Complex Powers of the Neumann Laplacian in Lipschitz Domains

Let L^q_r(Ω) be the usual scale of Sobolev spaces and let Δ_N be the Neumann Laplacian in an arbitrary Lipschitz domain Ω. We present an interpolation based approach to the following question: for what range of indices does map isomorphically onto L^q_r(Ω)/ℝ?     

‘Universal’ inequalities for the eigenvalues of the Hodge de Rham Laplacian

We obtain ‘universal’ inequalities for the eigenvalues of the Laplacian acting on differential forms of a Euclidean compact submanifold. These inequalities generalize the Yang inequality concerning the eigenvalues of the Dirichlet Laplacian of a bounded Euclidean domain.      

Hodge theorem for the natural projection of complex horizontal Laplacian on complex Finsler manifolds

Let M be a compact complex manifold with a complex Finsler metric F. We define a natural projection of complex horizontal Laplacian on M: it is independent of the fiber coordinate. By using Sobolev space theory and spectral resolution theory in Hilbert space, we prove the Hodge theorem for the natural projection of complex horizontal Laplacian on M.     

On the equivalence of heat kernel estimates and logarithmic Sobolev inequalities for the Hodge Laplacian

In this paper we consider the Hodge Laplacian on differential k-forms over smooth open manifolds MN, not necessarily compact. We find sufficient conditions under which the existence of a family of logarithmic Sobolev inequalities for the Hodge Laplacian is equivalent to the ultracontractivity of its heat operator.We will also show how to obtain a logarithmic Sobolev inequality for the Hodge Laplacian when there exists one for the Laplacian on functions. In the particular case of Ricci curvature bounded below, we use the Gaussian type bound for the heat kernel of the Laplacian on functions in order to obtain a similar Gaussian type bound for the heat kernel of the Hodge Laplacian. This is done via logarithmic Sobolev inequalities and under the additional assumption that the volume of balls of radius one is uniformly bounded below.     

On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains

Abstract

Let Ω be a bounded Lipschitz domain in ? ⁿ , n ≥ 3 with connected boundary. We study the Robin boundary condition ?u/?N + bu = f ∈ L ^p (?Ω) on ?Ω for Laplace's equation Δu = 0 in Ω, where b is a non-negative function on ?Ω. For 1 < p < 2 + ?, under suitable compatibility conditions on b, we obtain existence and uniqueness results with non-tangential maximal function estimate ‖(?u)*‖ _p  ≤ C‖f‖ _p , as well as a pointwise estimate for the associated Robin function. Moreover, the solution u is represented by a single layer potential.     

Function Spaces in Lipschitz Domains and Optimal Rates of Convergence for Sampling

Assume that we want to recover $f : \Omega \to {\bf C}$ in the $L_r$-quasi-norm ($0 < r \le \infty$) by a linear sampling method $$ S_n f = \sum_{j=1}^n f(x^j) h_j , $$ where $h_j \in L_r(\Omega )$ and $x^j \in \Omega$ and $\Omega \subset {\bf R}^d$ is an arbitrary bounded Lipschitz domain. We assume that $f$ is from the unit ball of a Besov space $B^s_{pq} (\Omega)$ or of a Triebel--Lizorkin space $F^s_{pq} (\Omega)$ with parameters such that the space is compactly embedded into $C(\overline{\Omega})$. We prove that the optimal rate of convergence of linear sampling methods is $$ n^{ -{s}/{d} + ({1}/{p}-{1}/{r})_+} , $$ nonlinear methods do not yield a better rate. To prove this we use a result from Wendland (2001) as well as results concerning the spaces $B^s_{pq} (\Omega) $ and $F^s_{pq}(\Omega)$. Actually, it is another aim of this paper to complement the existing literature about the function spaces $B^s_{pq} (\Omega)$ and $F^s_{pq} (\Omega)$ for bounded Lipschitz domains $\Omega \subset {\bf R}^d$. In this sense, the paper is also a continuation of a paper by Triebel (2002).     

