Finite Dirichlet Integral Research Articles

Functions with a finite dirichlet integral in a domain with cusp points on the boundary

Let Ω be a domain in ℝn, n >2, the boundary of which has a cusp point, pointing inside or outside the domain. The purpose of the paper is to characterize the traces on ∂Ω of the elements of the space H1(Ω) of functions with a finite Dirichlet integral. As a consequence one establishes the existence of a linear continuous extension operator H1 (Ω) →H1(ℝn) under the presence of an interior cusp point on ∂Ω. Theorems on domains with cusps are proved with the aid of results on cylindrical domains. In the space of functions with a finite Dirichlet integral in the exterior or the interior of the cylinder one introduces the norm, depending on a small parameter ɛ and generating a norm of the trace on ∂Ω as an element of the quotient space. The latter is placed in correspondence with an explicitly described norm of functions on the boundary, uniformly equivalent relative to ɛ. One constructs an operator of extension of functions from the exterior of the cylinder to Rn, preserving H1, whose norm is uniformly bounded relative to ɛ. For the optimal operator of extension from the inside of the cylinder one finds the asymptotic behavior of the norm as ɛ→0. From these results there follow similar theorems on functions with a finite Dirichlet integral inside and outside a thin closed tube (of width ɛ).

Cyclic vectors in the Dirichlet space

We study the Hilbert space of analytic functions with finite Dirichlet integral in the open unit disc. We try to identify the functions whose polynomial multiples are dense in this space. Theorems 1 and 2 confirm a special case of the following conjecture: if | f ( z ) | ⩾ | g ( z ) | |f(z)| \geqslant |g\,(z)| at all points and if g g is cyclic, then f f is cyclic. Theorems 3-5 give a sufficient condition ( f f is an outer function with some smoothness and the boundary zero set is at most countable) and a necessary condition (the radial limit can vanish only for a set of logarithmic capacity zero) for a function f f to be cyclic.

Beurling's ordinary value

Let n(w, f) be the number of w-points of f meromorphic in D = {|z| < 1}. Beurling defined the quantity and called w an ordinary value of f if < ∞. We shall consider the intermediate quantity ṉ(w, f) in the sense that n(w, f) ≤ ṉ(w, f) , and construct two bounded holo-morphic functions f1 and f2 of finite Dirichlet integrals in D for whichand.

Solvability of boundary and initial-boundary value problems for the Navier-Stokes equations in domains with noncompact boundaries

One considers boundary and Initial-boundary value problems for Navier-Stokes equations in domains with several “outlets” at infinity. It is shown that if one prescribes the limiting values of the pressure at infinity in those “outlets” which expand sufficiently fast, then these problems are solvable in the class of vectors with a finite Dirichlet integral.

The number of Dirichlet solutions of a fourth order differential equation

SynopsisIt is shown that the equation (p2y”)”–(p1y’)’+ p0y = 0 has exactly two linearly independent solutions on [0,∞) with finite Dirichlet integral when the coefficients are nonnegative and p2 satisfies a condition which includes all nondecreasing functions. An inequality for the Dirichlet form is derived and used to extend characterizations of the domains of certain self-adjoint operations associated with the differential expression to arbitrary symmetric boundary conditions at 0.

Nonnormal Dirichlet Quotients and Nonnormal Blaschke Quotients

There exists a nonnormal meromorphic function ${f_1}/{f_2}$ in $U = \{ |z| < 1\}$, where ${f_1}$ and ${f_2}$ both are holomorphic functions with finite Dirichlet integrals in U. For each $0 < \alpha < 1$, there exists a nonnormal meromorphic function ${B_1}/{B_2}$ in U, where ${B_1}$ and ${B_2}$ both are Blaschke products with finite $\alpha$-weighted Dirichlet integrals in U.

