Normal functions possessing angular boundary values almost everywhere

1974 ◽  
Vol 15 (6) ◽  
pp. 504-507
Author(s):  
V. I. Gavrilov
Keyword(s):  
Boundary Values ◽  
Normal Functions ◽  
Almost Everywhere ◽  
Angular Boundary

A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems

1994 ◽  
Vol 124 (3) ◽  
pp. 437-471 ◽  
Author(s):  
Gero Friesecke
Keyword(s):  
Solid Phase ◽  
Young Measure ◽  
Variational Problems ◽  
Boundary Values ◽  
Necessary And Sufficient Condition ◽  
Linear Deformation ◽  
Almost Everywhere ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Unique Limit

For scalar variational problemssubject to linear boundary values, we determine completely those integrandsW: ℝn→ ℝ for which the minimum is not attained, thereby completing previous efforts such as a recent nonexistence theorem of Chipot [9] and unifying a large number of examples and counterexamples in the literature.As a corollary, we show that in case of nonattainment (and providedWgrows superlinearly at infinity), every minimising sequence converges weakly but not strongly inW1,1(Ω) to a unique limit, namely the linear deformation prescribed at the boundary, and develops fine structure everywhere in Ω, that is to say every Young measure associated with the sequence of its gradients is almost-nowhere a Dirac mass.Connections with solid–solid phase transformations are indicated.

A POLAR DECOMPOSITION FOR HOLOMORPHIC FUNCTIONS ON A STRIP

2001 ◽  
Vol 33 (3) ◽  
pp. 309-319 ◽  
Author(s):  
KONRAD SCHMÜDGEN
Keyword(s):  
Holomorphic Function ◽  
Real Axis ◽  
Polar Decomposition ◽  
Holomorphic Functions ◽  
Heisenberg Algebra ◽  
Boundary Values ◽  
Operator Representation ◽  
The Real ◽  
Almost Everywhere

Let f be a holomorphic function on the strip {z ∈ [Copf ] : −α < Im z < α}, where α > 0, belonging to the class [Hscr ](α,−α;ε) defined below. It is shown that there exist holomorphic functions w1 on {z ∈ [Copf ] : 0 < Im z < 2α} and w2 on {z ∈ [Copf ] : −2α < Im z < 2α}, such that w1 and w2 have boundary values of modulus one on the real axis, and satisfy the relationsw1(z)=f(z-αi)w2(z-2αi) and w2(z+2αi)=f(z+αi)w1(z)for 0 < Im z < 2α, where f(z) := f(z). This leads to a ‘polar decomposition’ f(z) = uf(z + αi)gf(z) of the function f(z), where uf (z + αi) and gf(z) are holomorphic functions for −α < Im z < α, such that [mid ]uf(x)[mid ] = 1 and gf(x) [ges ] 0 almost everywhere on the real axis. As a byproduct, an operator representation of a q-deformed Heisenberg algebra is developed.

Two Results Concerning the Zeros of Functions with Finite Dirichlet Integral

1969 ◽  
Vol 21 ◽  
pp. 312-316 ◽  
Author(s):  
James G. Caughran
Keyword(s):  
Riemann Surface ◽  
Boundary Values ◽  
Hardy Class ◽  
Dirichlet Integral ◽  
Finite Area ◽  
Taylor Coefficients ◽  
Almost Everywhere ◽  
Integrable Functions ◽  
Zeros Of Functions ◽  
Image Position

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 < ∞. 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).

On an Extremal Problem Involving Harmonic Functions

1988 ◽  
Vol 31 (1) ◽  
pp. 63-69 ◽  
Author(s):  
E. J. P. Georg Schmidt
Keyword(s):  
Harmonic Function ◽  
Unique Solution ◽  
Extremal Problem ◽  
Open Subset ◽  
Harmonic Functions ◽  
Positive Measure ◽  
Normal Derivative ◽  
Regularity Conditions ◽  
Boundary Values ◽  
Almost Everywhere

AbstractGiven a domain D in R” and two specified points P0 and P1 in D we consider the problem of minimizing u(p1) over all functions harmonic in D with values between 0 and 1 normalised by the requirement u(P0) = 1/2. We show that when D is suitably regular the problem has a unique solution u* which necessarily takes on boundary values 0 or 1 almost everywhere on the boundary. In the process we prove that it is possible to separate P0 and P1by a harmonic function whose boundary value is supported in an arbitrary set of positive measure. These results depend on the fact that (under suitable regularity conditions) a harmonic function which vanishes on an open subset of the boundary has a normal derivative which is almost everywhere non-vanishing in that set.

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