Regularity of variational solutions to linear boundary value problems in Lipschitz domains

In a bounded Lipschitz domain in ?ⁿ, we consider a second-order strongly elliptic system with symmetric principal part written in divergent form. We study the Neumann boundary value problem in a generalized variational (or weak) setting using the Lebesgue spaces H _p ^σ (Ω) for solutions, where p can differ from 2 and σ can differ from 1. Using the tools of interpolation theory, we generalize the known theorem on the regularity of solutions, in which p = 2 and {σ ? 1} < 1/2, and the corresponding theorem on the unique solvability of the problem (Savaré, 1998) to p close to 2. We compare this approach with the nonvariational approach accepted in numerous papers of the modern theory of boundary value problems in Lipschitz domains. We discuss the regularity of eigenfunctions of the Dirichlet, Neumann, and Poincaré-Steklov spectral problems.     

Potential type operators and transmission problems for strongly elliptic second-order systems in Lipschitz domains

We consider a strongly elliptic second-order system in a bounded n-dimensional domain Ω⁺ with Lipschitz boundary Γ, n ≥ 2. The smoothness assumptions on the coefficients are minimized. For convenience, we assume that the domain is contained in the standard torus $ \mathbb{T}^n $ \mathbb{T}^n . In previous papers, we obtained results on the unique solvability of the Dirichlet and Neumann problems in the spaces H _p ^σ and B _p ^σ without use of surface potentials. In the present paper, using the approach proposed by Costabel and McLean, we define surface potentials and discuss their properties assuming that the Dirichlet and Neumann problems in Ω⁺ and the complementing domain Ω⁻ are uniquely solvable. In particular, we prove the invertibility of the integral single layer operator and the hypersingular operator in Besov spaces on Γ. We describe some of their spectral properties as well as those of the corresponding transmission problems.     

Boundary integral operators and boundary value problems for Laplace’s equation

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.     

To the theory of the Dirichlet and Neumann problems for strongly elliptic systems in Lipschitz domains

For strongly elliptic Systems with Douglis-Nirenberg structure, we investigate the regularity of variational solutions to the Dirichlet and Neumann problems in a bounded Lipschitz domain. The solutions of the problems with homogeneous boundary conditions are originally defined in the simplest L ₂-Sobolev spaces H ^σ . The regularity results are obtained in the potential spaces H _p ^σ and Besov spaces B _p ^σ . In the case of second-order Systems, the author’s results obtained a year ago are strengthened. The Dirichlet problem with nonhomogeneous boundary conditions is considered with the use of Whitney arrays.     

Mixed problems in a Lipschitz domain for strongly elliptic second-order systems

We consider mixed problems for strongly elliptic second-order systems in a bounded domain with Lipschitz boundary in the space ℝ ⁿ . For such problems, equivalent equations on the boundary in the simplest L ₂-spaces H ^s of Sobolev type are derived, which permits one to represent the solutions via surface potentials. We prove a result on the regularity of solutions in the slightly more general spaces H _p ^s of Bessel potentials and Besov spaces B _p ^s . Problems with spectral parameter in the system or in the condition on a part of the boundary are considered, and the spectral properties of the corresponding operators, including the eigenvalue asymptotics, are discussed.     

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.     

Strongly elliptic second-order systems with boundary conditions on a nonclosed Lipschitz surface

We consider boundary value problems and transmission problems for strongly elliptic second-order systems with boundary conditions on a compact nonclosed Lipschitz surface S with Lipschitz boundary. The main goal is to find conditions for the unique solvability of these problems in the spaces H ^s , the simplest L ₂-spaces of the Sobolev type, with the use of potential type operators on S. We also discuss, first, the regularity of solutions in somewhat more general Bessel potential spaces and Besov spaces and, second, the spectral properties of problems with spectral parameter in the transmission conditions on S, including the asymptotics of the eigenvalues.     

Nonlinear Neumann–Transmission Problems for Stokes and Brinkman Equations on Euclidean Lipschitz Domains

The purpose of this paper is to use a layer potential analysis and the Leray–Schauder degree theory to show an existence result for a nonlinear Neumann–transmission problem corresponding to the Stokes and Brinkman operators on Euclidean Lipschitz domains with boundary data in L ^p spaces, Sobolev spaces, and also in Besov spaces.     

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.     

Lipschitz Estimates in Almost‐Periodic Homogenization

We establish uniform Lipschitz estimates for second‐order elliptic systems in divergence form with rapidly oscillating, almost‐periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded C^1,α domains with either Dirichlet or Neumann data. The main results extend those in the periodic setting due to Avellaneda and Lin for interior and Dirichlet boundary estimates and later Kenig, Lin, and Shen for the Neumann boundary conditions. In contrast to these papers, our arguments are constructive (and thus the constants are in principle computable) and the results for the Neumann conditions are new even in the periodic setting, since we can treat nonsymmetric coefficients. We also obtain uniform W^1,p estimates.© 2016 Wiley Periodicals, Inc.     

Nonlocal Robin Laplacians and some remarks on a paper by Filonov on eigenvalue inequalities

The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators Θ which give rise to self-adjoint Laplacians −ΔΘ,Ω in L²(Ω;dnx) with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains Ω⊂Rn, n∈N, n?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 Ω, following an approach introduced by Filonov for this type of problems.     

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.     

