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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 .

We prove well-posedness results for the Dirichlet problem in Rn + for homogeneous, second order, constant complex coe cient elliptic systems with boundary data in generalized H older spaces C !(Rn1;CM) and in generalized Morrey-Campanato spaces E !;p(Rn1;CM) under certain assumptions on the growth function !. We also identify a class of growth functions ! for which C !(Rn1;CM) = E !;p(Rn1;CM) and for which the aforementioned well-posedness results are equivalent, in the sense that they have the same unique solution, satisfying natural regularity properties and estimates., Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in R&D" (SEV-2015-0554), European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT, Simons Foundation grant #281566

It is known (cf. Martell et al. in Rev Mat Iberoam 32(3):913–970, 2016) that the $$L^p$$ -Dirichlet problem for constant coefficient second-order systems satisfying the Legendre-Hadamard (strong) ellipticity condition is well posed in the upper half-space. However, this may fail if only weak ellipticity for the system in question is assumed. In this paper we shall show that the aforementioned failure is at a fundamental level, in the sense that there exist systems which are weakly elliptic (i.e., their characteristic matrix is invertible) for which the $$L^p$$ -Dirichlet problem in the upper half-space is not even Fredholm solvable.

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.

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

Let L be a second-order, homogeneous, constant (complex) coefficient elliptic system in \({\mathbb {R}}^n\). The goal of this article is to prove a Fatou-type result, regarding the a.e. existence of the nontangential boundary limits of any null-solution u of L in the upper half-space, whose nontangential maximal function satisfies an integrability condition with respect to the weighted Lebesgue measure (1 + |x′|n−1)−1dx′ in \({\mathbb {R}}^{n-1}\equiv \partial {\mathbb {R}}^n_{+}\). This is the best result of its kind in the literature. In addition, we establish a naturally accompanying integral representation formula involving the Agmon-Douglis-Nirenberg Poisson kernel for the system L. Finally, we use this machinery to derive well-posedness results for the Dirichlet boundary value problem for L in \({\mathbb {R}}^n_{+}\) formulated in a manner which allows for the simultaneous treatment of a variety of function spaces.

We prove that for any homogeneous, second-order, constant complex coefficient elliptic system L in ℝ, the Dirichlet problem in ℝ with boundary data in BMO.ℝ/ is well-posed in the class of functions u for which the Littlewood-Paley measure associated with u, namely dμ(x', t):=| ∇u(x',t)| t dx' dt; is a Carleson measure in ℝ. In the process we establish a Fatou-type theorem guaranteeing the existence of the pointwise nontangential boundary trace for smooth null-solutions u of such systems satisfying the said Carleson measure condition. In concert, these results imply that the space BMO(ℝ) can be characterized as the collection of nontangential pointwise traces of smooth null-solutions u to the elliptic system L with the property that μ is a Carleson measure in ℝ. We also establish a regularity result for the BMO-Dirichlet problem in the upper half-space, to the effect that the nontangential pointwise trace on the boundary of ℝ of any given smooth nullsolutions u of L in ℝ satisfying the above Carleson measure condition actually belongs to Sarason's space VMO.ℝ/ if and only if μ(T(Q))/|Q|→0 as |Q|→0, uniformly with respect to the location of the cubeQ⊂ℝ (where T(Q) is the Carleson box associated withQ, and |Q| denotes the Euclidean volume of Q). Moreover, we are able to establish the well-posedness of the Dirichlet problem in ℝ C for a system L as above in the case when the boundary data are prescribed in Morrey-Campanato spaces in ℝ. In such a scenario, the solution u is required to satisfy a vanishing Carleson measure condition of fractional order. By relying on these well-posedness and regularity results we succeed in producing characterizations of the space VMO as the closure in BMO of classes of smooth functions contained in BMO within which uniform continuity may be suitably quantified (such as the class of smooth functions satisfying a Hölder or Lipschitz condition). This improves on Sarason's classical result describing VMO as the closure in BMO of the space of uniformly continuous functions with bounded mean oscillations. In turn, this allows us to show that any Calderón-Zygmund operator T satisfying T (1) = 0 extends as a linear and bounded mapping from VMO (modulo constants) into itself. In turn, this is used to describe algebras of singular integral operators on VMO, and to characterize the membership to VMO via the action of various classes of singular integral operators., The first author would like to express his gratitude to the University of Missouri-Columbia (USA), for its support and hospitality while he was visiting this institution. The first author acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the \Severo Ochoa Programme for Centres of Excellence in R&D" (SEV-2015-0554). 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 the Simons Foundation grant #426669, the third author has been supported in part by the Simons Foundation grant #318658, 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 survey recent progress in a program which to date has produced [18]- [25], aimed at proving general Fatou-type results and establishing the well-posedness of a variety of boundary value problems in the upper half-space Rn + for second-order, homogeneous, constant complex coe cient, elliptic systems L, formulated in a manner that emphasizes pointwise nontangential boundary traces of the null-solutions of L in Rn +. 1, The rst author 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. He also acknowledges nancial support from the Spanish Ministry of Economy and Competitiveness, through the \Severo Ochoa Programme for Centres of Excellence in R&D" (SEV-2015-0554).

