Existence and uniqueness results in weighted spaces for Dirichlet problem in unbounded domains

Ricerche di Matematica - We study the Dirichlet problem for a second order linear elliptic partial differential equation with discontinuous coefficients in unbounded domains. We establish an...     

On theg-convergence of nonlinear elliptic operators related to the dirichlet problem in variable domains

A concept ofG-convergence of operatorsA _s:W _s W _s ^* to an operatorA:W W ^* is introduced and studied under a certain relationship between Banach spacesW _s,s=1,2, ..., and a Banach spaceW. It is shown that conditions establishing this relationship for abstract spaces are satisfied by the Sobolev spacesW ^k,m ( _s) andW ^k,m(), where { _s} is a sequence of perforated domains contained in a bounded region R ⁿ. Hence, the results obtained for abstract operators can be applied to the operators of the Dirichlet problem in the domains _s.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 948–962, July, 1993.     

The dam problem in unbounded domains

Summary The time dependent dam problem in a general unbounded domain is studied and an existence theorem is proved by means of a rather general approximation procedure. Moreover the propagation of the support of the solution is studied in a number of geometrical situations of interest. The corresponding stationary problem is also studied and an existence theorem is proved. Finally several one-dimensional examples are given in which the solution can be either written explicitely or found solving an ordinary differential equation.This work has been supported by MURST (Italy), by I.A.N.-C.N.R. of Pavia (Italy), and by SFB 123 of University of Heidelberg (Germany).     

Some counterexamples for the theory of sobolev spaces on bad domains

It is shown that some well-known properties of the Sobolev spaceL _p ^l () do not admit extension to the spaceL _p ^l () of the functions withl-th order derivatives inL _p (),l>1, without requirements to the domain . Namely, we give examples of such that

Generalized dirichlet problem in nonlinear potential theory

The operator extending the classical solution of the Dirichlet problem for the quasilinear elliptic equation divA (x,▽u)=0 akin to thep-Laplace equation is shown to be unique providedA obeys a specific order principle. The Keldych lemma is also generalized to this nonlinear setting. Part of this research was performed in 1988–1989 while a visitor at Indiana University, Bloomington, Indiana.     

Embeddings of Sobolev spaces on unbounded domains

Summary Piecewise polynomial and Fourier approximation of functions in the Sobolev spaces on unbounded domains Θ ⊂ Rⁿ are applied to the study of the type of compact embeddings into appropriate Lebesgue and Orlicz spaces. It is shown that if Θ and s satisfy certain conditions, the embeddings , m/n+1/q−1/p>0 and , Φ being an Orlicz function subordinate to both φ(t)=|t|^p exp |t|^n/(n−m) and Φ^σ(t)=exp |t|^σ−1, σ ⩾ 1, m/n>1/p, are of type l^s. One result dealing with multiplications maps from into L^q(Θ) is also obtained. Entrata in Redazione il 14 ottobre 1976.     

The Dirichlet problem for elliptic equations in weighted Sobolev spaces on unbounded domains of the plane

This paper deals with the Dirichlet problem for second order linear elliptic equations in unbounded domains of the plane in weighted Sobolev spaces. We prove an a priori bound and an existence and uniqueness result.     

Integral representations of functions and embeddings of sobolev spaces on cuspidal domains

Some classes of cuspidal domainsG ⊂ ℝ ⁿ are introduced, and embeddings of the formW _p ^(l) (G)↪L_q(G),l ∈ ℕ, for sobolev spaces are established. To this end, estimates of some integral operators are needed. These operators cannot be estimated via Riesz potentials or their anisotropic analogs. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 201–219, February, 1997. Translated by V. E. Nazaikinskii     

On the hilbert boundary value problem for holomorphic function in sobolev spaces

The holomorphic solution φ of the Hilbert boundary value problem Re (a + ib) φ = g on γ in a disk D bounded by the simple closed curve γ has been solved in the space of Hölder-continuously differentiable functions C₁, C_1,α (D) by many authors. Under the assumptionthat g belongs to the Slobodecky space W_s,p (γ), s = 1 ? 1/p,1 < p < ∞ it is shown here that the problem has a uniquesolution in the Sobolev space W_1,p: (D). An a-priori estimate for the norm of in W_{1 p}(D)is given.     

Embeddings of anisotropic Sobolev spaces on unbounded domains

Summary Integral representations of junctions in the anisotropic Sobolev spaces on unbounded domains are used in the study of the embeddings of these spaces into Lebesgue spaces. Estimates of entropy numbers of the embedding are obtained, where k, p and q satisfy certain conditions and where is a certain type of quasibounded domain,     

On the dirichlet problem for fully nonlinear parabolic equations

The paper describes a new class of fully nonlinear second-order parabolic equations. The peculiarity of this class in the nonlinear dependence of the equations both on the first-order time derivative and second-order spatial ones. Application of the classical continuity method to solve the first initial-boundary value problem for such equations is also discussed. Bibliography: 15 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 233, 1996, pp. 101–111.     

Weighted fractional Sobolev spaces as interpolation spaces in bounded domains

We characterize the real interpolation space between a weighted $L^p$ space and a weighted Sobolev space in arbitrary bounded domains in ${\mathbb{R}}^n$, with weights that are positive powers of the distance to the boundary.     

Maximum entropy solutions and moment problem in unbounded domains

The classical Stieltjes and Hamburger moment problems in the maximum entropy approach have been reconsidered. Incorrect conditions of existence, previously appeared in literature, have been reviewed and improved.     

Banach空间中无界域上n阶非线性微分—积分方程初值问题

应用锥理论研究Banach空间中无界域上n阶非线性Volterra型微分积分方程值问题，获得了最小非负解存在性结果。