Cluster sets for sobolev functions and quasiminimizers

In this paper, we study cluster sets and essential cluster sets for Sobolev functions and quasiharmonic functions (i.e., continuous quasiminimizers). We develop their basic theory with a particular emphasis on when they coincide and when they are connected. As a main result, we obtain that if a Sobolev function u on an open set Ω has boundary values f in Sobolev sense and f |_∂Ω is continuous at x ₀ ∈ ∂Ω, then the essential cluster set (u, x ₀,Ω) is connected. We characterize precisely in which metric spaces this result holds. Further, we provide some new boundary regularity results for quasiharmonic functions. Most of the results are new also in the Euclidean case.     

Controllability in a filled domain for the multidimensional wave equation with a singular boundary control

In accordance with the demands of the so-called local approach to inverse problems, the set of “waves” u^f (·, T) is studied, where u^f (x,t) is the solution of the initial boundary-value problem u_tt−Δu=0 in Ω×(0,T), u|_t<0=0, u|_∂Ω×(0,T)=f, and the (singular) control f runs over the class L²((0,T); H^−m (∂Ω)) (m>0). The following result is established. Let Ω^T={x ∈ Ω : dist(x, ∂Ω)<T)} be a subdomain of Ω ⊂ ℝⁿ (diam Ω<∞) filled with waves by a final instant of time t=T, let T_*=inf{T : Ω^T=Ω} be the time of filling the whole domain Ω. We introduce the notation D_m=Dom((−Δ)^m/2), where (−Δ) is the Laplace operator, Dom(−Δ)=H²(Ω)∩H ₀ ¹ (Ω);D_−m=(D_m)′;D_−m(Ω^T)={y∈D_−m:supp y ⋐ Ω^T. If T<T., then the reachable set R _m ^T ={u^t(·, T): f ∈ L₂((0,T), H^−m (∂Ω))} (∀m>0), which is dense in D_−m(Ω^T), does not contain the class C ₀ ^∞ (Ω^T). Examples of a ∈ C ₀ ^∞ , a ∈ R _m ^T , are presented. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 7–21. Translated by T. N. Surkova.     

Boundary behavior of the pluricomplex Green function

Let Ω be a bounded domain inC ⁿ . This paper deals with the study of the behavior of the pluricomplex Green functiong _Ω(z, w) when the polew tends to a boundary pointw ₀ of Ω. We find conditions on Ω which ensure that lim _w→wo g _Ω (z, w)=0, uniformly with respect toz on compact subsets of . Our main result is Theorem 5; it gives a sufficient condition for the above property to hold, formulated in terms of the existence of a plurisubharmonic peak function for Ω atw ₀ which satisfies a certain growth condition.     

A capacitary method for the asymptotic analysis of dirichlet problems for monotone operators

Given a non-linear elliptic equation of monotone type in a bounded open set Ω ⊂ Rⁿ, we prove that the asymptotic behaviour, asj → ∞, of the solutions of the Dirichlet problems corresponding to a sequence (Ω_j) of open sets contained in Ω is uniquely determined by the asymptotic behaviour, asj → ∞, of suitable non-linear capacities of the setsKΩ _j, whereK runs in the family of all compact subsets of Ω.     

Well-posedness and sharp uniform decay rates at the L 2(Ω)-Level of the Schrödinger equation with nonlinear boundary dissipation

We prove that the Schr?dinger equation defined on a bounded open domain of and subject to a certain attractive, nonlinear, dissipative boundary feedback is (semigroup) well-posed on L₂(Ω) for any n = 1, 2, 3, ..., and, moreover, stable on L₂(Ω) for n = 2, 3, with sharp (optimal) uniform rates of decay. Uniformity is with respect to all initial conditions contained in a given L₂(Ω)-ball. This result generalizes the corresponding linear case which was proved recently in [L-T-Z.2]. Both results critically rely—at the outset—on a far general result of interest in its own right: an energy estimate at the L₂(Ω)-level for a fully general Schr?dinger equation with gradient and potential terms. The latter requires a heavy use of pseudo-differential/micro-local machinery [L-T-Z.2, Section 10], to shift down the more natural H¹(Ω)-level energy estimate to the L₂(Ω)-level. In the present nonlinear boundary dissipation case, the resulting energy estimate is then shown to fit into the general uniform stabilization strategy, first proposed in [La-Ta.1] in the case of wave equations with nonlinear (interior and) boundary dissipation.     

