Search

Given a domain above a Lipschitz graph, we establish solvability results for strongly elliptic second-order systems in divergence-form, allowed to have lower-order (drift) terms, with L p L^p -boundary data for p p near 2 2 (more precisely, in an interval of the form ( 2 − ε , 2 ( n − 1 ) n − 2 + ε ) \big (2-\varepsilon ,\frac {2(n-1)}{n-2}+\varepsilon \big ) for some small ε > 0 \varepsilon >0 ). The main novel aspect of our result is that the coefficients of the operator do not have to be constant, or have very high regularity; instead they will satisfy a natural Carleson condition that has appeared first in the scalar case. A significant example of a system to which our result may be applied is the Lamé system for isotropic inhomogeneous materials. We show that our result applies to isotropic materials with Poisson ratio ν > 0.396 \nu >0.396 . Dealing with genuine systems gives rise to substantial new challenges, absent in the scalar case. Among other things, there is no maximum principle for general elliptic systems, and the De Giorgi–Nash–Moser theory may also not apply. We are, nonetheless, successful in establishing estimates for the square-function and the nontangential maximal operator for the solutions of the elliptic system described earlier, and use these as alternative tools for proving L p L^p solvability results for p p near 2 2 .

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

For an arbitrary open, nonempty, bounded set Ω⊂Rn, n∈N, and sufficiently smooth coefficients a,b,q, we consider the closed, strictly positive, higher-order differential operator AΩ,2m(a,b,q) in L2(Ω) defined on W02m,2(Ω), associated with the differential expression τ2m(a,b,q):=(∑j,k=1n(−i∂j−bj)aj,k(−i∂k−bk)+q)m,m∈N, and its Krein–von Neumann extension AK,Ω,2m(a,b,q) in L2(Ω). Denoting by N(λ;AK,Ω,2m(a,b,q)), λ>0, the eigenvalue counting function corresponding to the strictly positive eigenvalues of AK,Ω,2m(a,b,q), we derive the bound N(λ;AK,Ω,2m(a,b,q))≤Cvn(2π)−n(1+2m2m+n)n/(2m)λn/(2m),λ>0, where C=C(a,b,q,Ω)>0 (with C(In,0,0,Ω)=|Ω|) is connected to the eigenfunction expansion of the self-adjoint operator A˜2m(a,b,q) in L2(Rn) defined on W2m,2(Rn), corresponding to τ2m(a,b,q). Here vn:=πn/2/Γ((n+2)/2) denotes the (Euclidean) volume of the unit ball in Rn. Our method of proof relies on variational considerations exploiting the fundamental link between the Krein–von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of A˜2(a,b,q) in L2(Rn). We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension AF,Ω,2m(a,b,q) in L2(Ω) of AΩ,2m(a,b,q). No assumptions on the boundary ∂Ω of Ω are made.

We investigate closed, symmetric $L^2(\mathbb{R}^n)$-realizations $H$ of Schr\"odinger-type operators $(- \Delta +V)\upharpoonright_{C_0^{\infty}(\mathbb{R}^n \setminus \Sigma)}$ whose potential coefficient $V$ has a countable number of well-separated singularities on compact sets $\Sigma_j$, $j \in J$, of $n$-dimensional Lebesgue measure zero, with $J \subseteq \mathbb{N}$ an index set and $\Sigma = \bigcup_{j \in J} \Sigma_j$. We show that the defect, $\mathrm{def}(H)$, of $H$ can be computed in terms of the individual defects, $\mathrm{def}(H_j)$, of closed, symmetric $L^2(\mathbb{R}^n)$-realizations of $(- \Delta + V_j)\upharpoonright_{C_0^{\infty}(\mathbb{R}^n \setminus \Sigma_j)}$ with potential coefficient $V_j$ localized around the singularity $\Sigma_j$, $j \in J$, where $V = \sum_{j \in J} V_j$. In particular, we prove \[ \mathrm{def}(H) = \sum_{j \in J} \mathrm{def}(H_j), \] including the possibility that one, and hence both sides equal $\infty$. We first develop an abstract approach to the question of decoupling of deficiency indices and then apply it to the concrete case of Schr\"odinger-type operators in $L^2(\mathbb{R}^n)$. Moreover, we also show how operator (and form) bounds for $V$ relative to $H_0= - \Delta\upharpoonright_{H^2(\mathbb{R}^n)}$ can be estimated in terms of the operator (and form) bounds of $V_j$, $j \in J$, relative to $H_0$. Again, we first prove an abstract result and then show its applicability to Schr\"odinger-type operators in $L^2(\mathbb{R}^n)$. Extensions to second-order (locally uniformly) elliptic differential operators on $\mathbb{R}^n$ with a possibly strongly singular potential coefficient are treated as well., Comment: 33 pages