Nonnormal Dirichlet quotients and nonnormal Blaschke quotients

There exists a nonnormal meromorphic function f 1 / f 2 {f_1}/{f_2} in U = { | z | &gt; 1 } U = \{ |z| &gt; 1\} , where f 1 {f_1} and f 2 {f_2} both are holomorphic functions with finite Dirichlet integrals in U. For each 0 &gt; α &gt; 1 0 &gt; \alpha &gt; 1 , there exists a nonnormal meromorphic function B 1 / B 2 {B_1}/{B_2} in U, where B 1 {B_1} and B 2 {B_2} both are Blaschke products with finite α \alpha -weighted Dirichlet integrals in U.

Some inequalities for analytic functions with a finite Dirichlet integral on the unit disc

Some inequalities for analytic functions with a finite Dirichlet integral on the unit disc

Some problems of vector analysis and generalized formulations of boundary-value problems for the Navier-Stokes equations

We consider the problem of finding the restrictions on the domain Ω⊂R n,n=2,3, under which the space of the solenoidal vector fields from coincides with the space, the closure in W 12 (Ω) of the set of all solenoidal vectors from. We give domains Ω⊂Rn, for which the factor space has a finite nonzero dimension. A similar problem is considered for the spaces of solenoidal vectors with a finite Dirichlet integral. Based on this, one compares two generalized formulations of boundary-value problems for the Stokes and Navier-Stokes systems. The following auxiliary problems are studied:,.

The differential equation $\Delta x=2H(x\sb{u}\wedge x\sb{v})$ with vanishing boundary values

If $x(u,v)$ is a solution to the system $\Delta x = 2H({x_u} \wedge {x_v})$ on a bounded domain $G \subset {R^2}$ with finite Dirichlet integral and with $x = 0$ on $\partial G$, then $x \equiv 0$ for simply connected $G$, but for doubly-connected $G$ we construct nontrivial solutions.

Commuting self-adjoint partial differential operators and a group theoretic problem

In R n let Ω denote a Nikodym region (= a connected open set on which every distribution of finite Dirichlet integral is itself in L 2(Ω)) . The existence of n commuting self-adjoint operators H 1,…, H n in L 2(Ω) such that each H j is a restriction of −i β βx j (acting in the distribution sense) is shown to be equivalent to the existence of a set Λ ⊂ R n such that the restrictions to Ω of the functions exp i ∑ λ j x j form a total orthogonal family in L 2(Ω) . If it is required, in addition, that the unitary groups generated by H 1,…, H n act multiplicatively on L 2(Ω) , then this is shown to correspond to the requirement that Λ can be chosen as a subgroup of the additive group R n . The measurable sets Ω ⊂ R n (of finite Lebesgue measure) for which there exists a subgroup Λ ⊂ R n as stated are precisely those measurable sets which (after a correction by a null set) form a system of representatives for the quotient of R n by some subgroup Γ (essentially the dual of Λ).

ON STATIONARY SOLUTIONS OF THE PROBLEM OF FLOW PAST A BODY OF A VISCOUS INCOMPRESSIBLE FLUID

The stationary solutions of the problem of flow past a body with finite Dirichlet integral are considered. It is found that the vector velocity u(x) differs from its limit value u∞ by a quantity o(x-1). By the same token it is proved that any solution of the flow problem with finite Dirichlet integral possesses a wake outside which the vorticity is exponentially small. Bibliography: 16 items.

Extremal length and harmonic functions on Riemann surfaces

Expressions for several conformally invariant pseudometrics on a Riemann surface R R are given in terms of three new forms of reduced extremal distance. The pseudometrics are defined by means of various subclasses of the set of all harmonic functions on R R having finite Dirichlet integral. The reduced extremal distance between two points is defined on R R , on the Alexandroff one-point compactification of R R and on the Kerékjártó-Stoïlow compactification of R R . These reduced extremal distances are computed in terms of harmonic functions having specified singularities and boundary behavior. The key to establishing this connection with harmonic functions is a general theorem dealing with extremal length on a compact bordered Riemann surface and its extensions to noncompact bordered surfaces. These results are used to obtain new tests for degeneracy in the classification theory of Riemann surfaces. Finally, some of the results are illustrated for a hyperbolic simply connected Riemann surface.

