A bilinear estimate for biharmonic functions in Lipschitz domains

We show that a bilinear estimate for biharmonic functions in a Lipschitz domain Ω is equivalent to the solvability of the Dirichlet problem for the biharmonic equation in Ω. As a result, we prove that for any given bounded Lipschitz domain Ω in _boxclose^d{\mathbb{R}^{d}} and 1 < q < ∞, the solvability of the L ^q Dirichlet problem for Δ ² u = 0 in Ω with boundary data in WA ^1,q (∂Ω) is equivalent to that of the L ^p regularity problem for Δ ² u = 0 in Ω with boundary data in WA ^2,p (∂Ω), where \frac1p + \frac1q=1{\frac{1}{p} + \frac{1}{q}=1}. This duality relation, together with known results on the Dirichlet problem, allows us to solve the L ^p regularity problem for d ≥ 4 and p in certain ranges.     

Maximum principles for the polyharmonic equation on Lipschitz domains

The Agmon-Miranda maximum principle for the polyharmonic equations of all orders is shown to hold in Lipschitz domains in ℝ³. In ℝn,n≥4, the Agmon-Miranda maximum principle andL ^p-Dirichlet estimates for certainp>2 are shown to fail in Lipschitz domains for these equations. In particular if 4≤n≤2m+1 theL ^p Dirichlet problem for Δ ^m fails to be solvable forp>2(n−1)/(n−3). Supported in part by the NSF.     

Square roots of elliptic second order divergence operators on strongly lipschitz domains:<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> theory

We prove the Kato conjecture for square roots of elliptic second order non-self-adjoint operators in divergence formL = -div(A∇) on strongly Lipschitz domains in ℝⁿ, n≥2, subject to Dirichlet or to Neumann boundary conditions. The method relies on a transference procedure from the recent positive result on ℝⁿ in [2].     

Boundary Problems for Harmonic Functions and Norm Estimates for Inverses of Singular Integrals in Two Dimensions

In this paper, we establish sharp well-posedness results for tangential derivative problems for the Laplacian with data in L ^p , 1 < p < ∞, on curvilinear polygons. Furthermore, we produce norm estimates/formulas for inverses of singular integral operators relevant for the Dirichlet, Neumann, tangential derivative, and transmission boundary value problems associated with the Laplacian in a distinguished subclass of Lipschitz domains in two dimensions. Our approach relies on Calderón-Zygmund theory and Mellin transform techniques.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> solvability of the stationary stokes problem in domains with conical singularity in any dimension

We consider the Dirichlet boundary value problem for the Stokes operator with L ^p data in any dimension in domains with conical singularity (not necessarily a Lipschitz graph). We establish the solvability of the problem for all p ∈ (2 − ε, ∞] and also in C(D) for the data in C( [`(D)] ) C\left( {\overline D } \right) . Bibliography: 14 titles. In memory of Michael Sh. Birman     

Local a priori estimates in Lp for first order linear operators with nonsmooth coefficients

We prove local a priori estimates inL ^p , 1<p<∞, for first-order linear operators that satisfy the Nirenberg-Treves condition (p) and whose coefficients have Lipschitz continuous derivatives of order one. When the number of variables is two, only Lipschitz continuity of the coefficients is assumed. This extends toL ^p spaces estimates that were previously known forp=2. Examples show that the regularity required from the coefficients is essentially minimal. Research partially supported by CNPq.     

The Estimates for Sharp Maximal Functions of Multilinear Strongly Singular Integral Operators

In this paper, the author obtains that the multilinear operators of strongly singular integral operators and their dual operators are bounded from some L^p（R^n） to L^p（R^n） when the m-th order derivatives of A belong to L^p（R^n） for r large enough. By this result, the author gets the estimates for the Sharp maximal functions of the multilinear operators with the m-th order derivatives of A being Lipschitz functions. It follows that the multilinear operators are （L^p, L^p）-type operators for 1 〈 p 〈 ∞.     