We show that the boundedness of the Hardy-Littlewood maximal operator on a Kothe function space X and on its Kothe dual X is equivalent to the well-posedness of the X-Dirichlet and X-Dirichlet problems in Rn + in the class of all second-order, homogeneous, elliptic systems, with constant complex coefficients. As a consequence, we obtain that the Dirichlet problem for such systems is well-posed for boundary data in Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versions. We also discuss a version of the aforementioned result which contains, as a particular case, the Dirichlet problem for elliptic systems with data in the classical Hardy space H1, and the Beurling-Hardy space HAp for p € (1,∞). Based on the well-posedness of the Lp-Dirichlet problem we then prove the uniqueness of the Poisson kernel associated with such systems, as well as the fact that they generate a strongly continuous semigroup in natural settings. Finally, we establish a general Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for null-solutions of such systems., The First author has been supported in part by MINECO Grant MTM2010-16518, ICMAT Severo Ochoa project SEV-2011-0087. He also acknowledges that the research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT. The second author has been supported in part by a Simons Foundation grant #200750, the third author has been supported in part by US NSF grant #0547944, while the fourth author has been supported in part by the Simons Foundation grant #281566, and by a University of Missouri Research Leave grant

We consider layer potentials associated to elliptic operators $Lu=-{\rm div}(A \nabla u)$ acting in the upper half-space $\mathbb{R}^{n+1}_+$ for $n\geq 2$, or more generally, in a Lipschitz graph domain, where the coefficient matrix $A$ is $L^\infty$ and $t$-independent, and solutions of $Lu=0$ satisfy interior estimates of De Giorgi/Nash/Moser type. A "Calder\'on-Zygmund" theory is developed for the boundedness of layer potentials, whereby sharp $L^p$ and endpoint space bounds are deduced from $L^2$ bounds. Appropriate versions of the classical "jump-relation" formulae are also derived. The method of layer potentials is then used to establish well-posedness of boundary value problems for $L$ with data in $L^p$ and endpoint spaces.

One of the principal topics of this paper concerns the realization of self-adjoint operators $L_{\Theta, \Om}$ in $L^2(\Om; d^n x)^m$, $m, n \in \bbN$, associated with divergence form elliptic partial differential expressions $L$ with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains $\Om \subset \bbR^n$. In particular, we develop the theory in the vector-valued case and hence focus on matrix-valued differential expressions $L$ which act as $$ Lu = - \biggl(\sum_{j,k=1}^n\partial_j\bigg(\sum_{\beta = 1}^m a^{\alpha,\beta}_{j,k}\partial_k u_\beta\bigg) \bigg)_{1\leq\alpha\leq m}, \quad u=(u_1,...,u_m). $$ The (nonlocal) Robin-type boundary conditions are then of the form $$ \nu \cdot A D u + \Theta \big[u\big|_{\partial \Om}\big] = 0 \, \text{on $\partial \Om$}, $$ where $\Theta$ represents an appropriate operator acting on Sobolev spaces associated with the boundary $\partial \Om$ of $\Om$, $\nu$ denotes the outward pointing normal unit vector on $\partial\Om$, and $Du:=\bigl(\partial_j u_\alpha\bigr)_{\substack{1\leq\alpha\leq m 1\leq j\leq n}}$. Assuming $\Theta \geq 0$ in the scalar case $m=1$, we prove Gaussian heat kernel bounds for $L_{\Theta, \Om}$ by employing positivity preserving arguments for the associated semigroups and reducing the problem to the corresponding Gaussian heat kernel bounds for the case of Neumann boundary conditions on $\partial \Om$. We also discuss additional zero-order potential coefficients $V$ and hence operators corresponding to the form sum $L_{\Theta, \Om} + V$., Comment: 45 pages; small corrections are made in this version

