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Let Ω ⊂ ℝ n be a bounded Lipschitz domain, whose boundary decomposes into two disjoint pieces Σ t , Σ n ⊆ ∂Ω, which meet at an angle 0 with the property that if |2 - p| < e, then the following holds. Consider a vector field u with components u 1 ,..., u n ∈ L p (Ω) such that and curl u =(∂ j u k - ∂ k u j ) 1≤j,k≤n ∈ L p (Ω). Set ν · u = Σ n j =1 ν j u j and ν × u = (ν j u k - ν k u j ) 1≤j,k≤n . Then the following are equivalent: (i) (ν · u) |Σ t ∈ L p (Σ t ) and (v × u) |Σ n ∈ L p (Σ n ); (ii) ν · u ∈ L p (∂Ω); (iii) ν × u ∈ L p (∂Ω). Moreover, if either condition holds, then u belongs to the Besov space B p,max(p,2) 1/p (Ω). In fact, similar results are valid for differential forms of arbitrary degree. This generalizes earlier work dealing with the case when Σ t = O or Σ n = O.

We study the analyticity of the semigroup generated by the Stokes operator equipped with Neumann-type boundary conditions onLpL^pspaces in Lipschitz domains. Our strategy is to regularize this operator by considering the Hodge Laplacian, which has the additional property that it commutes with the Leray projection.

We identify the Banach envelope of Hardy type spaces \(\mathcal{H}^{p}(\Omega)\), \(\frac{m-1}{m} < p < 1\), of monogenic functions in an arbitrary Lipschitz domain \(\Omega\, \subset\,\mathbb{R}^{m}\) in the form of a suitable Bergman type space of monogenic functions.

We study the extent to which classical trace and extension theorems for scalar-valued functions can be extended to differential forms of higher-degree. For maximum applicability, this is done in the context of Lipschitz subdomains of Riemannian manifolds, and on the scale of Besov and Triebel-Lizorkin spaces. A key ingredient in this regard is the boundary De Rham complex, which we consider in the geometric and analytic setting above. Applications to Hodge-decompositions and interpolation with constraints are also presented.

In this paper we study the self-adjoint Krein–von Neumann realization A k of the perturbed Laplacian \(-\Delta\;+\;V\) in a bounded Lipschitz domain \(\Omega\;\subset\;\mathbb{R}^n\). We provide an explicit and self-contained description of the domain of A k in terms of Dirichlet and Neumann boundary traces, and we establish a Weyl asymptotic formula for the eigenvalues of A k.

Given a domain D in R^d with mild geometric measure theoretic assumptions on its boundary, we show that boundedness of the principal value Riesz tranforms (witn kernel of homogeneity -(d-1)) on H\"older spaces of order alpha on the boundary of D is equivalent to D being a Lyapunov domain of order alpha (i.e., the boundary of D is an hypersurface of class 1+alpha). Another equivalent condition involving Riesz transforms on D is discussed. We also prove that on Lyapunov domains of order alpha the higher order Riesz transforms associated with an odd polynomial are bounded on the H\"older space of order alpha on the boundary of D. Finally, a limiting case of the above results dealing with VMO and Semmes-Kenig-Toro domains is considered., Comment: 55 pages, exposition improved, references added

We introduce certain Sobolev-Besov spaces which are particularly well adapted for measuring the smoothness of data and solutions of mixed boundary value problems in Lipschitz domains. In particular, these are used to obtain sharp well-posedness results for the Poisson problem for the Laplacian with mixed boundary conditions on bounded Lipschitz domains which satisfy a suitable geometric condition introduced by R.Brown in (1994). In this context, we obtain results which generalize those by D.Jerison and C.Kenig (1995) as well as E.Fabes, O.Mendez and M.Mitrea (1998). Applications to Hodge theory and the regularity of Green operators are also presented.

We explore the extent to which a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with the Schrodinger operator on a half-line to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrodinger equation, generalizes to higher dimensions. In this multi-dimensional extension the half-line is replaced by an open set Ω ⊂ ℝ n , n = 2, 3, where Ω has a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on ∂Ω and invokes the corresponding Dirichlet-to-Neumann map. As a result, we succeed in reducing a certain ratio of modified Fredholm perturbation determinants perturbation associated with operators in L 2 (Ω d n x ) to modified Fredholm determinants associated with operators in L 2 (∂Ω; d n −l σ), n = 2, 3.