The asymptotics of the solution to the Neumann spectral problem in a domain of the “dense-comb” type

Convergence theorems and asymptotic estimates (as ε → 0) are proved for the eigenvalues and the eigenfunctions of the Neumann problem in a dense singular junction Ω _ɛ of a domain Ω₀ and a large number N of thin cylinders with thickness of order ε=lN⁻¹, where l is the total length of common boundaries for Ω₀ and the cylinders in question. Bibliography: 27 titles. We dedicate the present paper to Olga Arsenievna Oleinik, as a symbol of our deep respect and gratitude Translated from Trudy Seminara imeni I G. Petrovskogo, No. 19. pp. 000-000. 0000.     

The Singularities of the Distance Function Near Convex Boundary Points

In an open bounded set Ω, we consider the distance function from ∂Ω associated to a Riemannian metric with C ^1,1 coefficients. Assuming that Ω is convex near a boundary point x ₀, we show that the distance function is differentiable at x ₀ if and only if there exists the tangent space to ∂Ω at x ₀. Furthermore, if the distance function is not differentiable at x ₀ then there exists a Lipschitz continuous curve, with initial point at x ₀, such that the distance function is not differentiable along such a curve.      

Estimates for the hyperbolic metric of the punctured plane and applications

The hyperbolic metrich ^Ω of the twice punctured complex plane Ω is studied. A new recursive algorithm for computing the density λ_Ω ofh ^Ω is given. For a proper subdomainG of Ω we answer a question of G. Martin concerning quasiconformal mappings ofG that can be extended to the complement ofG as the identity map.     

Regularity of a free boundary problem

Let u be the Newtonian potential of a real analytic distribution in an open set Ω. In this paper we assume u is analytically continuable from the complement of Ω into some neighborhood of a point x₀ ∈ ∂Ω, and we study conditions under which the analytic continuation implies that ∂Ω is a real analytic hypersurface in some neighborhood of x₀.     

Equivalences between bloch type spaces and BOM spaces on the hyperbolic domains inC

For an analytic function f on the hyperbolic domain Ω inC, the following conclusions are obtained: (i) f∈B(Ω)=BMO A(Ω,m) if and only ifRef∈B_h(Ω)=BMOH(Ω,m). (ii) QB_h(Ω)=B_h(Ω)(BMOH _n(Ω,m)=BMOH(Ω,m)) if and only ifC(Ω)=inf{λ_Ω(z)·δ_Ω(z):z∈Ω}>0. Also, some applications to automorphic functions are considered. This research was supported by the Doctoral Program Foundation of Institute of Higher Education.     

Green functions and eigenfunction expansions for the square of the Laplace–Beltrami operator on plane domains

For a certain class of domains Ω⊂ℂ with smooth boundary and Δtilde;_Ω=w ²Δ the Laplace–Beltrami operator with respect to the Poincaré metric ds ²=w(z)^-2 dz dz on Ω, we (1) show that the Green function for the biharmonic operator Δtilde;_Ω ², with Dirichlet boundary data, is positive on Ω×Ω; and (2) obtain an eigenfunction expansion for the operator Δtilde;_Ω, which reduces to the ordinary non-Euclidean Fourier transform of Helgason for Ω=𝔻 (the unit disc). In both cases the proofs go via uniformization, and in (1) we obtain a Myrberg-like formula for the corresponding Green function. Finally, the latter formula as well as the eigenfunction expansion are worked out more explicitly in the simplest case of Ω an annulus, and a result is established concerning the convergence of the series ∑ _ω∈G (1-|ω0|²) ^s for G the covering group of the uniformization map of Ω and 0<s<1. Received: August 21, 2000?Published online: October 30, 2002 RID="*" ID="*"The first author was supported by GA AV CR grants no. A1019701 and A1019005.     