The Rate of Convergence for the Finite Element Method

The character of the proper refinement of the elements (mesh) around the boundary is studied. It is shown that it is possible to obtain the highest (optimal) rate of convergence by this refinement.

An extremal harmonic function

surface 3x=h-'(X) and D(u; X) for the Dirichlet integral over the region ix bounded by a and fi. The main result of this paper is the inequality: maxhJ|Qx=m(h; X)=D(h; X)<m(u; X)<D(u; X), for uEEHo(Q), u?? h. Thus h minimizes the mean and the Dirichlet integral in the class Ho(Q). This inequality can be used in the classification of locally Euclidean spaces. In particular, we shall show that OG- OHOM = OHOD = OHOB. Here OG is the class of spaces V possessing no Green's function, and OHOK, K = M, D, B is the class of V on which there exist no nonconstant HoK functions, that is, harmonic functions which vanish on the border of a boundary neighborhood of V and which are of finite mean, of finite Dirichlet integral, or bounded, respectively.

Continuous boundary values of analytic functions of several complex variables

Our method is elementary, and was suggested by the proofs of the Hartogs' Extension Theorem given by Bochner [I] and Martinelli [6]. We remark that the differential forms co appearing in the theorem are a natural generalization to several variables of the analytic differentials which play the corresponding role in the well-known one-variable theorem. By appealing to the regularity theorem of Kohn [4] for the 5-Neumann problem we obtain as a corollary that if the Levi form on r has at least one positive eigenvalue at each point, thenfEA (r) if and only if f is a weak solution of the tangential Cauchy-Riemann equations. (With this hypothesis on r, it follows that if D is connected, then so is r.) This theorem was obtained by Fichera [2] under the additional hypotheses that r is connected and f is the boundary value of a function with finite Dirichlet integral. His proof is based on an approximation theorem for harmonic functions in. the Dirichlet norm. Kohn and Rossi [5] have shown that the theorem holds for CI functions when D is a finite domain on a complex manifold and the Levi form satisfies the above condition on r. They also obtain extension theorems for (p, q) forms.

Two Results Concerning the Zeros of Functions with Finite Dirichlet Integral

A function f, analytic in the unit disk, is said to have finite Dirichlet integral if1Geometrically, this is equivalent to f mapping the disk onto a Riemann surface of finite area. The class of Dirichlet integrable functions will be denoted by . The condition above can be restated in terms of Taylor coefficients; if f(z) = Σanzn, then if and only if Σn|an|2 &lt; ∞. Thus, is contained in the Hardy class H2.In particular, every such function has boundary valuesalmost everywhere and log |f(eiθ)| ∊ L1(dθ).The zeros zn of a function must satisfy the Blaschke conditionand f(s) = B(z)F(z), where F(z) has no zeros andis the Blaschke product with zeros zn; see (5).

Regular operators and spaces of harmonic functions with finite Dirichlet integral on open Riemann surfaces

Regular operators and spaces of harmonic functions with finite Dirichlet integral on open Riemann surfaces

Finite Interpolation for Analytic Functions with Finite Dirichlet Integrals

Finite Interpolation for Analytic Functions with Finite Dirichlet Integrals

Finite interpolation for analytic functions with finite Dirichlet integrals

PROOF. The necessity of the condition R (IOAD is evident. We have to show the solvability of (1) under the condition R GOAD. Since R has finite genus, R GOAD implies the existence of a nonconstant ABDfunction F(z) on R. Let R* be a closed Riemann surface which contains R as a subsurface. Choose a point Po in R{1, 12, . . . , rnI such that F(to) 5 F(?k) (k = 1, 2, . .. , n). For each fixed k (k = 1, 2, ... , n), by Riemann-Roch's theorem, there exists a meromorphic function rk(z) on R* such that rk(z) has a simple pole at rk and a pole of order nk at to and regular on R*-{ -o, 0k }. Let mk be the order of zero of the function 1flL (F(z) F(?j))m+l at rk and let s=max {mknk; k=1, 2, * ,n}. Put