Ωϕ\left( \left\vert \nabla u\right\vert \right) , \] where $Ω$ is a bounded domain of a complete Riemannian manifold $\mathcal{M}.$ We also study the asymptotic Dirichlet problem when $Ω=\mathcal{M}$ is a Cartan-Hadamard manifold. Our aim is to present a unified approach to this problem which comprises the classical examples of the $p-$Laplacian ($ϕ(s)=s^{p}$, $p>1)$ and the minimal surface equation ($ϕ(s)=\sqrt{1+s^{2}}$). Our approach does not use the direct method of the Calculu…
Dirichlet--Neumann duality for the Basic Spectrum of Riemannian Submersions: A Supersymmetric Perspective
研究纤维具有基本平均曲率的黎曼浸没的谱几何，通过限制拉普拉斯-贝尔特拉米算子于基本函数空间，将谱问题简化为基流形上的加权拉普拉斯问题，并利用超对称量子力学建立基本Dirichlet与Neumann谱在变换S↦1/S下的对偶性。
In this article, we prove Talenti's comparison theorem for Poisson equation on complete noncompact Riemannian manifold with nonnegative Ricci curvature. Furthermore, we obtain the Faber-Krahn inequality for the first eigenvalue of Dirichlet Laplacian, $L^1$- and $L^\infty$-moment spectrum, especially Saint-Venant theorem for torsional rigidity and a reverse Hölder inequality for eigenfunctions of Dirichlet Laplacian.
A Riemannian Extension of a Quadrature Surface Free Boundary Problem: Stability and Optimality
该研究将求积曲面自由边界问题从欧氏环境扩展至紧黎曼流形，建立其紧性、狄利克雷问题稳定性、一阶最优性条件及相关比较原理，为该框架的黎曼扩展提供了严格依据。
In this paper, we study a first Dirichlet eigenfunction of the weighted $p$-Laplacian on a bounded domain in a complete weighted Riemannian manifold. By constructing gradient estimates for a first eigenfunction, we obtain some relationships between weighted $p$-Laplacian first eigenvalues. As an immediate application, we also obtain some eigenvalue comparison results between the first Dirichlet eigenvalue of the weighted Laplacian, the first clamped plate eigenvalue and the first buckling eigenvalue.
Boundary determination of the Riemannian metric from Cauchy data for the Stokes equations
该研究针对带光滑边界的紧致连通黎曼流形，证明Stokes方程的柯西数据可唯一确定流形边界上黎曼度量的所有阶偏导数，为相关逆问题研究提供理论支撑。
This paper deals with symmetry phenomena for solutions of the Dirichlet problem involving semilinear PDEs on Riemannian domains. We shall present a rather general framework where the symmetry problem can be formulated and provide some evidence that this framework is completely natural by pointing out some results for stable solutions. The case of manifolds with density, and corresponding weighted Laplacians, is inserted in the picture from the very beginning.
In this paper, the elastic Dirichlet-to-Neumann map $Ξ_g$ is studied for the stationary elasticity system in a compact Riemannian manifold $(Ω,g)$ with smooth boundary $\partial Ω$. By overcoming methodological difficulties, we explicitly get matrix-valued full symbol for the elastic Dirichlet-to-Neumann map $Ξ_g$. We prove that for a strong convex or extendable real-analytic manifold with boundary, the elastic Dirichlet-to-Neumann map $Ξ_g$ uniquely determines the metric $g$ of $Ω$ in the sense of isometry, thereby solving an open problem for the uniqueness of the metric under real-analytic s…
We make systematic developments on Lawson-Osserman constructions relating to the Dirichlet problem (over unit disks) for minimal surfaces of high codimension in their 1977 Acta paper. In particular, we show the existence of boundary functions for which infinitely many analytic solutions and at least one nonsmooth Lipschitz solution exist simultaneously. This newly-discovered amusing phenomenon enriches the understanding on the Lawson-Osserman philosophy.
In this paper, we prove the existence of martingale solutions to the stochastic heat equation taking values in a Riemannian manifold, which admits Wiener (Brownian bridge) measure on the Riemannian path (loop) space as an invariant measure using a suitable Dirichlet form. Using the Andersson-Driver approximation, we heuristically derive a form of the equation solved by the process given by the Dirichlet form. Moreover, we establish the log-Sobolev inequality for the Dirichlet form in the path space. In addition, some characterizations for the lower bounds of the Ricci curvature are presented r…
We derive a priori second order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under some very general structure conditions. We treat both equations on closed manifolds, and the Dirichlet problem on manifolds with boundary without any geometric restrictions to the boundary except being smooth and compact. As applications of these estimates we obtain results on regularity and existence.
Given two Jordan curves in a Riemannian manifold, a minimal surface of annulus type bounded by these curves is described as the harmonic extension of a critical point of some functional (the Dirichlet integral) in a certain space of boundary parametrizations. The $H^{2,2}$-regularity of the minimal surface of annulus type will be proved by applying the critical points theory and Morrey's growth condition.
