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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.

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

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 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 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 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 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 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.

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.

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.

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 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.

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 \(\alpha \in {\bigl ( 0,[\mathrm{log}_{2}C_{\rho }]^{-1}\bigr ]}\).

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 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 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 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.

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 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.

In this paper we establish a boundary behavior theory for H p spaces of Clifford valued functions monogenic in a Lipschitz domain in the case ρ>1 The action of a natural class of integral operators with monogenic kernels is studied on these spaces and several applications to PDE's in non-smooth domains 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.

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

In this chapter we study the implications of our general metrization theory at the level of quasimetric spaces, with special emphasis on analytical aspects. More specifically, we study the nature of Holder functions on quasimetric spaces by proving density, embeddings, separation, and extension theorems. We also quantify the richness of such spaces by introducing and studying a notion of index that interfaces tightly with the critical exponent beyond which the Holder spaces become trivial. Other applications are targeted to Hardy spaces on spaces of homogeneous type, regularized distance, Whitney decompositions, and partitions of unity, as well as the Gromov–Pompeiu–Hausdorff distance.

In this chapter the goal is to explore the implications of our general metrization theory to aspects of functional analysis in nonlocally convex topological vector spaces. Some of the concrete topics studied here deal with the completeness and separability of such spaces, as well as with the issues of pointwise convergence and the Fatou property in the case of function spaces. Various other applications to analysis on Boolean algebras are also considered.

We study the regularity of the higher dimensional Cauchy integral operator and of other closely related operators in terms of the weighted Sobolev-Besov spaces L p,s(∂Ω) on the Lipschitz hypersurface in ∂Ω and some appropriate weighted Hardy spaces on the non-smooth domain Ω in

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

The aim of this article is twofold. On the one hand, we study the well-posedness of the Lame system $${-\mu\Delta-\mu'\nabla{\rm div} }$$ in L q (Ω), where Ω is an open subset of $${{\mathbb{R}}^n}$$ satisfying mild regularity assumptions and the Lame moduli μ, μ′ are such that μ > 0 and μ + μ′ > 0. On the other hand, we prove the analyticity of the semigroup generated by the Lame operator as well as the maximal regularity property for the time-dependent Lame system equipped with a homogeneous Dirichlet boundary condition based on off-diagonal estimates.

We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrodinger operators describing nonlocal interactions in $L^2(\Omega; d^n x)$, $n\geq 2$, where $\Omega$ is an open set with a compact, nonempty boundary satisfying certain regularity conditions. As an application we describe a reduction of a certain ratio of Fredholm perturbation determinants associated with operators in $L^2(\Omega; d^n x)$ to Fredholm perturbation determinants associated with operators in $L^2(\partial\Omega; d^{n-1}\sigma)$. This leads to an extension of a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with a Schr\"odinger operator on the half-line $(0,\infty)$, in the case of local interactions, to a Wronski determinant of appropriate distributional solutions of the underlying Schrodinger equation., Comment: 18 pages

We develop the theory of layer potentials and related singular integral operators as a tool to study a variety of elliptic boundary problems on a family of domains introduced by Semmes [105, 106] and Kenig and Toro[64-66], which we call regular Semmes-Kenig-Toro domains. This extends the classic work of Fabes, Jodeit, and Riviere in several ways. For one, the class of domains that are locally graphs of functions whose gradients have vanishing mean oscillation (VMO 1 domains), which in turn contains the class of C 1 domains. In addition, we study not only the Dirichlet and Neumann boundary problems but also a variety of others. Furthermore, we treat not only constant coefficient operators but also operators with variable coefficients, including operators on manifolds.

The aim of this paper is to prove the boundedness of a category of integral operators mapping functions from Besov spaces on the boundary of a Lipschitz domain \(\Omega \subseteq \mathbb{R}^n \) into functions belonging to weighted Sobolev spaces in Ω. The model we have in mind is the Poisson integral operator $$(PIf)(x): = - \int_{\partial \Omega } {\partial _{v(y)} G(x,y)f(y)d\sigma (y),} {\rm }x \in \Omega ,$$ where G(·; ·) is the Green function for the Dirichlet Laplacian in Ω, \(\partial _v \) is the normal derivative, and σ is the surface area on \(\partial \Omega \), in the case where \(\Omega \subseteq \mathbb{R}^n \) is a bounded Lipschitz domain satisfying a uniform exterior ball condition.

Recall that a Lipschitz domain is a domain whose boundary is locally given by graphs of Lipschitz functions. The formulation of, respectively, the Dirichlet and regularity problems for the Laplacian in a Lipschitz domain \(\Omega \subset \mathbb{R}^n \) is $$ \begin{array}{*{20}c} {(D_\Delta )_p \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\Delta u = 0} & {{\rm in}\,\Omega ,} \\ \end{array}} \\ {\mathcal{N}u \in L^p (\partial \Omega ),} \\ {u|_{\partial \Omega } = f \in L^p (\partial \Omega ),} \\ \end{array}} \right.} & {(R_\Delta )_p \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\Delta u = 0} & {{\rm in}\,\Omega ,} \\ \end{array}} \\ {\mathcal{N}u \in L^p (\partial \Omega ),} \\ {u|_{\partial \Omega } = f \in L_1^p (\partial \Omega ),} \\ \end{array}} \right.} \\ \end{array} $$ (24.1) .