We explore the extent to which basic differential operators (such as Laplace–Beltrami, Lame, Navier–Stokes, etc.) and boundary value problems on a hypersurface in ℝn can be expressed globally, in terms of the standard spatial coordinates in ℝn. The approach we develop also provides, in some important cases, useful simplifications as well as new interpretations of classical operators and equations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We study the Poisson problem Au - Vu = F with Dirichlet and Neumann boundary conditions on a Lipschitz domain Ω in a compact Riemannian manifold equipped with a rough metric tensor. We seek a solution u in the weighted Sobolev space W 1,α p (Ω):= {u ∈ L p loc (Ω) | dist(·,∂Ω) α [|u| + |∇u|] ∈ L p (Ω)} when F belongs to an analogous space W -1,α p (Ω) in the case of the Dirichlet boundary condition, and when F ∈ W 1,-α p (Ω)*, 1/p + 1/p' = 1, in the case of the Neumann boundary condition. We take Dirichlet boundary data in the Besov space B p,p s (∂Ω), with s=1-α-1/p, and obtain sharp results on the range of indices (s, 1/p) for which this problem is well posed, and a parallel result for the Neumann boundary condition. These results are related to the Sobolev-Besov estimates obtained in [11], [5], and [20]-[21]. They also complement certain results in [19], whose reading inspired and motivated the current work.

We study the Dirichlet and Neumann boundary value problems for the Laplacian in a Lipschitz domain \({\Omega}\), with boundary data in the Besov space \({B_{s}^{p,p} (\partial\Omega).}\) The novelty is to identify a way of measuring smoothness for the solution u that allows us to consider the case p 1 was treated.

We examine solutions u = PIf to Δu − Vu = 0 on a Lipschitz domain Ω in a compact Riemannian manifold M, satisfying u = f on ∂Ω, with particular attention to ranges of (s, p) for which one has Besov-to-L p -Sobolev space results of the form and variants, when the metric tensor on M has limited regularity, described by a Holder or a Dini-type modulus of continuity. We also discuss related estimates for solutions to the Neumann problem.

We prove a general integration by parts formula for first-order differential operators on submanifolds of arbitrary codimension of a given Riemannian manifold. The differential operators in question are assumed to satisfy a suitable tangentiality condition. Several particular cases, particularly relevant to applications to PDE arising in mathematical physics are discussed. Copyright © 2004 John Wiley & Sons, Ltd.

We consider the linearized version of the stationary Navier-Stokes equations on a subdomain Ω of a smooth, compact Riemannian manifold M. The emphasis is on regularity: the boundary of Ω is assumed to be only C1 and even Lipschitz, and the data are selected from appropriate Sobolev-Besov scales. Our approach relies on the method of boundary integral equations, suitably adapted to the variable-coefficient setting we are considering here. Applications to the stationary, nonlinear Navier-Stokes equations in this context are also discussed.

Given an open, bounded, connected domain Ω ⊂ R n , let GD, GN be the solution operators for the Poisson equation for the Laplacian in Ω with homogeneous Dirichlet and Neumann boundary conditions, respectively.The aim of this note is to describe the sharp ranges of indices for which GD, GN are smoothing operators of order two, on Besov and Triebel-Lizorkin scales in arbitrary Lipschitz domains.This builds on the work of many people who have dealt with homogeneous/inhomogenous problems for the Laplacian with Dirichlet/Neumann boundary conditions in Lipschitz domains; an excellent account can be found in [23].Earlier results emphasized homogeneous problems with boundary data exhibiting an integer amount of smoothness.Such estimates underpin the entire work here and play the role of end-point/limiting cases in our theory. The main theorems we prove extend the work of D.Jerison and C.Kenig [20], whose methods and results are largely restricted to the case p ≥ 1, and answer the open problem # 3.2.21 on p.121 in C. Kenig’s book [23] in the most complete fashion.When specialized to Hardy spaces (viewed as a subclass of Triebel-Lizorkin scale), our results provide a solution of a (strengthened form of a) conjecture made by D.-C.Chang, S.Krantz and E.Stein regarding the regularity of the Green potentials on Hardy spaces in Lipschitz domains.Cf. p.130 of [5] where the authors write: “ For some applications it would be desirable to find minimal smoothness conditions on ∂Ω in order for our analysis of the Dirichlet and Neumann problems to remain valid. We do not know whether C 1+e boundary is sufficient in order to obtain [Hardy space] estimates for the Dirichlet problem when p is near 1. [...]