We announce a local $T(b)$ theorem, an inductive scheme, and $L^p$ extrapolation results for $L^2$ square function estimates related to the analysis of integral operators that act on Ahlfors-David regular sets of arbitrary codimension in ambient quasi-metric spaces. The inductive scheme is a natural application of the local $T(b)$ theorem and it implies the stability of $L^2$ square function estimates under the so-called big pieces functor. In particular, this analysis implies $L^p$ and Hardy space square function estimates for integral operators on uniformly rectifiable subsets of the Euclidean space., 9 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 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 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.

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.

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.

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 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 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.

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. [...]

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.

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 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.

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 ?

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.

Let D D be a Dirac type operator on a compact manifold M {\mathcal {M}} and let Σ \Sigma be a Lipschitz submanifold of codimension one partitioning M {\mathcal {M}} into two Lipschitz domains Ω ± \Omega _{\pm } . Also, let H ± p ( Σ , D ) {\mathcal {H}}^{p}_{\pm }(\Sigma ,D) be the traces on Σ \Sigma of the ( L p L^{p} -style) Hardy spaces associated with D D in Ω ± \Omega _{\pm } . Then ( H − p ( Σ , D ) , H + p ( Σ , D ) ) ({\mathcal {H}}^{p}_{-}(\Sigma ,D),{\mathcal {H}}^{p}_{+}(\Sigma ,D)) is a Fredholm pair of subspaces for L p ( Σ ) L^{p}(\Sigma ) (in Kato’s sense) whose index is the same as the index of the Dirac operator D D considered on the whole manifold M {\mathcal {M}} .

We investigate the stability of Fredholm properties on interpolation scales of quasi-Banach spaces. This analysis is motivated by problems arising in PDE’s and several applications are presented.

We study boundary value problems for the time-harmonic form of the Maxwell equations, as well as for other related systems of equations, on arbitrary Lipschitz domains in the three-dimensional Euclidean space. The main goal is to develop the corresponding theory for Lp-integrable bounday data for optimal values of p’s. We also discuss a number of relevant applications in electromagnetic scattering.

A layer potential based approach for boundary value problems for harmonic differential forms in nonsmooth domains is developed. This allows a complete and unified treatment of several fundamental problems in potential theory.

The principal aim of this short note is to extend a recent result on Gaussian heat kernel bounds for self-adjoint $L^2(\Om; d^n x)$-realizations, $n\in\bbN$, $n\geq 2$, of divergence form elliptic partial differential expressions $L$ with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains $\Om \subset \bbR^n$, where $$ Lu = - \sum_{j,k=1}^n\partial_j a_{j,k}\partial_k u. $$ The (nonlocal) Robin-type boundary conditions are then of the form $$ \nu\cdot A\nabla u + \Theta\big[u\big|_{\partial\Om}\big]=0 \, \text{on} \, \partial\Omega, $$ where $\Theta$ represents an appropriate operator acting on Sobolev spaces associated with the boundary $\partial \Om$ of $\Om$, and $\nu$ denotes the outward pointing normal unit vector on $\partial\Om$., Comment: 14 pages

We give new proofs to the Lusin area integral theorem of Dahlberg. Our techniques rely on the theory of elliptic boundary value problems on nonsmooth domains and are shown to extend to other important cases, including systems of equations.

(1994). A Generalization of a Theorem of Euler. The American Mathematical Monthly: Vol. 101, No. 1, pp. 55-58.

In the first part of the paper, we give a satisfactory definition of the Stokes operator in Lipschitz domains in $ {\mathbb{R}^n} $ when boundary conditions of Neumann type are considered. We then proceed to establish optimal global Sobolev regularity results for vector fields in the domains of fractional powers of this Neumann–Stokes operator. Finally, we study the existence, regularity, and uniqueness of mild solutions of the Navier–Stokes system with Neumann boundary conditions. Bibliography: 43 titles. Illustrations: 2 figures.