Szegö and Bergman projections on non-smooth planar domains

We establish L^p regularity for the Szegö and Bergman projections associated to a simply connected planar domain in any of the following classes: vanishing chord arc; Lipschitz; Ahlfors-regular; or local graph (for the Szegö projection to be well defined, the local graph curve must be rectifiable). As applications, we obtain L^p regularity for the Riesz transforms, as well as Sobolev space regularity for the non-homogeneous Dirichlet problem associated to any of the domains above and, more generally, to an arbitrary proper simply connected domain in the plane.     

Duality and well-posedness in convex interpolation ∗)

We consider the problem for convex interpolation with minimal L_p norm of the second derivative, 1 < p < +α. Convergence of a class of dual methods is established and numerical results are presented. It is proved that if p 2 then the solution of the problem is locally Lipschitz with respect to the data in the uniform metric.     

The multidirectional Neumann problem in ℝ4

The Neumann problem as formulated in Lipschitz domains with L^p boundary data is solved for harmonic functions in any compact polyhedral domain of ℝ⁴ that has a connected 3-manifold boundary. Energy estimates on the boundary are derived from new polyhedral Rellich formulas together with a Whitney type decomposition of the polyhedron into similar Lipschitz domains. The classical layer potentials are thereby shown to be semi-Fredholm. To settle the onto question a method of continuity is devised that uses the classical 3-manifold theory of E. E. Moise in order to untwist the polyhedral boundary into a Lipschitz boundary. It is shown that this untwisting can be extended to include the interior of the domain in local neighborhoods of the boundary. In this way the flattening arguments of B. E. J. Dahlberg and C. E. Kenig for the H¹_at Neumann problem can be extended to polyhedral domains in ℝ⁴. A compact polyhedral domain in ℝ⁶ of M. L. Curtis and E. C. Zeeman, based on a construction of M. H. A. Newman, shows that the untwisting and flattening techniques used here are unavailable in general for higher dimensional boundary value problems in polyhedra.     

Closed convex hull of the family of multivalently close-to-convex functions of order β

In this paper the closed convex hulls of the compact familiesC _β(p), of multivalently close to convex functions of order β andV ₀ ^k (p), of multivalent functions of bounded boundary rotation, have been determined, respectively for β≥1 andk≥2(p+1)/p. Extreme points of these convex hulls are partially characterised. For a fixed pointz ₀∈D={z:|z|<1}, a new familyC _β(p, z₀) is defined through Montel normalisation and its closed convex hull is also foud. Sharp coefficient estimates for functions subordinate to or majorised by some function inC _β(p) orC' _β(p) are discussed for β>0. It is shown that iff is subordinate to some function inC _β(p) then each Taylor coefficient off is dominated by the corresponding coefficient of the function .     

Boundary value problems for the Laplacian in convex and semiconvex domains

We study the fully inhomogeneous Dirichlet problem for the Laplacian in bounded convex domains in Rn, 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).     

Error estimates for the numerical approximation of a quaslinear Neumann problem under minimal regularity of the data

The finite element based approximation of a quasilinear elliptic equation of non monotone type with Neumann boundary conditions is studied. Minimal regularity assumptions on the data are imposed. The consideration is restricted to polygonal domains of dimension two and polyhedral domains of dimension three. Finite elements of degree k ≥ 1 are used to approximate the equation. Error estimates are established in the L ²(Ω) and H ¹(Ω) norms for convex and non-convex domains. The issue of uniqueness of a solution to the approximate discrete equation is also addressed.     

Convex polynomial and spline approximation inL p,O<p<∞

We prove that a convex functionf ∈ L _p[−1, 1], 0<p<∞, can be approximated by convex polynomials with an error not exceeding Cω ₃ ^ϕ (f,1/n)_p where ω ₃ ^ϕ (f,·) is the Ditzian-Totik modulus of smoothness of order three off. We are thus filling the gap between previously known estimates involving ω ₃ ^ϕ (f,1/n)_p, and the impossibility of having such estimates involving ω₄. We also give similar estimates for the approximation off by convexC ⁰ andC ¹ piecewise quadratics as well as convexC ² piecewise cubic polynomials. Communicated by Dietrich Braess     

Estimates for Green function and Poisson kernels of higher-order Dirichlet boundary value problems

Pointwise estimates are derived for the kernels associated to the polyharmonic Dirichlet problem on bounded smooth domains. As a consequence, one obtains optimal weighted L^p-L^q-regularity estimates for weights involving the distance function.     