Spin c geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds

From the existence of parallel spinor fields on Calabi-Yau, hyper-Kähler or complex flat manifolds, we deduce the existence of harmonic differential forms of different degrees on their minimal Lagrangian submanifolds. In particular, when the submanifolds are compact, we obtain sharp estimates on their Betti numbers which generalize those obtained by Smoczyk in [49]. When the ambient manifold is Kähler-Einstein with positive scalar curvature, and especially if it is a complex contact manifold or the complex projective space, we prove the existence of Kählerian Killing spinor fields for some particular spin ^c structures. Using these fields, we construct eigenforms for the Hodge Laplacian on certain minimal Lagrangian submanifolds and give some estimates for their spectra. These results also generalize some theorems by Smoczyk in [50]. Finally, applications on the Morse index of minimal Lagrangian submanifolds are obtained.     

Diffraction at Corners for the Wave Equation on Differential Forms

Our study of a basic model for incompressible two-phase flows with phase transitions consistent with thermodynamics in the case of constant but non-equal densities of the phases, begun in [23 Prüss , J. , Shimizu , S. (2012). On well-posedness of incompressible two-phase flows with phase transition: The case of non-equal densities. J. Evol. Eqs. 12:917–941.[Crossref], [Web of Science ®] , [Google Scholar]], is continued. We extend our well-posedness result to general geometries, study the stability of the equilibria of the problem, and show that a solution which does not develop singularities exists globally. Moreover, if its limit set contains a stable equilibrium it converges to this equilibrium as time goes to infinity, in the natural state manifold for the problem in an L _p -setting.     

Lipschitz Continuity and Interpolation Properties of Some Classes of Increasing Functions

We study local Lipschitz continuity and interpolation properties of some classes of increasing functions defined on the cone Rⁿ ₊₊ of n-vectors with positive coordinates. We also study the so-called self-conjugate increasing positively homogeneous functions.     

On the lowest eigenvalue of the Hodge Laplacian on compact,negatively curved domains

We study the first eigenvalue of the Laplacian acting on differential forms on a compact Riemannian domain, for the absolute or relative boundary conditions. We prove a series of lower bounds when the domain is starlike or p-convex and the ambient manifold has pinched negative curvature. The bounds are sharp for starlike domains. We then compute the asymptotics of the first eigenvalue of hyperbolic balls of large radius. Finally, we give lower bounds also for Euclidean domains.      

Extrapolation for the L p Dirichlet Problem in Lipschitz Domains

Let L be a second-order linear elliptic operator with complex coefficients. It is shown that if the Lp Dirichlet problem for the elliptic system L(u) = 0 in a fixed Lipschitz domain Ω in...     

Elliptic boundary problems on manifolds with polycylindrical ends

We investigate general Shapiro-Lopatinsky elliptic boundary value problems on manifolds with polycylindrical ends. This is accomplished by compactifying such a manifold to a manifold with corners of in general higher codimension, and we then deal with boundary value problems for cusp differential operators. We introduce an adapted Boutet de Monvel's calculus of pseudodifferential boundary value problems, and construct parametrices for elliptic cusp operators within this calculus. Fredholm solvability and elliptic regularity up to the boundary and up to infinity for boundary value problems on manifolds with polycylindrical ends follows.     

A proof of the trace theorem of Sobolev spaces on Lipschitz domains

A complete proof of the trace theorem of Sobolev spaces on Lipschitz domains has not appeared in the literature yet. The purpose of this paper is to give a complete proof of the trace theorem of Sobolev spaces on Lipschitz domains by taking advantage of the intrinsic norm on . It is proved that the trace operator is a linear bounded operator from to for .

     

Existence and multiplicity of solutions for nonlocal systems involving fractional Laplacian with non-differentiable terms

In this paper, we devote to the study of the existence and multiplicity of solutions of nonlocal systems involving fractional Laplacian with non-differentiable terms using some extended critical point theorems for locally Lipschitz function on product spaces.     

用单层位势求解尖角区域上的Dirichlet外问题

采用Kress变换以及处理第一类奇异核的积分方法,运用Nystrom方法利用单层位势求解尖角区域上的Dirichlet外问题.给出具体的算法和数值例子,通过数值例子可以看出用单层位势求解尖角区域上的Dirichlet外问题与用单双层结合求解所得的结果基本上一致,说明这种方法是有效的和可行的.