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Two Weight Estimates for the Single Layer Potential on Lipschitz Surfaces with Small Lipschitz Constant
Abstract
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.
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The simple and double layer potentials for second order linear strongly elliptic differential operators on Lipschitz domains are studied and it is shown that in a certain range of Sobolev spaces, results on continuity and regularity can be obtained without using either Calderon’s theorem on the -continuity of the Cauchy integral on Lipschitz curves [J. L. Journe, “Calderon-Zygmuno operators, pseudo-differential operators and the Cauchy integral of Calderon,” in Lecture Notes in Math. 994, Springer-Verlag, Berlin, 1983] or Dahlberg’s estimates of harmonic measures [“On the Poisson integral for Lipschitz and domains,” Studio Math., 66 (1979), pp. 7–24]. The operator of the simple layer potential and of the normal derivative of the double layer potential are shown to be strongly elliptic in the sense that they satisfy Carding inequalities in the respective energy norms. As an application, error estimates for Galerkin approximation schemes for integral equations of the first kind are derived.
We consider the following equation for the Riesz potential of order one: Uniqueness is proved in the class of solutions for which the integral is absolutely convergent for almost every x. We also prove an existence result and derive an asymptotic formula for solutions near the origin. Our analysis is carried out in local L p -spaces and Sobolev spaces, which allows us to obtain optimal results concerning the class of right-hand sides and solutions. We also apply our results to weighted L p -spaces and homogenous Sobolev spaces.
For a locally convex space
with the topology given by a family {p(┬; α)} α ∈ ω of seminorms, we study the existence and uniqueness of fixed points for a mapping
defined on some set
. We require that there exists a linear and positive operatorK, acting on functions defined on the index set Ω, such that for everyu,
Under some additional assumptions, one of which is the existence of a fixed point for the operator
, we prove that there exists a fixed point of
. For a class of elements satisfyingK
n
(p)u;┬))(α) → 0 asn → ∞, we show that fixed points are unique. This class includes, in particular, the class for which we prove the existence of fixed points.
We consider several applications by proving existence and uniqueness of solutions to first and second order nonlinear differential equations in Banach spaces. We also consider pseudodifferential equations with nonlinear terms.
"The book under review is the first systematic and self-contained presentation of a theory of arbitrary order ordinary differential equations with unbounded operator coefficients in a Hilbert or Banach space … . this is an excellent book, that contains recent results of the topic, deep theoretical results and various applications to PDE-s. It is warmly recommended to specialists in ODE-s, PDE-s, functional analysis." (Jeno Hegedus, Acta Scientiarum Mathematicarum, Vol. 72, 2006)
This paper considers to the equation [\int_{S} \frac{U(Q)}{|P-Q|^{N-1}} dS(Q)
= F(P), P \in S,] where the surface S is the graph of a Lipschitz function \phi
on R^N, which has a small Lipschitz constant. The integral in the left-hand
side is the single layer potential corresponding to the Laplacian in R^{N+1}.
Let \Lambda(r) be a Lipschitz constant of \phi on the ball centered at the
origin with radius 2r. Our analysis is carried out in local L^p-spaces and
local Sobolev spaces, where 1 < p < \infty, and results are presented in terms
of \Lambda(r). Estimates of solutions to the equation are provided, which can
be used to obtain knowledge about the behaviour of the solutions near a point
on the surface. The estimates are given in terms of seminorms. Solutions are
also shown to be unique if they are subject to certain growth conditions. Local
estimates are provided and some applications are supplied.
For the computation of the local singular behaviour of an homogeneous anisotropic clastic field near the three-dimensional vertex subjected to displacement boundary conditions, one can use a boundary integral equation of the first kind whose unkown is the boundary stress. Mellin transformation yields a one - dimensional integral equation on the intersection curve 7 of the cone with the unit sphere. The Mellin transformed operator defines the singular exponents and Jordan chains, which provide via inverse Mellin transformation a local expansion of the solution near the vertex. Based on Kondratiev's technique which yields a holomorphic operator pencil of elliptic boundary value problems on the cross - sectional interior and exterior intersection of the unit sphere with the conical interior and exterior original cones, respectively, and using results by Maz'ya and Kozlov, it can be shown how the Jordan chains of the one-dimensional boundary integral equation are related to the corresponding Jordan chains of the operator pencil and their jumps across γ. This allows a new and detailed analysis of the asymptotic behaviour of the boundary integral equation solutions near the vertex of the cone. In particular, the integral equation method developed by Schmitz, Volk and Wendland for the special case of the elastic Dirichlet problem in isotropic homogeneous materials could be completed and generalized to the anisotropic case.
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.
Differential Equations with Operator Coefficients Author's personal copy Two Weight Estimates for the Single Layer Potential on Lipschitz Surfaces
- V Kozlov
- V Maz 'ya
Kozlov, V., Maz'ya, V.: Differential Equations with Operator Coefficients. Springer, Berlin (1999) Author's personal copy Two Weight Estimates for the Single Layer Potential on Lipschitz Surfaces