Solvability of Elliptic Boundary Value Problems without Standard Landesman-Lazer Condition

Abstract In this paper we prove a very general result concerning solvability of the resonant problem: Δu + λ_κ u + g(x, u) = h (x); u = 0, x ∈∂Ω, which immediately gives three generalized Landesman-Lazer conditions. The most interesting application of the general result is concerned with the problem when λ_κ = λ₁, in which case we prove solvability results for it under conditions which are not the standard Landesman-Lazer condition or only partly enjoy it. Furthermore, we propose a new sign condition and give a comprehensive extension of a main result of Figueiredo and Ni.     

Multivalued Analytic Continuation of the Cauchy Transform

In this paper we introduce the notion of multivalued analytic continuation of the Cauchy transforms. Many difficulties arise because the continuation is not single-valued. Our main result asserts that if χ_Ω has a multivalued analytic continuation, then the free boundary ∂Ω has zero Lebesgue measure. Here χ_Ω is the characteristic function of a domain Ω and ∂Ω is its boundary. We also discuss the connections between this notion, quadrature domains and approximations of analytic functions with single-valued integrals by rational functions. The last problem is related to the existence of a continuous function g and a closed connected set K such that the gradient of g vanishes on K, nevertheless g is not constant on K. Mathematics Subject Classifications (2000) Primary 31A25, 31B20; secondary 30E10, 35J05, 41A20.     

A local version of the two-circles theorem

A necessary and sufficient condition is given so that in a domain Ω there are no functions whose average over all balls contained in Ω of radiir ₁,r ₂ vanish except the zero function. Partially supported by NSF grant DMS-8401356 and by NSF grant OJR 85-OV-108 through the Systems Research Center of the University of Maryland.     

On quasi-similarity of subnormal operators

For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) in L2(v) and let Sv denote the operator of multiplication by the independent variable z on R2(K, v), that is, Svf = zf for every f∈R2(K, v). SupposeΩis a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Letσdenote a harmonic measure forΩ. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2(Ω,σ). We show that for a given v supported onΩ, Sv is quasi-similar to Sσif and only if v/■Ω■σ and log(dv/dσ)∈L1(σ). Our result extends a well-known result of Clary on the unit disk.     

Exact controllability for the wave equation with variable coefficients

We consider in this paper the evolution systemy″−Ay=0, whereA =∂ _i(a_ij∂_j) anda _ij ∈C ¹ (ℝ⁺;W ^1,∞ (Ω)) ∩W ^1,∞ (Ω × ℝ⁺), with initial data given by (y ₀,y ₁) ∈L ²(Ω) ×H ⁻¹ (Ω) and the nonhomogeneous conditiony=v on Γ ×]0,T[. Exact controllability means that there exist a timeT>0 and a controlv such thaty(T, v)=y′(T, v)=0. The main result of this paper is to prove that the above system is exactly controllable whenT is “sufficiently large”. Moreover, we obtain sharper estimates onT.     

Boundaries of singularity sets, removable singularities, and CR-invariant subsets of CR-manifolds

Let Ω be a bounded strictly pseudoconvex domain in ℂⁿ, n ≥ 3, with boundary ∂Ω, of class C². A compact subset K is called removable if any analytic function in a suitable small neighborhood of ∂Ω K extends to an analytic function in Ω. We obtain sufficient conditions for removability in geometric terms under the condition that K is contained in a generic C² -submanifold M of co-dimension one in ∂Ω. The result uses information on the global geometry of the decomposition of a CR-manifold into CR-orbits, which may be of some independent interest. The minimal obstructions for removability contained in M are compact sets K of two kinds. Either K is the boundary of a complex variety of co-dimension one in Ω or it is an exceptional minimal CR-invariant subset of M, which is a certain analog of exceptional minimal sets in co-dimension one foliations. It is shown by an example that the latter possibility may occur as a nonremovable singularity set. Further examples show that the germ of envelopes of holomorphy of neighborhoods of ∞Ω K for K ⊂ M may be multisheeted. A couple of open problems are discussed.     

