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Layer potentials and boundary value problems for Laplacian in Lipschitz domains with data in quasi-Banach Besov spaces
Abstract
We study the Dirichlet and Neumann boundary value problems for the Laplacian in a Lipschitz domain Ω, with boundary data in the Besov space B s p,p (∂Ω). The novelty is to identify a way of measuring smoothness for the solution u which allows us to consider the case p<1. This is accomplished by using a Besov-based non-tangential maximal function in place of the classical one. This builds on the works of D. Jerison and C. E. Kenig [J. Funct. Anal. 130, No. 1, 161–219 (1995; Zbl 0832.35034)] where the case p>1 was treated.
... Let us point out that although the range (1.3) is sharp in the class of Lipschitz domains if one insists that 1 ≤ p ≤ ∞, there are suitable versions of (1.2) which continue to hold for certain pairs of indices (s, p) with p < 1. In our earlier work [13], we have produced such an extension of the main result in [11] based on a modification of the non-tangential maximal operator (1.4), i.e. ...
... is the trace of u on ∂Ω, and C is a Cauchy-type singular integral operator whose properties closely mirror those of the harmonic double layer (cf. §4 in [13] for details in the case when p > 1). Then (4.15) follows from the mapping properties of C established in Theorem 1.2, in concert with the estimate ...
... Now, the implication ∆u = 0 in Ω, N ∞ s (u) ∈ L p (∂Ω), u ∂Ω = 0 =⇒ u ≡ 0 in Ω (6.9) has been proved in Theorem 6.1 of [13], granted that s, p are as in (1.16)-(1.17). Clearly, this takes care of the uniqueness statement in Theorem 1.1. ...
The purpose of the present paper is to continue the program of study of elliptic boundary value problems on Lipschitz domains with boundary data in quasi-Banach Besov spaces B p,p s (∂Ω), initiated in [13]. Introducing a modified square function which is well-adapted for handling data with a fractional amount of smoothness, we establish the well-posedness of the Dirichlet and Neumann boundary problems for the Laplacian in Lipschitz domains, for a range of indices which includes values of p less than 1. An important ingredient in this regard is establishing suitable square-function estimates for singular integral of potential type.
... In the recent paper [5], authors studied Dirichlet and Neumann problems for (1.3) with data in quasi-Banach Besov spaces. They used Besov based non-tangential maximal function in place of the classical one to achieve similar results to those in [3] and [4] for f ∈ B s p,p with p < 1. ...
... They used Besov based non-tangential maximal function in place of the classical one to achieve similar results to those in [3] and [4] for f ∈ B s p,p with p < 1. For detailed information on smoothness properties of (1.3) we refer to [5] and the references therein. ...
Here we study Dirichlet and Neumann problems for a special Helmholtz equation on an annulus. Our main aim is to measure smoothness of solutions for the boundary datum in Besov spaces. We shall use operator theory to solve this problem. The most important advantage of this technique is that it enables to consider equations in vector-valued settings. It is interesting to note that optimal regularity of this problem will be a special case of our main result.
... We remark that in Lipschitz domains, there is an extensive literature concerning solution of the Regularity, Neumann, and Dirichlet problems by way of layer potentials (with boundary data in L p ) for classical linear elliptic PDE arising in mathematical physics, e.g., Laplace's equation, Maxwell's equation, Stokes and Láme systems of equations (see , [2], [3], [4], [13]). More recently layer potential solutions to these problems have been studied for Laplace's equation in domains beyond Lipschitz domains and in Lipschitz domains with boundary data in certain Besov spaces (see [7] and [14] for references). To compare our results with those cited above, we shall need some notation. ...
... Theorems 1.4-1.6 can be thought of as weak versions for Besov spaces of the Regularity, Neumann, and Dirichlet problems with boundary data in a Besov space. For Lipschitz domains it follows from the results in [14] that analogues of Theorems 1.4-1.6, hold for boundary data in a variety of other Besov and Hardy spaces. ...
In this paper, we define boundary single and double layer potentials for Laplace’s equation in certain bounded domains with
d-Ahlfors regular boundary, considerably more general than Lipschitz domains. We show that these layer potentials are invertible
as mappings between certain Besov spaces and thus obtain layer potential solutions to the regularity, Neumann, and Dirichlet
problems with boundary data in these spaces.