The Morse and Maslov indices for Schrödinger operators

We study the spectrum of Schrödinger operators with matrixvalued potentials, utilizing tools from infinite-dimensional symplectic geometry. Using the spaces of abstract boundary values, we derive relations between the Morse and Maslov indices for a family of operators on a Hilbert space obtained by perturbing a given self-adjoint operator by a smooth family of bounded self-adjoint operators. The abstract results are applied to the Schrödinger operators with θ-periodic, Dirichlet, and Neumann boundary conditions. In particular, we derive an analogue of the Morse-Smale Index Theorem for multi-dimensional Schrödinger operators with periodic potentials. For quasi-convex domains in Rⁿ, we recast the results, connecting the Morse and Maslov indices using the Dirichlet and Neumann traces on the boundary of the domain.     

Homogenization of elliptic boundary value problems in Lipschitz domains

In this paper we study the L ^p boundary value problems for \({\mathcal{L}(u)=0}\) in \({\mathbb{R}^{d+1}_+}\) , where \({\mathcal{L}=-{\rm div} (A\nabla )}\) is a second order elliptic operator with real and symmetric coefficients. Assume that A is periodic in x _d+1 and satisfies some minimal smoothness condition in the x _d+1 variable, we show that the L ^p Neumann and regularity problems are uniquely solvable for 1 < p < 2 + δ. We also present a new proof of Dahlberg’s theorem on the L ^p Dirichlet problem for 2 ? δ < p < ∞ (Dahlberg’s original unpublished proof is given in the Appendix). As the periodic and smoothness conditions are imposed only on the x _d+1 variable, these results extend directly from \({\mathbb{R}^{d+1}_+}\) to regions above Lipschitz graphs. Consequently, by localization techniques, we obtain uniform L ^p estimates for the Dirichlet, Neumann and regularity problems on bounded Lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization. The results on the Neumann and regularity problems are new even for smooth domains.     

Mixed exterior Laplace's problem

In [C. Amrouche, V. Girault, J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator. An approach in weighted Sobolev spaces, J. Math. Pures Appl. 76 (1997) 55-81], authors study Dirichlet and Neumann problems for the Laplace operator in exterior domains of Rn. This paper extends this study to the resolution of a mixed exterior Laplace's problem. Here, we give existence, unicity and regularity results in Lp's theory with 1<p<∞, in weighted Sobolev spaces.     

Noncontinuous data boundary value problems for Schrödinger equation in Lipschitz domains

The noncontinuous data boundary value problems for Schrödinger equations in Lipschitz domains and its progress are pointed out in this paper. Particularly, the L ^p boundary value problems with p > 1, and H ^p boundary value problems with p < 1 have been studied. Some open problems about the Besov-Sobolev and Orlicz boundary value problems are given.     

Full asymptotics and Laurent series of layer potentials for Laplace's equation on the half‐space

We combine the calculus of conormal distributions, in particular the Pull‐Back and Push‐Forward Theorems, with the method of layer potentials to solve the Dirichlet and Neumann problems on half‐spaces. We obtain full asymptotic expansions for the solutions, show that boundary layer potential operators are elements of the full b‐calculus and give a new proof of the classical jump relations. En route, we improve Siegel and Talvila's growth estimates for the modified layer potentials in the case of polyhomogeneous boundary data. The techniques we use here can be generalised to geometrically more complex settings, as for instance the exterior domain of touching domains or domains with fibred cusps. This work is intended to be a first step in a longer program aiming at understanding the method of layer potentials in the setting of certain non‐Lipschitz singularities that can be resolved in the sense of Melrose using manifolds with corners and at applying a matching asymptotics ansatz to singular perturbations of related problems.     

Second-order properly elliptic boundary value problems on irregular plane domains

Conditions are presented under which properly elliptic second-order boundary value problems are well posed on irregular plane domains. The coefficients can be discontinuous. The results include known results for coercive forms, and also reduce to known results on proper ellipticity when the coefficients and domain are smooth. The main tool is an “inverse five-lemma” which relates the Neumann problem on a plane domain to a related modified Dirichlet problem. This inverse five-lemma can be used in a variety of settings. We show how it can be used to translate results of Grisvard on the index of Dirichlet operators in Sobolev spaces $\text{H}{}^{\mathrm{s}}\text{(Ω)}$ to results on Neumann operators, and examine the implications for regularity.     

Fractional powers of operators corresponding to coercive problems in Lipschitz domains

Let Ω be a bounded Lipschitz domain in ? ⁿ , n ? 2, and let L be a second-order matrix strongly elliptic operator in Ω written in divergence form. There is a vast literature dealing with the study of domains of fractional powers of operators corresponding to various problems (beginning with the Dirichlet and Neumann problems) with homogeneous boundary conditions for the equation Lu = f, including the solution of the Kato square root problem, which arose in 1961. Mixed problems and a class of problems for higher-order systems have been covered as well. We suggest a new abstract approach to the topic, which permits one to obtain the results that we deem to be most important in a much simpler and unified way and cover new operators, namely, classical boundary operators on the Lipschitz boundary Γ = ?Ω or part of it. To this end, we simultaneously consider two well-known operators associated with the boundary value problem.     

Well-posedness and regularity of boundary feedback parabolic systems

A general second order parabolic equation is considered with both Dirichlet or mixed (in particular, Neumann) input function acting on the boundary S of the bounded spatial domain Ω. The distinctive new feature is that the input function is demanded to be expressed in feedback form, i.e. as a linear operator (of finite dimensional range) of the solution, continuous from $\text{H}{}^{\mathrm{s}}\text{(Ω)}$ into L_p(S), for some non-negative real s and for p ? 1. Well posedness and regularity results of the resulting closed loop system are established in appropriate functions spaces. The results are illustrated by examples of physical interest.