On homogenization of the mixed boundary-value problem for the heat equation in a domain whose part contains channels arranged in a periodic way

In this paper, we study the asymptotic behavior of the solutionsu _ε (ε is a small parameter) of boundaryvalue problems for the heat equation in the domain Ω_ε=Ω⁻∪Ω _ε ⁺ ∪γ one part of which (Ω _ε ⁺ ) contains ε-periodically situated channels with diameters of order ε and the other part of which (Ω⁺) is a homogeneous medium; γ=∂Ω _ε ⁺ ∩∂Ω⁺. On the boundary of the channels the Neumann boundary condition is posed, and on ∂Ω_ε∩∂Ω the Dirichlet boundary condition is prescribed. The homogenized problem is the Dirichlet problem in Ω with the transmission condition on γ. The estimates for the difference betweenu _ε and the solution of the homogenized problem are obtained. Bibliography: 14 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 20, pp. 27–47, 1997.     

The solvability of the initial boundary-value problem for equations of motion of a viscous compressible barotropic liquid in the spacesW 2 l+1,l/2+1 (Q T )

A new method of estimating the solutions of the Navier-Stokes equations for a viscous compressible barotropic fluid in a bounded domain Ω⊂ℝ³ is suggested, which makes it possible to investigate the problem for the whole scale of anisotropic spaces W ₂ ^l+2,l/2+1 (Q_T), Q_T=Ω×(0,T), for arbitrary l>1/2. Bibliography: 10 titles. To dear Olga Alexandrovna Ladyzenskaya on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 177–186. Translated by V. A. Solonnikov.     

Decay Rates of Interactive Hyperbolic-Parabolic PDE Models with Thermal Effects on the Interface

We consider coupled PDE systems comprising of a hyperbolic and a parabolic-like equation with an interface on a portion of the boundary. These models are motivated by structural acoustic problems. A specific prototype consists of a wave equation defined on a three-dimensional bounded domain Ω coupled with a thermoelastic plate equation defined on Γ ₀—a flat surface of the boundary \partial Ω . Thus, the coupling between the wave and the plate takes place on the interface Γ ₀. The main issue studied here is that of uniform stability of the overall interactive model. Since the original (uncontrolled) model is only strongly stable, but not uniformly stable, the question becomes: what is the ``minimal amount' of dissipation necessary to obtain uniform decay rates for the energy of the overall system? Our main result states that boundary nonlinear dissipation placed only on a suitable portion of the part of the boundary which is complementary to Γ <sub>0</sub>, suffices for the stabilization of the entire structure. This result is new with respect to the literature on several accounts: (i) thermoelasticity is accounted for in the plate model; (ii) the plate model does not account for any type of mechanical damping, including the structural damping most often considered in the literature; (iii) there is no mechanical damping placed on the interface Γ <sub>0</sub>; (iv) the boundary damping is nonlinear without a prescribed growth rate at the origin; (v) the undamped portions of the boundary \partial Ω are subject to Neumann (rather than Dirichlet) boundary conditions, which is a recognized difficulty in the context of stabilization of wave equations, due to the fact that the strong Lopatinski condition does not hold. The main mathematical challenge is to show how the thermal energy is propagated onto the hyperbolic component of the structure. This is achieved by using a recently developed sharp theory of boundary traces corresponding to wave and plate equations, along with the analytic estimates recently established for the co-continuous semigroup associated with thermal plates subject to free boundary conditions. These trace inequalities along with the analyticity of the thermoelastic plate component allow one to establish appropriate inverse/ recovery type estimates which are critical for uniform stabilization. Our main result provides ``optimal' uniform decay rates for the energy function corresponding to the full structure. These rates are described by a suitable nonlinear ordinary differential equation, whose coefficients depend on the growth of the nonlinear dissipation at the origin. \par Accepted 12 May 2000. Online publication 6 October 2000.