... From Costabel and Wendland [8,Remark 3.15] (see also Steinbach and Wendland [28] and Mayboroda and Mitrea [24]), we deduce the validity of the following. The kernels and cokernels of ± 1 2 I +K ∂Ω are also known; they are independent of τ . ...
In contrast with the well-known methods of matching asymptotics and multiscale (or compound) asymptotics, the " functional analytic approach " of Lanza de Cristoforis (Analysis 28, 2008) allows to prove convergence of expansions around interior small holes of size for solutions of elliptic boundary value problems. Using the method of layer potentials, the asymptotic behavior of the solution as tends to zero is described not only by asymptotic series in powers of , but by convergent power series. Here we use this method to investigate the Dirichlet problem for the Laplace operator where holes are collapsing at a polygonal corner of opening . Then in addition to the scale there appears the scale . We prove that when is irrational, the solution of the Dirichlet problem is given by convergent series in powers of these two small parameters. Due to interference of the two scales, this convergence is obtained, in full generality, by grouping together integer powers of the two scales that are very close to each other. Nevertheless, there exists a dense subset of openings (characterized by Diophantine approximation properties), for which real analyticity in the two variables and holds and the power series converge unconditionally. When is rational, the series are unconditionally convergent, but contain terms in log .
... From Costabel and Wendland [8,Remark 3.15] (see also Steinbach and Wendland [28] and Mayboroda and Mitrea [24]), we deduce the validity of the following. The kernels and cokernels of ± 1 2 I + K ∂Ω are also known; they are independent of τ . ...
In contrast with the well-known methods of matching asymptotics and multiscale (or compound) asymptotics, the "functional analytic approach" of Lanza de Cristoforis (Analysis 28, 2008) allows to prove convergence of expansions around interior small holes of size for solutions of elliptic boundary value problems. Using the method of layer potentials, the asymptotic behavior of the solution as tends to zero is described not only by asymptotic series in powers of , but by convergent power series. Here we use this method to investigate the Dirichlet problem for the Laplace operator where holes are collapsing at a polygonal corner of opening . Then in addition to the scale there appears the scale = ^{/}. We prove that when / is irrational, the solution of the Dirichlet problem is given by convergent series in powers of these two small parameters. Due to interference of the two scales, this convergence is obtained, in full generality, by grouping together integer powers of the two scales that are very close to each other. Nevertheless, there exists a dense subset of openings (characterized by Diophantine approximation properties), for which real analyticity in the two variables and holds and the power series converge unconditionally. When / is rational, the series are unconditionally convergent, but contain terms in log .
Sobolev spaces become the established and universal language of partial differential equations and mathematical analysis. Among a huge variety of problems where Sobolev spaces are used, the following important topics are in the focus of this volume: boundary value problems in domains with singularities, higher order partial differential equations, local polynomial approximations, inequalities in Sobolev-Lorentz spaces, function spaces in cellular domains, the spectrum of a Schrodinger operator with negative potential and other spectral problems, criteria for the complete integrability of systems of differential equations with applications to differential geometry, some aspects of differential forms on Riemannian manifolds related to Sobolev inequalities, Brownian motion on a Cartan-Hadamard manifold, etc. Two short biographical articles on the works of Sobolev in the 1930's and foundation of Akademgorodok in Siberia, supplied with unique archive photos of S. Sobolev are included.
This article considers two weight estimates for the single layer potential — corresponding to the Laplace operator in R
N+1 — on Lipschitz surfaces with small Lipschitz constant. We present conditions on the weights to obtain solvability and uniqueness results in weighted Lebesgue spaces and weighted homogeneous Sobolev spaces, where the weights are assumed to be radial and doubling. In the case when the weights are additionally assumed to be differentiable almost everywhere, simplified conditions in terms of the logarithmic derivative are presented, and as an application, we prove that the operator corresponding to the single layer potential in question is an isomorphism between certain weighted spaces of the type mentioned above. Furthermore, we consider several explicit weight functions. In particular, we present results for power exponential weights which generalize known results for the case when the single layer potential is reduced to a Riesz potential, which is the case when the Lipschitz surface is given by a hyperplane.