Interpolation error estimates in W 1,p for degenerate Q 1 isoparametric elements

Optimal order error estimates in H ¹, for the Q ₁ isoparametric interpolation were obtained in Acosta and Durán (SIAM J Numer Anal37, 18–36, 1999) for a very general class of degenerate convex quadrilateral elements. In this work we show that the same conlusions are valid in W ^1,p for 1≤ p < 3 and we give a counterexample for the case p ≥ 3, showing that the result cannot be generalized for more regular functions. Despite this fact, we show that optimal order error estimates are valid for any p ≥ 1, keeping the interior angles of the element bounded away from 0 and π, independently of the aspect ratio. We also show that the restriction on the maximum angle is sharp for p ≥ 3.     

A multidirectional Dirichlet problem

B.E.J. Dahlberg’s theorems on the mutual absolute continuity of harmonic and surface measures, and on the unique solvability of the Dirichlet problem for Laplace’s equation with data taken in L^p spaces p > 2 ? δ are extended to compact polyhedral domains of ?ⁿ. Consequently, for q < 2 + δ, Dahlberg’s reverse Hölder inequality for the density of harmonic measure is established for compact polyhedra that additionally satisfy the Harnack chain condition. It is proved that a compact polyhedral domain satisfies the Harnack chain condition if its boundary is a topological manifold. The double suspension of the Mazur manifold is invoked to indicate that perhaps such a polyhedron need not itself be a manifold with boundary; see the footnote in Section 9. A theorem on approximating compact polyhedra by Lipschitz domains in a certain weak sense is proved, along with other geometric lemmas.     

On Gaussian Marginals of Uniformly Convex Bodies

Recently, Bo’az Klartag showed that arbitrary convex bodies have Gaussian marginals in most directions. We show that Klartag’s quantitative estimates may be improved for many uniformly convex bodies. These include uniformly convex bodies with power type 2, and power type p>2 with some additional type condition. In particular, our results apply to all unit-balls of subspaces of quotients of L _p for 1<p<∞. The same is true when L _p is replaced by S _p ^m , the l _p -Schatten class space. We also extend our results to arbitrary uniformly convex bodies with power type p, for 2≤p<4. These results are obtained by putting the bodies in (surprisingly) non-isotropic positions and by a new concentration of volume observation for uniformly convex bodies. Supported in part by BSF and ISF.     

A fast interface solver for the biharmonic Dirichlet problem on polygonal domains

Summary. In this paper we propose and analyze an efficient discretization scheme for the boundary reduction of the biharmonic Dirichlet problem on convex polygonal domains. We show that the biharmonic Dirichlet problem can be reduced to the solution of a harmonic Dirichlet problem and of an equation with a Poincaré-Steklov operator acting between subspaces of the trace spaces. We then propose a mixed FE discretization (by linear elements) of this equation which admits efficient preconditioning and matrix compression resulting in the complexity . Here is the number of degrees of freedom on the underlying boundary, is an error reduction factor, or for rectangular or polygonal boundaries, respectively. As a consequence an asymptotically optimal iterative interface solver for boundary reductions of the biharmonic Dirichlet problem on convex polygonal domains is derived. A numerical example confirms the theory. Received September 1, 1995 / Revised version received February 12, 1996     

Hardy spaces of the conjugate Beltrami equation

We study Hardy spaces of solutions to the conjugate Beltrami equation with Lipschitz coefficient on Dini-smooth simply connected planar domains, in the range of exponents 1<p<∞. We analyse their boundary behaviour and certain density properties of their traces. We derive on the way an analog of the Fatou theorem for the Dirichlet and Neumann problems associated with the equation div(σ∇u)=0 with Lp boundary data.