Remarks on Nonlinear Neumann Problems in Periodic Domains

We study a semilinear elliptic equation Au = f(x, u) with nonlinear Neumann boundary condition Bu = φ(ξ, u) in an unbounded domain Ω ? ?ⁿ, the boundary of which is defined by periodic functions. We assume that f and φ and the coefficients of the operators are asymptotically periodic in the space variables. Our main result states the existence of an asymptotically decaying, nontrivial solution of this problem with minimal energy. The proof is based on the concentration-compactness principle.     

Faber-Krahn Inequality for Anisotropic Eigenvalue Problems with Robin Boundary Conditions

In this paper we study the main properties of the first eigenvalue λ ₁(Ω) and its eigenfunctions of a class of highly nonlinear elliptic operators in a bounded Lipschitz domain Ω ? ? _n , assuming a Robin boundary condition. Moreover, we prove a Faber-Krahn inequality for λ ₁(Ω).     

Spectral theory and iterative methods for the Maxwell system in nonsmooth domains

We study spectral properties of boundary integral operators which naturally arise in the study of the Maxwell system of equations in a Lipschitz domain Ω ? ?³. By employing Rellich‐type identities we show that the spectrum of the magnetic dipole boundary integral operator (composed with an appropriate projection) acting on L²(?Ω) lies in the exterior of a hyperbola whose shape depends only on the Lipschitz constant of Ω. These spectral theory results are then used to construct generalized Neumann series solutions for boundary value problems associated with the Maxwell system and to study their rates of convergence (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The \bar \partial -problem on q-pseudoconvex domains with applications

For a q-pseudoconvex domain Ω in ? ⁿ , 1 ≤ q ≤ n, with Lipschitz boundary, we solve the $\bar \partial $ -problem with exact support in Ω. Moreover, we solve the $\bar \partial $ -problem with solutions smooth up to the boundary over Ω provided that it has smooth boundary. Applications are given to the solvability of the tangential Cauchy-Riemann equations on the boundary.     

On the solvability of the jump problem in clifford analysis

Let Ω be a bounded open and oriented connected subset of ? ⁿ which has a compact topological boundary Γ, let C be the Dirac operator in ? ⁿ , and let ?_0,n be the Clifford algebra constructed over the quadratic space ? ⁿ . An ?_0,n -valued smooth function f : Ω → ?_0,n in Ω is called monogenic in Ω if Df = 0 in Ω. The aim of this paper is to present the most general condition on Γ obtained so far for which a Hölder continuous function f can be decomposed as F ⁺ ? F ^? = f on Γ, where the components F ^± are extendable to monogenic functions in Ω^± with Ω⁺ := Ω, and Ω^? := ? ⁿ \ (Ω ? Γ), respectively.     

On the Uniform Poincaré Inequality

We give a proof of the Poincaré inequality in W ^{1, p} (Ω) with a constant that is independent of Ω ? , where  is a set of uniformly bounded and uniformly Lipschitz domains in ? ⁿ . As a byproduct, we obtain the following: The first non vanishing eigenvalues λ₂(Ω) of the standard Neumann (variational) boundary value problem on Ω for the Laplace operator are bounded below by a positive constant if the domains Ω vary and remain uniformly bounded and uniformly Lipschitz regular.     

The $$\overline \partial $$-Neumann operator on Lipschitz q-pseudoconvex domains

On a bounded q-pseudoconvex domain Ω in ? ⁿ with a Lipschitz boundary, we prove that the \(\overline \partial \)-Neumann operator N satisfies a subelliptic (1/2)-estimate on Ω and N can be extended as a bounded operator from Sobolev (?1/2)-spaces to Sobolev (1/2)-spaces.     

Hyperbolic Equations with Growing Coefficients in Unbounded Domains

Let Ω ? ? ⁿ , n?≥?2, be an unbounded domain with a smooth (possibly noncompact) star-shaped boundary Γ. For the first mixed problem for a hyperbolic equation with an unbounded coefficient with power growth at infinity, the large-time behavior of the solutions is studied. Estimates for the resolvent of the spectral problem are obtained for various values of the parameters.     

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.     

Zero products of Toeplitz operators with harmonic symbols

On the Bergman space of the unit ball in Cⁿ, we solve the zero-product problem for two Toeplitz operators with harmonic symbols that have continuous extensions to (some part of) the boundary. In the case where symbols have Lipschitz continuous extensions to the boundary, we solve the zero-product problem for multiple products with the number of factors depending on the dimension n of the underlying space; the number of factors is n+3. We also prove a local version of this result but with loss of a factor.     

Heat kernel bounds for elliptic partial differential operators in divergence form with Robin-type boundary conditions

One of the principal topics of this paper concerns the realization of self-adjoint operators L _Θ,Ω in L ²(Ω; d ⁿ x) ^m , m, n ∈ ?, associated with divergence form elliptic partial differential expressions L with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains Ω ? ? ⁿ . In particular, we develop the theory in the vector-valued case and hence focus on matrix-valued differential expressions L which act as $$Lu = - \left( {\sum\limits_{j,k = 1}^n {\partial _j } \left( {\sum\limits_{\beta = 1}^m {a_{j,k}^{\alpha ,\beta } \partial _k u_\beta } } \right)} \right)_{1 \leqslant \alpha \leqslant m} , u = \left( {u_1 , \ldots ,u_m } \right).$$ The (nonlocal) Robin-type boundary conditions are then of the form $$v \cdot ADu + \Theta [u|_{\partial \Omega } ] = 0{\text{ on }}\partial \Omega ,$$ where Θ represents an appropriate operator acting on Sobolev spaces associated with the boundary ?Ω of Ω, ν denotes the outward pointing normal unit vector on ?Ω, and $Du: = \left( {\partial _j u_\alpha } \right)_{_{1 \leqslant j \leqslant n}^{1 \leqslant \alpha \leqslant m} } .$ Assuming Θ ≥ 0 in the scalar case m = 1, we prove Gaussian heat kernel bounds for L _Θ,Ω, by employing positivity preserving arguments for the associated semigroups and reducing the problem to the corresponding Gaussian heat kernel bounds for the case of Neumann boundary conditions on ?Ω. We also discuss additional zero-order potential coefficients V and hence operators corresponding to the form sum L _Θ,Ω + V.     