We have seen in Sect. 4.3 that \(H_{\alpha }^{p}(X)\) and \(\tilde{H}_{\alpha }^{p}(X)\) can be identified with L p (X) whenever p ∈ (1, ∞] and \(\alpha \in {\bigl ( 0,[\mathrm{log}_{2}C_{\rho }]^{-1}\bigr ]}\).

The main goal of this chapter is to rework, in a sharp and relatively self-contained fashion, some of the most fundamental tools used in the area of analysis on quasi-metric spaces. Many of the results presented in this section are of independent interest and will be found useful in a plethora of subsequent applications.

This chapter is dedicated to the exploration of the molecular and ionic theory of H p (X) in the setting of d-AR spaces. As a motivation for this topic, suppose one is concerned with the behavior of a bounded linear operator T: L 2(X, μ) → L 2(X, μ).

The main goal of this chapter is to identify criteria guaranteeing that a given linear operator \(T: L^{q}(X,\mu ) \rightarrow \mathcal{B}_{1}\) with q ≥ 1, extends as a bounded operator \(T: H^{p}(X) \rightarrow \mathcal{B}_{2}\) for p as in ( 7.262).

In this paper we provide an answer to a question found in "Groupoid metrization theory," With applications to analysis on quasi-metric spaces and functional analysis,, namely when given a quasi-metric $\rho$, if one examines all quasi-metrics which are pointwise equivalent to $\rho$, does there exist one which is most like an ultrametric (or, equivalently, exhibits an optimal amount of Hölder regularity)? The answer, in general, is negative, which we demonstrate by constructing a suitable Rolewicz--Orlicz space.

The 1960s and 1970s saw the birth of a new scale of spaces in the Euclidean setting known as Besov spaces, \(B_{s}^{p,q}\big(\mathbb{R}^{d}\big)\), and Triebel-Lizorkin spaces, \(F_{s}^{p,q}\big(\mathbb{R}^{d}\big)\), where the parameters \(s \in \mathbb{R}\) and p, q ∈ (0, ∞] measure the “smoothness” and, respectively, the “size” of a given distribution in these spaces.

Let \(\mathcal{M}\) be a Riemannian manifold and suppose \(\varOmega \subset \mathcal{M}\) is a regular SKT domain. In this setting, we prove the well-posedness of the Regularity boundary value problem for the Laplace–Beltrami operator associated with the metric on \(\mathcal{M}\), with boundary data from the L p -based Sobolev space of order one on ∂ Ω. This is achieved using potential theoretic methods and complements work in [HoMiTa10], where the Dirichlet problem has been treated.

The main goal here is to develop a Calderon-Zygmund theory for singular integral operators of boundary layer potential type naturally associated with second-order elliptic systems on Riemannian manifolds which is effective in the treatment of boundary value problems in rough domains.

The main goal of this chapter is to introduce Hardy spaces in the context of d-Ahlfors-regular quasi-metric spaces by defining H p (X) as a collection of distributions whose maximal belongs to L p (X). This is in the spirit of the pioneering work of C.

We study the applicability of the method of layer potentials in the treatment of boundary value problems for the Laplace-Beltrami operator on Lipschitz sub-domains of Riemannian manifolds, in the case when the metric tensor g j k d x j ⊗ d x k g_{jk} dx_j\otimes dx_k has low regularity. Under the assumption that \[ | g j k ( x ) − g j k ( y ) | ≤ C ω ( | x − y | ) , |g_{jk}(x)-g_{jk}(y)|\leq C\,\omega (|x-y|), \] where the modulus of continuity ω \omega satisfies a Dini-type condition, we prove the well-posedness of the classical Dirichlet and Neumann problems with L p L^p boundary data, for sharp ranges of p p ’s and with optimal nontangential maximal function estimates.

Let L be a second order, (variable coefficient) elliptic differential operator and let u ∈ B p,p α (Ω), 1 0, satisfy Lu = 0 in the Lipschitz domain Ω. We show that u can exhibit more regularity on Besov scales for which smoothness is measured in L τ with r < p. Similar results are valid for functions representable in terms of layer potentials.

We consider two basic potential theoretic problems in Riemannian manifolds: Hodge decompositions and Maxwell's equations. Here we are concerned with smoothness and integrability assumptions. In the context of L p forms in Lipschitz domains, we show that both are well posed provided that 2− e p e , for some e >0, depending on the domain. Our approach is constructive (in the sense that we produce integral representation formulas for the solutions) and emphasizes the intimate connections between the two problems at hand. Applications to other related PDEs, such as boundary problems for the Hodge Dirac operator, are also presented.