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.

We show that operators of layer potential type on surfaces that are locally graphs of Lipschitz functions with gradients in vmo are equal, modulo compacts, to pseudodifferential operators (with rough symbols), for which a symbol calculus is available. We build further on the calculus of operators whose symbols have coefficients in [math] vmo, and apply these results to elliptic boundary problems on domains with such boundaries, which in turn we identify with the class of Lipschitz domains with normals in vmo. This work simultaneously extends and refines classical work of Fabes, Jodeit and Rivière, and also work of Lewis, Salvaggi and Sisto, in the context of [math] surfaces.

We prove that given any k∈N, for each open set Ω⊆Rn and any closed subset D of Ω¯ such that Ω is locally an (e,δ)-domain near ∂Ω∖D, there exists a linear and bounded extension operator Ek,D mapping, for each p∈[1,∞], the space WDk,p(Ω) into WDk,p(Rn). Here, with O denoting either Ω or Rn, the space WDk,p(O) is defined as the completion in the classical Sobolev space Wk,p(O) of (restrictions to O of) functions from Cc∞(Rn) whose supports are disjoint from D. In turn, this result is used to develop a functional analytic theory for the class WDk,p(Ω) (including intrinsic characterizations, boundary traces and extensions results, interpolation theorems, among other things) which is then employed in the treatment of mixed boundary value problems formulated in locally (e,δ)-domains. Finally, we also prove extension results on the scales of Besov and Bessel potential spaces on (e,δ)-domains with partially vanishing traces on Ahlfors regular sets and explore some of the implications of such extension results.

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

We study the infinitesimal generator of the Poisson semigroup in L associated with homogeneous, second-order, strongly elliptic systems with constant complex coefficients in the upper-half space, which is proved to be the Dirichlet-to-Normal mapping in this setting. Also, its domain is identified as the linear subspace of the L-based Sobolev space of order one on the boundary of the upper-half space consisting of functions for which the Regularity problem is solvable. Moreover, for a class of systems containing the Lamé system, as well as all second-order, scalar elliptic operators, with constant complex coefficients, the action of the infinitesimal generator is explicitly described in terms of singular integral operators whose kernels involve first-order derivatives of the canonical fundamental solution of the given system. Furthermore, arbitrary powers of the infinitesimal generator of the said Poisson semigroup are also described in terms of higher order Sobolev spaces and a higher order Regularity problem for the system in question. Finally, we indicate how our techniques may be adapted to treat the case of higher order systems in graph Lipschitz domains., The first author acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554). 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 # 426669, the third author has been supported in part by the Simons Foundation grant #318658, while the fourth author has been supported in part by the Simons Foundation grant # 281566, and by a University of Missouri Research Leave grant.

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 discuss geometrical scenarios guaranteeing that functions defined on a given set may be extended to the entire ambient, with preservation of the class of regularity. This extends to arbitrary quasi-metric spaces work done by E.J. McShane in the context of metric spaces, and to geometrically doubling quasi-metric spaces work done by H. Whitney in the Euclidean setting. These generalizations are quantitatively sharp.

We are concerned with establishing completeness and separability criteria for large classes of topological vector spaces which are typically non-locally convex, including Lebesgue-like spaces, Lorentz spaces, Orlicz spaces, mixed-normed spaces, tent spaces, and discrete Triebel–Lizorkin and Besov spaces. For vector spaces of measurable functions we also derive pointwise convergence results. Our approach relies on abstract capacitary estimates and works in certain cases of interest even in the absence of a background measure space and/or of a vector space structure.

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 $\partial\Omega$.

The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators $\Theta$ which give rise to self-adjoint Laplacians $-\Delta_{\Theta, \Omega}$ in $L^2(\Omega; d^n x)$ with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains $\Omega\subset\bbR^n$, $n\in\bbN$, $n\geq 2$. Second, we extend Friedlander's inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains $\Omega$, following an approach introduced by Filonov for this type of problems., Comment: 23 pages, added Remark 5.4

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.