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We introduce and study Hardy spaces consisting of Clifford algebra-valued functions annihilated by perturbed Dirac operators in exterior uniformly rectifiable (UR) domains, and which radiate at infinity. In this context, we establish a higher dimensional version of Cauchy’s vanishing theorem, whose proof makes use of the properties of Cauchy-like operators in exterior UR domains, a sharp version of the Divergence Theorem in exterior Ahlfors regular domains, and a good understanding of the nature of various radiation conditions and properties of the far field pattern for Clifford algebra-valued null-solutions of the Helmholtz operator.

The principal aim of this paper is to derive an abstract form of the third Green identity associated with a proper extension $T$ of a symmetric operator $S$ in a Hilbert space $\mathfrak H$, employing the technique of quasi boundary triples for $T$. The general results are illustrated with couplings of Schr\"{o}dinger operators on Lipschitz domains on smooth, boundaryless Riemannian manifolds., Comment: 26 pages

The aim of this article is threefold. Firstly, we study Hardy spaces, hpL(Ω), associated with an operator L which is either the Dirichlet Laplacian ∆D or the Neumann Laplacian ∆N on a bounded Lipschitz domain Ω in Rn, for 0 < p ≤ 1. We obtain equivalent characterizations of these function spaces in terms of maximal functions and atomic decompositions. Secondly, we establish regularity results for the Green operators, regarded as the inverse of the Dirichlet and Neumann Laplacians, in the context of Hardy spaces associated with these operators on a bounded semiconvex domain Ω in Rn. Thirdly, we study relations between the Hardy spaces associated with operators and the standard Hardy spaces hr(Ω) and h p z(Ω), then establish regularity of the Green operators for the Dirichlet problem on a bounded semiconvex domain Ω in Rn, and for the Neumann problem on a bounded convex domain Ω in Rn, in the context of the standard Hardy spaces hr(Ω) and hz(Ω). This gives a new solution to the conjecture made by D.-C. Chang, S. Krantz and E.M. Stein regarding the regularity of Green operators for the Dirichlet and Neumann problems on hr(Ω) and h p z(Ω), respectively, for all n n+1 < p ≤ 1.

We prove that an open, proper, nonempty subset of ${\mathbb{R}}^n$ is a locally Lyapunov domain if and only if it satisfies a uniform hour-glass condition. The limiting cases are as follows: Lipschitz domains may be characterized by a uniform double cone condition, and domains of class may be characterized by a uniform two-sided ball condition. We discuss a sharp generalization of the Hopf–Oleinik boundary point principle for domains satisfying an interior pseudoball condition, for semi-elliptic operators with singular drift and obtain a sharp version of the Hopf strong maximum principle for second order, nondivergence form differential operators with singular drift. Bibliography: 66 titles. Illustrations: 7 figures.

We prove L p-bounds for the Riesz transforms d/ √ −∆, δ/ √ −∆ associated with the Hodge-Laplacian ∆ = −δd − dδ equipped with absolute and relative boundary conditions in a Lipschitz subdomain Ω of (smooth) Riemannian manifold M, for p in a certain interval depending on the Lipschitz character of the domain.

We discuss a sharp generalization of the Hopf-Oleinik boundary point principle (BPP) for domains satisfying an interior pseudo-ball condition, for non-divergence form, semi-elliptic operators with ...

We study the basic properties of the Laplace–Beltrami operator on Lipschitz surfaces, as well as abstract Lipschitz manifolds, including mapping and invertibility properties on scales of Sobolev spaces, being the infinitesimal generator of an analytic semigroup, the nature of the spectrum and the regularity of eigenfunctions. Much of this analysis is carried out in the more general case of second order, divergence-form, strongly elliptic differential operators with bounded, measurable, complex matrix-valued coefficients. Bibliography: 34 titles.

We establish the well-posedness of the inhomogeneous Dirichlet problem for Δ2 in arbitrary Lipschitz domains in $ {\mathbb{R}^3} $ , with data from Besov–Triebel–Lizorkin spaces, for the optimal range of indices. The main novel contribution is to allow for certain nonlocally convex spaces to be considered, and to establish integral representations for the solution. Bibliography: 57 titles. Illustrations: 1 figure.

We study the fully inhomogeneous Dirichlet problem for the Laplacian in bounded convex domains in R n , when the size/smoothness of both the data and the solution are measured on scales of Besov and Triebel–Lizorkin spaces. As a preamble, we deal with the Dirichlet and Regularity problems for harmonic functions in convex domains, with optimal nontangential maximal function estimates. As a corollary, sharp estimates for the Green potential are obtained in a variety of contexts, including local Hardy spaces. A substantial part of this analysis applies to bounded semiconvex domains (i.e., Lipschitz domains satisfying a uniform exterior ball condition).

We study the Dirichlet problem, in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order with bounded, complex-valued coefficients. A sharp corollary of our main solvability result is that the operator of this problem performs an isomorphism between weighted Sobolev spaces when its coefficients and the unit normal of the boundary belong to the space VMO.

In this paper, we prove a $H^1$-coercive estimate for differential forms of arbitrary degrees in semi-convex domains of the Euclidean space. The key result is an integral identity involving a boundary term in which the Weingarten matrix of the boundary intervenes, established for any Lipschitz domain $\Omega\subseteq \mathcal{R}^n$ whose outward unit normal $\nu$ belongs to $L^{n-1}_1(\partial\Omega)$, the $L^{n-1}$-based Sobolev space of order one on $\partial\Omega$.

We prove the well-posedness of the mixed problem for the Stokes system in a class of Lipschitz domains inRn{\mathbb {R}}^n,n≥3n\geq 3. The strategy is to reduce the original problem to a boundary integral equation, and we establish certain new Rellich-type estimates which imply that the intervening boundary integral operator is semi-Fredholm. We then prove that its index is zero by performing a homotopic deformation of it onto an operator related to the Lamé system, which has recently been shown to be invertible.

The goal of this work is to study the inhomogeneous Dirichlet problem for the Stokes system in a Lipschitz domain Ω ⊆ ℝ n , n⩾2. Our main result is that this problem is well posed in Besov-Triebel-Lizorkin spaces, provided that the unit normal ν to Ω has small mean oscillation.

We study the interplay between the geometry of Hardy spaces and functional analytic properties of singular integral operators (SIO’s), such as the Riesz transforms as well as Cauchy–Clifford and harmonic double-layer operator, on the one hand and, on the other hand, the regularity and geometric properties of domains of locally finite perimeter. Among other things, we give several characterizations of Euclidean balls, their complements, and half-spaces, in terms of the aforementioned SIO’s.

We study the Maxwell system with mixed boundary conditions in a Lipschitz domain Ω in ℝ 3 . It is assumed that two disjoint, relatively open subsets ∑ e , ∑ h of ∂Ω such that ∑ e ∩ ∑ h = ∂∑ e = ∂∑ h have been fixed, and one prescribes the tangential components of the electric and magnetic fields on ∑ e and ∑ h , respectively. Under suitable geometric assumptions on ∂Ω, ∑ e and ∑ h , we prove that this boundary value problem is well-posed when L P -estimates for the nontangential maximal function are sought, with p near 2. A higher-dimensional version of this result is established as well, in the language of differential forms. This extends earlier work by R. Brown and by the author and collaborators.

We study generalized polar decompositions of densely defined closed linear operators in Hilbert spaces and provide some applications to relatively (form) bounded and relatively (form) compact perturbations of self-adjoint, normal, and m-sectorial operators.

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 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.

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 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 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

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 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 ?

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.