Domains In Manifolds Research Articles

Layer Potentials and Poisson Problems for the Nonsmooth Coefficient Brinkman System in Sobolev and Besov Spaces

In this paper we obtain well-posedness results for Poisson problems with Dirichlet, Neumann, or mixed boundary conditions for the Brinkman system with measurable coefficients and data in $$L^p$$ -based Sobolev and Besov spaces in Lipschitz domains on a compact Riemannian manifold M of dimension $$m\ge 2$$ . For the mixed problem we refer to partially vanishing traces on Ahlfors regular sets. We exploit the continuity property of an operator related to the variational formulation of such a boundary value problem on complex interpolation scales of $$L^p$$ -based Sobolev spaces defined on M or on a Lipschitz domain of M, $$p\in (1,\infty )$$ , and the property that this operator is an isomorphism for $$p=2$$ . Then the stability of the quality of being isomorphism on complex interpolation scales leads to the extension of the well-posedness results of analyzed boundary value problems from $$p\!=\!2$$ to p in a neighborhood of 2. First, we focus on a variational approach that reduces boundary problems of transmission, Dirichlet and mixed type for the Brinkman system to equivalent mixed variational formulations with data in $$L^p$$ -based Sobolev and Besov spaces. For $$p=2$$ , such a mixed variational formulation is well-posed. The mixed variational formulation is further expressed in terms of a linear continuous operator on $$H^{1,q}\times L^q$$ -Sobolev spaces for any $$q\in (1,\infty )$$ , which is also invertible on the solution space corresponding to $$q=2$$ . Working on complex interpolation scales allows us to extend the invertibility of the operator for $$q=2$$ to a neighborhood of 2, and then to extend the well-posedness result to $$L^p$$ -based Sobolev spaces with p in a neighborhood of 2. Well-posedness results for the analyzed transmission problems allow us to define the layer potentials for the nonsmooth coefficient Brinkman system and to obtain their properties in $$L^p$$ -based Sobolev and Besov spaces. Then the solution of the Poisson problem of Dirichlet type is constructed explicitly in terms of such layer potentials. Finally, the Poisson problem of Neumann type is also analyzed and the corresponding well-posedness result in $$L^p$$ -based Sobolev and Besov spaces is also obtained. In addition, we determine the unique solution of the Neumann problem in the case $$p=2$$ , by using a layer potential approach. We extend the well-posedness results obtained in Kohr and Wendland (Boundary value problems for the Brinkman system with measurable coefficients in Lipschitz domains on compact Riemannian manifolds: a variational approach, 2018) for boundary problems for the nonsmooth coefficient Brinkman system in $$L^2$$ -based Sobolev spaces on Lipschitz domains in compact Riemannian manifolds to a more general setting of $$L^p$$ -based Sobolev and Besov spaces.

Singularities of mean convex level set flow in general ambient manifolds

We prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds. Our estimates give control – exponential in time – for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove a stumbling block that has been left after the work of White [16,17,20], and Haslhofer–Kleiner [9], and thus allow us to extend the structure theory for mean convex level set flow to general ambient manifolds of arbitrary dimension.

Comparison Results, Exit Time Moments, and Eigenvalues on Riemannian Manifolds with a Lower Ricci Curvature Bound

We study the relationship between the geometry of smoothly bounded domains in complete Riemannian manifolds and the associated sequence of $L^1$-norms of exit time moments for Brownian motion. We establish bounds for Dirichlet eigenvalues and, for closed manifolds, we establish a comparison result for elements of the moment sequence.

Proper quasi-homogeneous domains in flag manifolds and geometric structures

In this paper we study domains in flag manifolds which are bounded in an affine chart and whose projective automorphism group acts co-compactly. In contrast to the many examples in real projective space, we will show that no examples exist in many flag manifolds. Moreover, in the cases where such domains can exist, we show that they satisfy a natural convexity condition and have an invariant metric which generalizes the Hilbert metric. As an application we give some restrictions on the developing map for certain (G,X)-structures.

The PPW conjecture in curved spaces

In Euclidean ([1]) and Hyperbolic ([5]) space, and the round hemisphere ([2]), geodesic balls maximize the gap λ2−λ1 of Dirichlet eigenvalues, among domains with fixed λ1. We prove an upper bound on λ2−λ1 for domains in manifolds with certain curvature bounds. The inequality is sharp on geodesic balls in spaceforms.