We study the well-posedness of the half-Dirichlet and Poisson problems for Dirac operators in three-dimensional Lipschitz domains, with a special emphasis on optimal Lebesgue and Sobolev-Besov estimates. As an application, an elliptization procedure for the Maxwell system is devised. Copyright © 2002 John Wiley & Sons, Ltd.

We identify the optimal range of coefficients s, p for which differential forms with coefficients in the Sobolev space L p s (Ω) admit natural Hodge decompositions in arbitrary two and three dimensional Lipschitz domains Ω . To cite this article: D. Mitrea, M. Mitrea, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 109–112

For an arbitrary nonempty, open set $\Omega \subset \mathbb{R}^n$, $n \in \mathbb{N}$, of finite (Euclidean) volume, we consider the minimally defined higher-order Laplacian $(- \Delta)^m\big|_{C_0^{\infty}(\Omega)}$, $m \in \mathbb{N}$, and its Krein--von Neumann extension $A_{K,\Omega,m}$ in $L^2(\Omega)$. With $N(\lambda,A_{K,\Omega,m})$, $\lambda > 0$, denoting the eigenvalue counting function corresponding to the strictly positive eigenvalues of $A_{K,\Omega,m}$, we derive the bound $$ N(\lambda,A_{K,\Omega,m}) \leq (2 \pi)^{-n} v_n |\Omega| \{1 + [2m/(2m+n)]\}^{n/(2m)} \lambda^{n/(2m)}, \quad \lambda > 0, $$ where $v_n := \pi^{n/2}/\Gamma((n+2)/2)$ denotes the (Euclidean) volume of the unit ball in $\mathbb{R}^n$. The proof relies on variational considerations and exploits the fundamental link between the Krein--von Neumann extension and an underlying (abstract) buckling problem., Comment: 22 pages. Considerable improvements made

We initiate the study of the finiteness condition $\int_{\Omega}u(x)^{-\beta}\,dx\leq C(\Omega,\beta)0$ for which the aforementioned condition holds under various hypotheses on the smoothness of $\Omega$ and demands on the nature of the constant $C(\Omega,\beta)$. Classes of domains for which our analysis applies include bounded piecewise $C^1$ domains in ${\mathbb{R}}^n$, $n\geq 2$, with conical singularities (in particular polygonal domains in the plane), polyhedra in ${\mathbb{R}}^3$, and bounded domains which are locally of class $C^2$ and which have (finitely many) outwardly pointing cusps. For example, we show that if $u_N$ is the solution of the Saint Venant problem in the regular polygon $\Omega_N$ with $N$ sides circumscribed by the unit disc in the plane, then for each $\beta\in(0,1)$ the following asymptotic formula holds: % {eqnarray*} \int_{\Omega_N}u_N(x)^{-\beta}\,dx=\frac{4^\beta\pi}{1-\beta} +{\mathcal{O}}(N^{\beta-1})\quad{as}\,\,N\to\infty. {eqnarray*} % One of the original motivations for addressing the aforementioned issues was the study of sublevel set estimates for functions $v$ satisfying $v(0)=0$, $\nabla v(0)=0$ and $\Delta v\geq c>0$.

The Navier-Stokes equations are a system of nonlinear evolution equations modeling the flow of a viscous, incompressible fluid. One ingredient in the analysis of this system is the stationary, linear system known as the Stokes system, a boundary value problem (BVP) that will be described in detail in the next section. Layer potential methods in smoothly bounded domains in Euclidean space have proven useful in the analysis of the Stokes system, starting with work of Odqvist and Lichtenstein, and including work of Solonnikov and many others. See the discussion in Chapter III of [10] and in [17], for the case of flow in regions with smooth boundary. A treatment based on the modern language of pseudodifferential operators can be found in [18]. In 1988, E. Fabes, C. Kenig and G. Verchota [6], extended this classical layer potential approach to cover Lipschitz domains in Euclidean space. In [6] the main result concerning the (constant coefficient) Stokes system on Lipschitz domains with connected boundary in Euclidean space, is the treatment of the L-Dirichlet boundary value problem (and its regular version). To achieve this, the authors solve certain auxiliary Neumann type problems and then exploit the duality between these and the original BVP’s at the level of boundary integral operators. P. Deuring and W. von Wahl [4] made use of the analysis in [6] to demonstrate the short-time existence of solutions to the Navier-Stokes equations in bounded Lipschitz domains in threedimensional Euclidean space. It was necessary in [4] to include the hypothesis that the boundary be connected. The hypothesis that the boundary be connected pervaded much work on the application of layer potentials to analysis on Lipschitz domains. It was certainly natural to speculate that this restriction was an artifact of the methods used and not ∗Partly supported by NSF grant DMS-9870018. †Partially supported by NSF grant DMS-9877077. 1991Mathematics Subject Classification. Primary 35Q30, 76D05, 35J25; Secondary 42B20, 45E05.