In this paper, we establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary established by Fraser. Our theory also includes as a special case the min-max theory for Plateau problem of minimal disks, which can be used to generalize the famous work by Morse-Thompkins and Shiffman on minimal surfaces in $\mathbf{R}^n$ to the Riemannian setting. More precisely, we generalize the min-max construction of minimal surfaces using harmonic replace…
Determining Lamé coefficients by elastic Dirichlet-to-Neumann map on a Riemannian manifold
本文研究在黎曼流形上通过弹性 Dirichlet-to-Neumann 映射反演 Lamé 系数的问题，给出了该映射完整符号的显式表达，并证明其在边界上唯一确定系数的所有阶偏导数，在实解析条件下可唯一确定整个流形上的 Lamé 系数。
Given a complete $n$-dimensional Riemannian manifold $M$, we study the existence of vertical graphs in $M\times\mathbb{R}$ with prescribed mean curvature $H=H(x,z)$. Precisely, we prove that the Dirichlet problem for the vertical mean curvature equation in a smooth bounded domain $Ω\subset M$ has solution for arbitrary smooth boundary data if $(n-1)\mathcal{H}_{\partialΩ}(y)\geq n\sup\limits_{z\in\mathbb{R}}\left|{H(y,z)}\right|$ for each $y\in\partialΩ$ provided the function $H$ also satisfies $\mathrm{Ricc}_x\geq n\sup\limits_{z\in\mathbb{R}}\left\|\nabla_x H(x,z)\right\|-\dfrac{n^2}{n-1}\in…
In this paper, we investigate the Dirichlet problem of Laplacian on complete Riemannian manifolds. By constructing new trial functions, we obtain a sharp upper bound of the gap of the consecutive eigenvalues in the sense of the order, which affirmatively answers to a conjecture proposed by Chen-Zheng-Yang. In addition, we also exploit the closed eigenvalue problem of Laplacian and obtain a similar optimal upper bound. As some important examples, we investigate the eigenvalues of the eigenvalue problem of the Laplacian on the unit sphere and cylinder, compact homogeneous Riemannian manifolds wi…
Existence and uniqueness of $t$-graphs of prescribed mean curvature in Heisenberg groups
本文研究任意维Riemannian Heisenberg群中$t$-图的给定平均曲率方程，刻画无Dirichlet边界数据时经典解的存在唯一性条件，并推广至非常数平均曲率及次Riemannian情形。
We shall discuss the inhomogeneous Dirichlet problem for: $f(x,u, Du, D^2u) = ψ(x)$ where $f$ is a "natural" differential operator, with a restricted domain $F$, on a manifold $X$. By "natural" we mean operators that arise intrinsically from a given geometry on $X$. An important point is that the equation need not be convex and can be highly degenerate. Furthermore, the inhomogeneous term can take values at the boundary of the restricted domain $F$ of the operator $f$. A simple example is the real Monge-Ampère operator ${\rm det}({\rm Hess}\,u) = ψ(x)$ on a riemannian manifold $X$, where ${\rm…
Overdetermined elliptic problems on model Riemannian manifolds
本文在模型黎曼流形的超定椭圆问题背景下，证明了满足常数Dirichlet和Neumann边界条件的环形扇形域必为球面扇形且解具径向对称性，将经典刚性现象推广至更一般的框架。
Two Riemannian manifolds are said to be isospectral if the associated Laplace-Belttrami operators have the same eigenvalue spectrum. If the manifolds have boundary, one specifies DIrichlet or Neumann isospectrality depending on the boundary conditions imposed on the eigenfunctions. We construct continuous families of (Neumann and Dirichlet) isospectral metrics which have different local geometry on manifolds with boundary in every dimension greater than 6 and also new examples of pairs of closed isospectral manifolds with different local geometry. These examples illustrate for the first time t…
To a closed Riemannian manifold, we associate a set of (special values of) a family of Dirichlet series, indexed by functions on the manifold. We study the meaning of equality of two such families of spectral Dirichlet series under pullback along a map. This allows us to give a spectral characterization of when a smooth diffeomorphism between Riemannian manifolds is an isometry, in terms of equality along pullback. We also use the invariant to define the (spectral) length of a map between Riemannian manifolds, where a map of length zero between manifolds is an isometry. We show that this lengt…
We try to present an estimate relating the first Dirichlet and Neumann eigenvalues of a compact bordered Riemannian surface.
In this article we prove a reverse Hölder inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains.
Maximum principles in unbounded Riemannian domains
该研究针对无界黎曼区域上带狄利克雷边界条件的二阶椭圆算子，结合环境流形与微分算子两类假设路径，开展最大值原理的证明研究，全文分为两个独立部分。
We lay foundations of the subject in the title, on which we build in another paper devoted to isometries in spaces of Kähler metrics.
The aim of this paper is the study of the geodesic distance in operator groups with several Riemannian metrics. More precisely we study the geodesic distance in self-adjoint operator groups with the left invariant Riemannian metric induced by the infinite trace and extend known results about the completeness of some classical Banach-Lie groups to this general class.
In this paper, we study eigenvalue of linear fourth order elliptic operators in divergence form with Dirichlet boundary condition on a bounded domain in a compact Riemannian manifolds with boundary (possibly empty) and find a general inequality for them. As an application, by using this inequality, we study eigenvalues of this operator on compact domains of complete submanifolds in a Euclidean space.
We associate to any Riemannian symmetric space (of finite or infinite dimension) a L$^*$-algebra, under the assumption that the curvature operator has a fixed sign. L$^*$-algebras are Lie algebras with a pleasant Hilbert space structure. The L$^*$-algebra that we construct is a complete local isomorphism invariant and allows us to classify Riemannian symmetric spaces with fixed-sign curvature operator. The case of nonpositive curvature is emphasized.
In this paper we will investigate the global properties of complete Hilbert manifolds with upper and lower bounded sectional curvature. We shall prove the Focal Index Lemma that we will allow us to extend some classical results of finite dimensional Riemannian geometry such as Rauch and Berger Theorems and the Topogonov Theorem in the class of manifolds on which the Hopf-Rinow Theorem holds.
On the existence and prolongation of infinitesimal isometries on special sub-Riemannian manifolds
本文研究特殊次黎曼流形上无穷小等距的局部构造与延拓问题，利用 Nomizu 引入并经作者修改的 $\is^*$-正则点和 $\is$-正则点条件，给出在给定点附近构造并延拓无穷小等距的判据。
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