Asymptotically radial solutions to an elliptic problem on expanding annular domains in Riemannian manifolds with radial symmetry

We consider the boundary value problem $$\left \{ \textstyle\begin{array}{l} \Delta_{\mathbf{g}} u +u^{p}= 0 \quad\mbox{in } \Omega_{R}, \\ u=0 \quad \mbox{on } \partial\Omega_{R}, \end{array}\displaystyle \right . $$ $\Omega_{R}$ being a smooth bounded domain diffeomorphic to the expanding domain $A_{R}:=\{x \in M, R< r(x)< R+1\}$ in a Riemannian manifold M of dimension $n \geq2$ endowed with the metric ${\mathbf{g}}=dr^{2}+S^{2}(r)g_{{\mathbb{S}}^{n-1}}$ . After recalling a result about existence, uniqueness, and non-degeneracy of the positive radial solution when $\Omega_{R}=A_{R}$ , we prove that there exists a positive non-radial solution to the aforementioned problem on the domain $\Omega_{R}$ . Such a solution is close to the radial solution to the corresponding problem on $A_{R}$ .

On the linear independence of truncated hierarchical generating systems

Motivated by the necessity to perform adaptive refinement in geometric design and numerical simulation, the construction of hierarchical splines from generating systems spanning nested spaces has been recently studied in several publications. Linear independence can be guaranteed with the help of the local linear independence of the spline basis at each level. The present paper extends this framework in several ways. Firstly, we consider spline functions that are defined on domain manifolds, while the existing constructions are limited to domains that are open subsets of Rd. Secondly, we generalize the approach to generating systems containing functions which are not necessarily non-negative. Thirdly, we present a more general approach to guarantee linear independence and present a refinement algorithm that maintains this property. The three extensions of the framework are then used in several relevant applications: doubly hierarchical B-splines, hierarchical Zwart-Powell elements, and three different types of hierarchical subdivision splines.

Gap theorems for Ricci-harmonic solitons

In this paper, using estimates for the generalized Ricci curvature, we shall give some gap theorems for Ricci-harmonic solitons with compact domain manifolds by showing some necessary and sufficient conditions for the solitons to be harmonic-Einstein. Our results may be regarded as generalizations of recent works by H. Li, and M. Fernandez-Lopez and E. Garcia-Rio.

On the transmission problems for the Oseen and Brinkman systems on Lipschitz domains in compact Riemannian manifolds

The purpose of this work is to show the well‐posedness in L2‐Sobolev spaces of the Poisson‐transmission problem for the Oseen and Brinkman systems on complementary Lipschitz domains in a compact Riemannian manifold. The Oseen system appears as a perturbation of order one of the Stokes system, given in terms of the Levi‐Civita connection, while the Brinkman system is a zero order perturbation of the Stokes system. The technical details of this paper rely on the layer potential theory for the Stokes system and the invertibility of some perturbed zero index Fredholm operators by a first order differential operator given in terms of the Levi‐Civita connection. The compactness of this differential operator requires to restrict ourselves to low dimensional compact Riemannian manifolds.

Estimates for eigenvalues of fourth-order elliptic operators in divergence form

In this paper, we study the eigenvalue problem of fourth-order elliptic operators in divergence form with weight on compact Riemannian manifolds with boundary (possibly empty) and get two types of general inequalities for them. By using these general inequalities, we can obtain some universal inequalities for eigenvalues of fourth-order elliptic operators in divergence form with weight on bounded domains in complete Riemannian manifolds. Furthermore, we also obtain some estimates for lower order eigenvalues of fourth-order elliptic operators in divergence form with weight on compact submanifolds in Euclidean spaces.

Exit radii of submanifolds from cylindrical domains in warped product manifolds

Let UB(p0; ρ1) × f MV be a cylindrically bounded domain in a warped product manifold := MB × fMV and let M be an isometrically immersed submanifold in . The purpose of this paper is to provide explicit radii of the geodesic balls of M which first exit from UB(p0; ρ1) × fMV for the case in which the mean curvature of M is sufficiently small and the lower bound of the Ricci curvature of M does not diverge to –∞ too rapidly at infinity.

A lower diameter bound for compact domain manifolds of shrinking Ricci-harmonic solitons

In this paper, we shall give a lower diameter bound for compact domain manifolds of shrinking Ricci-harmonic solitons. Our result may be regarded as a generalization to Ricci-harmonic geometry of the recent works by Fernández-López and García-Río (Q. J. Math. 61, 319-327, 2010), Futaki and Sano (Asian J. Math. 17, 17-32, 2013), and Futaki et al. (Ann. Global Anal. Geom. 44, 105-114, 2013).