We prove new mean value formulas for harmonic functions in polyhedra. Our approach is elementary and utilizes vanishing formulas for holomorphic functions of one complex variable and monogenic functions of one quaternionic or Clifford variable.

We develop a function theory associated with Dirac type operators on Lipschitz subdomains of Riemannian manifolds. The main emphasis is on Hardy spaces and boundary value problems, and our aim is to identify the geometric and analytic assumptions guaranteeing the validity of basic results from complex function theory in this general setting. For example, we study Plemelj-Calderon-Seeley-Bojarski type splittings of Cauchy boundary data into traces of ‘inner’ and ‘outer’ monogenics and show that this problem has finite index. We also consider Szego projections and the corresponding Lp-decompositions. Our approach relies on an extension of the classical Calderon-Zygmund theory of singular integral operators which allow one to consider Cauchy type operators with variable kernels on Lipschitz graphs. In the second part, where we explore connections with Maxwell's equations, the main novelty is the treatment of the corresponding electro-magnetic boundary value problem by recasting it as a ‘half’ Dirichlet problem for a suitable Dirac operator.

Let Lr(Ω) be the usual scale of Sobolev spaces and let ∆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 (−∆N )− r 2 +iδ map {f ∈ Lq(Ω); ∫ Ω f = 0} isomorphically onto Lr(Ω)/IR ?

Recall that in the case of the Dirichlet problem for the Laplace operator ∂2 x +∂ 2 y in Ω ⊆ R2, one prescribes the whole trace of a harmonic function in, say, L2(∂Ω). On the other hand, for the Cauchy-Riemann operator ∂x + i∂y, natural boundary problems are obtained by prescribing “half” of the trace of the analytic function in L2(∂Ω). Such half-Dirichlet problems arise when, for example, one prescribes the boundary values of the real part of an holomorphic function. This type of phenomenon has a natural extension to the higher dimensional setting in the context of Clifford algebras and Dirac operators. The main goal of this paper is to continue the line of investigation initiated in [Mi], [Mi2] (and further pursued in [McMM], [McM], [Mi3]) aimed at solving elliptic boundary problems in nonsmooth domains via tools originating in Clifford analysis. More specifically, here we shall study certain half-Dirichlet type problems for Dirac operators in nonsmooth Euclidean domains. Since no symbolic calculus for singular integral operators in the irregular context is available so far, one is forced to study such problems on an individual basis. Furthermore, it is well known that there are topological obstructions to the existence of local elliptic boundary conditions for general Dirac type operators (cf., e.g., [Gi]). Here we focus the discussion on the classical (full, as opposed to chiral) Dirac operator D := ∑ j ej∂j in Rm. Let Ω be an arbitrary Lipschitz subdomain of Rm. A specific example of a boundary value problem which fits the above description is

We consider layer potentials associated with the Hodge-Laplacian acting on differential forms of arbitrary degree defined on Lipschitz subdomains of a Riemannian manifold. The main emphasis is on the interplay between the mapping properties of such layer potentials and the topology of the underlying domain.

With “hat” denoting the Banach envelope (of a quasi-Banach space) we prove that Open image in new window if 0

(1.1) gj(x) := x− yj |x− yj |n , x ∈ Rn \ {yj}. Let Hs,p denote the usual scale of Lp-Sobolev spaces, and denote by div and Div the divergence of 1-tensors (i.e., vector fields) and 2-tensors, respectively. Given a divergence-free vector field u on Ω, possessing a certain regularity, e.g., u ∈ Hs,p(Ω), we want to investigate two closely related problems. One is to extend u to a divergence-free vector field ũ defined on a neighborhood of Ω (in fact, defined on Rn \{yj}), such that ũ is as smooth as u. The second is to produce an anti-symmetric 2-tensor field v such that (1.2) u = Div v + ∑

We present a simple, Clifford algebra-based approach to several key results in the theory of Maxwell's equations in non-smooth subdomains of R m . Among other things, we give new proofs to the boundary energy estimates of Rellich type for Maxwell's equations in Lipschitz domains from [20, 10], discuss radiation conditions and the case of variable wave number.