This article considers two weight estimates for the single layer potential
--- corresponding to the Laplace operator in --- on
Lipschitz surfaces with small Lipschitz constant. We present conditions on the
weights to obtain solvability and uniqueness results in weighted Lebesgue
spaces and weighted homogeneous Sobolev spaces, where the weights are assumed
to be radial and doubling. In the case when the weights are additionally
assumed to be differentiable almost everywhere, simplified conditions in terms
of the logarithmic derivative are presented, and as an application, we prove
that the operator corresponding to the single layer potential in question is an
isomorphism between certain weighted spaces of the type mentioned above.
Furthermore, we consider several explicit weight functions. In particular, we
present results for power exponential weights which generalize known results
for the case when the single layer potential is reduced to a Riesz potential,
which is the case when the Lipschitz surface is given by a hyperplane.
Let the nonnegative singular potential V belong to the reverse Hölder class on , and let (n − 1)/n p ≤ 2, we establish the solvability and derivative estimates for the solutions to the Neumann problem and the regularity problem
of the Schrödinger equation − Δu + Vu = 0 in a connected Lipschitz domain Ω, with boundary data in the Hardy space and the modified Hardy–Sobolev space related to the potential V. To deal with the H
p
regularity problem, we construct a new characterization of the atomic decomposition for space. The invertibility of the boundary layer potentials on Hardy spaces and Hölder spaces are shown in this paper.
This article has been retracted by the Chief Editor.Reason: This article was not intended to be published in the Journal of Differential Equations.
Let L be the generator of an analytic semigroup whose heat kernel satisfies an upper bound of Poisson type acting on L2(X) where X is a (possibly non-doubling) space of polynomial upper bound on volume growth. The aim of this paper is to introduce a new class of Besov spaces associated with the operator L so that when L is the Laplace operator −Δ or its square root acting on the Euclidean space Rn, the new Besov spaces are equivalent to the classical Besov spaces. Depending on the choice of L, the new Besov spaces are natural settings for generic estimates for certain singular integral operators such as the fractional powers Lα. Since our approach does not require the doubling volume property of the underlying space, it is applicable to any subset Ω of Euclidean spaces without any smoothness requirement on the boundary. We will also develop a number of Calderón reproducing formulas which play an important role in the theory of function spaces and are of independent interest. As an application, we study Besov spaces associated with Schrödinger operators with non-negative potentials satisfying reverse Hölder estimates.
Given an open, bounded, connected domain Ω ⊂ Rn, let GD, GN be the solu- tion 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 op- erators 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 ho- mogeneous/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 C1+ε boundary is sufficient in order to obtain (Hardy space) estimates for the Dirichlet problem when p is near 1. (...)
How to Measure Smoothness.- Atoms and Pointwise Multipliers.- Wavelets.- Spaces on Lipschitz Domains, Wavelets and Sampling Numbers.- Anisotropic Function Spaces.- Weighted Function Spaces.- Fractal Analysis: Measures, Characteristics, Operators.- Function Spaces on Quasi-metric Spaces.- Function Spaces on Sets.
On donne des resultats optimaux pour la resolubilite du probleme de Neumann dans des domaines de Lipschitz a donnees dans L p . On obtient des resultats du point extreme correspondant pour des espaces de Hardy
Existence and uniqueness is established for the solution to the inhomogeneous Neumann problem for Laplace's equation in Lipschitz domains with data in Lp Sobolev spaces.
If the “hat” denotes the Banach envelope, we show that for
then
For 0<p, q<1, one has
(i)
for 0<p<1,1≤q< ⋡
(ii)
and for 11≤p< ⋡,0<q<1
(iii)
We present applications of these results to global regularity of PDE’s in non-smooth domains, emphasizing the three-dimensional Lame system of elastostatics (Theorem 10).