Radial and non-radial solutions to an elliptic problem on annular domains in Riemannian manifolds with radial symmetry

We show existence and uniqueness of positive radial solutions to{Δgu+λu+up=0in A,u=0on ∂A, with λ<0, A being an annular domain in a Riemannian manifold M of dimension n endowed with the metric dr2+S2(r)gSn−1. Secondly we show that there exist positive non-radial solutions arising by bifurcation from the radial solution. p and λ are the bifurcation parameters.

Eigenvalues of the buckling problem of arbitrary order and of the polyharmonic operator on Ricci flat manifolds

In this paper, we study eigenvalues of the buckling problem of arbitrary order and of the polyharmonic operator on bounded domains in Ricci flat manifolds supporting a special function and obtain universal bounds on the (k+1)th eigenvalue in terms of the first k eigenvalues independent of the domains.

Poisson-Transmission Problems for $$L^{\infty }$$ L ∞ -Perturbations of the Stokes System on Lipschitz Domains in Compact Riemannian Manifolds

The purpose of this work is to show the well-posedness in \(L^2\)-Sobolev spaces for a Poisson-transmission problem involving \(L^{\infty }\)-perturbations of the Stokes system on complementary Lipschitz domains in compact Riemannian manifolds. The technical details rely on the layer potential theory for the Stokes system and the invertibility of some perturbed zero index Fredholm operators by suitable compact operators. The compact part is provided by the \(L^{\infty }\)-perturbations of the Stokes system. An existence result for a related semilinear Poisson-transmission problem is also formulated.

Counting nodal domains in Riemannian manifolds

The number of nodal domains associated to eigenfunctions of the Laplacian are counted. Asymptotic estimates are derived as the eigenvalues approach infinity. Earlier results by other authors are improved and generalized.

Neumann-transmission problems for pseudodifferential Brinkman operators on Lipschitz domains in compact Riemannian manifolds

The aim of this paper is twofold. On the one handwe construct Neumann-transmission kernels forpseudodifferential Brinkman operators. They are used to provide simple representations of thesolution to some transmission problems for the pseudodifferential Brinkman operator.On the other hand, we showthe well-posedness of a Neumann-transmission problem for twopseudodifferential Brinkman operators on adjacent Lipschitz domainsin a compact Riemannian manifold, with boundary data in some $L^p$, Sobolev or Besov spaces. We rely on thelayer potential theory in order to obtain an explicit representation of the solution to this problem.Compactness and invertibility results of associated layer potentialoperators on $L^p$, Sobolev and Besov spaces are also presented.

Dirichlet‐transmission problems for pseudodifferential Brinkman operators on Sobolev and Besov spaces associated to Lipschitz domains in Riemannian manifolds

AbstractIn this paper we use a layer potential analysis to show the existence of solutions for a Dirichlet‐transmission problem for pseudodifferential Brinkman operators on Sobolev and Besov spaces associated to Lipschitz domains in compact boundaryless Riemannian manifolds. Compactness and invertibility properties of corresponding layer potential operators on Lp, Sobolev, or Besov scales are also obtained.

Monotonicity formulae and holomorphicity of harmonic maps between Kähler manifolds

In this paper, we introduce the stress-energy tensors of the partial energies E'(f) and E"(f) of maps between Kaehler manifolds. Assuming the domain manifolds poss some special exhaustion functions, we use these stress-energy tensors to establish some monotonicity formulae of the partial energies of pluriharmonic maps into any Kaehler manifolds and harmonic maps into Kaehler manifolds with strongly semi-negative curvature respectively. These monotonicity inequalities enable us to derive some holomorphicity and Liouville type results for these pluriharmonic maps and harmonic maps. We also use the stress-energy tensors to investigate the holomorphic extension problem of CR maps.

Partial energies monotonicity and holomorphicity of Hermitian pluriharmonic maps

In this paper, we introduce the notion of Hermitian pluriharmonic maps from Hermitian manifold into Kahler manifold. Assuming the domain manifolds possess some special exhaustion functions and the vecotor field V = JMδJM satisfies some decay conditions, we use stress-energy tensors to establish some monotonicity formulas of partial energies of Hermitian pluriharmonic maps. These monotonicity inequalities enable us to derive some holomorphicity for these Hermitian pluriharmonic maps.