Boundary criterion for integral operators

Integral operators of the form \(L_K^{ - 1} f(x) = \int\limits_\Omega {K(x,t)f(t)dt}\) for the case of a finite domain Ω ? Rⁿ with smooth boundary ?Ω are considered. Conditions on the real kernel K(x, t) of an integral operator under which this operator satisfies a well-defined boundary condition for the corresponding differential equation are found. The application of the results is demonstrated on the example of a Sturm–Liouville equation, for which the derivation of the general form of well-posed boundary value problems is presented.     

Strong Solutions of the Navier–Stokes System in Lipschitz Bounded Domains

Let Ω be a bounded domain in ?³ with a connected Lipschitz boundary ?Ω. Consider the Navier-Stokes system.     

Necessary and Sufficient Conditions for the Solvability of the Neumann and the Regularity Problems in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> of Some Schrödinger Equations on Lipschitz Domains

Let n ≥? 3and Ω be a bounded Lipschitz domain in \(\mathbb {R}^{n}\). Assume that the non-negative potential V belongs to the reverse Hölder class \(RH_{n}(\mathbb {R}^{n})\) and p ∈ (2, ∞). In this article, two necessary and sufficient conditions for the unique solvability of the Neumann and the Regularity problems of the Schrödinger equation ? Δu + V u =? 0 in Ω with boundary data in L ^p , in terms of a weak reverse Hölder inequality with exponent p and the unique solvability of the Neumann and the Regularity problems with boundary data in some weighted L ² space, are established. As applications, for any p ∈ (1, ∞), the unique solvability of the Regularity problem for the Schrödinger equation ? Δu + V u =?0 in the bounded (semi-)convex domain Ω with boundary data in L ^p is obtained.     

Cauchy integral decomposition for harmonic forms

Let ann-dimensional differential form Ω be defined at points of aC ¹-smooth boundary π of a domainG ? ? ⁿ . Under what condition can Ω be represented as Ω = Ω₊ + Ω₊ + Ω_-, where Ω_± are forms insideG and outsideG, harmonic in the sense of Hodge? A necessary condition is that both restrictions Ω{inπ and *Ω{inπ be closed in the sense of currents. This condition, with an additional smoothness assumption, turns out to be sufficient as well. This is an analogue of the Cauchy integral decomposition of functions in the plane.     

Hermitian decomposition of continuous functions on a fractal surface

In this paper the Théodoresco transform is used to show that, under additional assumptions, each Hölder continuous function f defined on the boundary Γ of a fractal domain Ω ? ?²ⁿ can be expressed as f = Ψ⁺ ? Ψ^?, where Ψ^± are Hölder continuous functions on Γ and Hermitian monogenically extendable to Ω and to ?²ⁿ ? (Ω ∪ Γ) respectively.     

Spectral properties for layer potentials associated to the Stokes equation in Lipschitz domains

In this paper, we consider the spectral properties of the double layer potentials K and \({\tilde{K}}\) related to the traction boundary value problem and the slip boundary value problem, respectively, of the Stokes equations in a bounded Lipschitz domain Ω in R ⁿ . We show the invertibility of λI ? K and \({\lambda I - \tilde{K}}\) in L ²(?Ω) for \({\lambda \in {\bf R}{\setminus} [-\frac 12, \frac12]}\). As an application, we study the transmission problems of the Stokes equations.     

Intrinsic characterizations of Besov spaces on Lipschitz domains

The aim of this paper is to study the equivalence between quasi‐norms of Besov spaces on domains. We suppose that the domain Ω ? ?ⁿ is a bounded Lipschitz open subset in ?ⁿ. First, we define Besov spaces on Ω as the restrictions of the corresponding Besov spaces on ?ⁿ. Then, with the help of equivalent and intrinsic characterizations (the Peetre‐type characterization 3.10 and the characterization via local means 3.13) of these spaces, we get another equivalent and intrinsic quasi‐norm using, this time, generalized differences and moduli of smoothness. We extend the well‐known characterization of Besov spaces on ?ⁿ described in Theorem 2.4 to the case of Lipschitz domains.     

The resolvent of a singularly perturbed elliptic operator

Let A and B be uniformly elliptic operators of orders 2m and 2n, respectively, m > n. We consider the Dirichlet problems for the equations (?^{2(m ? n)}A + B + λ²ⁿI)u_? = f and (B + λ²ⁿI)u = f in a bounded domain Ω in R^k with a smooth boundary ?Ω. The estimate is derived. This result extends the results of [7, 9, 10, 12, 14, 15, 18]by giving estimates up to the boundary, improving the rate of convergence in ?, using lower norms, and considering operators of higher order with variable coefficients. An application to a parabolic boundary value problem is given.     

Szegö's limit theorem: The higher-dimensional matrix case

The continuous version of Szegö's theorem gives the first two terms of the asymptotics as α → ∞ of the determinants of certain convolution operators on L₂(0, α) with scalar-valued kernels. Generalizations are known if the kernel is matrix valued or if the interval (0, α) is replaced by αΩ with Ω a bounded set in Rⁿ with smooth boundary. This paper treats the higher-dimensional matrix case. The coefficient in the interesting (second) term is an integral over the contangent bundle of ?Ω of the correponding coefficients of one-dimensional problems.