We study a representation formula of the form for a distribution ƒ on n. This formula is obtained by discretizing and localizing a standard Littlewood-Paley decomposition. The map taking ƒ to the sequence , with Q running over the dyadic cubes in n, is called the ϑ-transform. The functions ϑQ and ψQ have a particularly simple form. Moreover, most of the familiar distribution spaces (Lp-spaces, 1 < p < +∞, Hp spaces, 0 < p ⩽ 1, Sobolev and potential spaces, BMO, Besov and Triebel-Lizorkin spaces) are characterized by the magnitude of the ϑ-transform. This enables us to carry out a discrete Littlewood-Paley theory on the sequence spaces corresponding to these distribution spaces. The sequence space norms depend only on magnitudes; cancellation is accounted for in the ϑQ's and ψQ's. Consequently, analysis on the sequence space level is often easy. With this we can simplify, extend, and unify a variety of results in harmonic analysis. We obtain conditions for the boundedness of linear operators on these distribution spaces by considering corresponding conditions for matrices on the associated sequence spaces. Applications include a general version of the Hörmander (Fourier) multiplier theorem and results for kernel operators of Calderón-Zygmund type. We discuss certain other, more general, decomposition methods, including the “smooth atomic decomposition,” and the “generalized ϑ-transform.” The smooth atomic decomposition yields a simple method for dealing with restriction and extension phenomena for hyperplanes in n. We also consider pointwise multipliers. For the characteristic function of a domain, we obtain boundedness results for a general class of domains which properly includes Lipschitz domains. Several interpolation methods are easily analyzed via the sequence spaces. For real interpolation, we obtain, among other things, an extension to the case p = 0. This in turn gives a new approach to the traditional atomic decomposition of Hardy spaces.
We show that on a starshaped domain Ω in Cn (actually on a somewhat larger, biholomorphically invariant class) the Lp-Sobolev spaces of analytic functions form an interpolation scale for both the real and complex methods, for each $p, 0
For D, a bounded Lipschitz domain in Rn, n ⩾ 2, the classical layer potentials for Laplace's equation are shown to be invertible operators on L2(∂D) and various subspaces of L2(∂D). For 1 < p ⩽ 2 and data in Lp(∂D) with first derivatives in Lp(∂D) it is shown that there exists a unique harmonic function, u, that solves the Dirichlet problem for the given data and such that the nontangential maximal function of ▽u is in Lp(∂D). When n = 2 the question of the invertibility of the layer potentials on every Lp(∂D), 1 < p < ∞, is answered.
We study inhomogeneous boundary value problems for the Laplacian in arbitrary Lipschitz domains with data in Sobolev–Besov spaces. As such, this is a natural continuation of work in [Jerison and Kenig,J. Funct. Anal.(1995), 16–219] where the inhomogeneous Dirichlet problem is treated via harmonic measure techniques. The novelty of our approach resides in the systematic use of boundary integral methods. In this regard, the key results are establishing the invertibility of the classical layer potential operators on scales of Sobolev–Besov spaces on Lipschitz boundaries for optimal ranges of indices. Applications toLp-based Helmholtz type decompositions of vector fields in Lipschitz domains are also presented.
We continue a program to develop layer potential techniques for PDE on Lipschitz domains in Riemannian manifolds. Building on Lp and Hardy space estimates established in previous papers, here we establish Sobolev and Besov space estimates on solutions to the Dirichlet and Neumann problems for the Laplace operator plus a potential, on a Lipschitz domain in a Riemannian manifold with a metric tensor smooth of class C1+γ, for some γ>0. We treat the inhomogeneous problem and extend it to the setting of manifolds results obtained for the constant-coefficient Laplace operator on a Lipschitz domain in Euclidean space, with the Dirichlet boundary condition, by D. Jerison and C. Kenig.
Function spaces of type Bs pq and Fs pq cover as special cases classical and fractional Sobolev spaces, classical Besov spaces, Holder- zgmund spaces and inhomogeneous Hardy spaces. In the last or 3 decades they haven been studied preferably on Rn and in mooth bounded domains in Rn including numerous applications to pseudodi_erential operators, elliptic boundary value problems etc. To a lesser extent spaces of this type have been considered in Lipschitz domains. But in recent times there is a growing interest to study and to use spaces of this type in Lipschitz domains and on their boundaries. It is the aim of this paper to deal with function spaces of Bs pq and Fs pq type in Lipschitz domains and on Lipschitz manifolds in a systematic (although not comprehensive) way: We describe and comment on known results, seal some gaps, give new proofs, and add a few new results of relevant aspects.
We investigate 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.
Characteristic functions as pointwise multipliers
- H Triebel
- H. Triebel