17 research outputs found
Malmheden's theorem revisited
In 1934 H. Malmheden discovered an elegant geometric algorithm for solving
the Dirichlet problem in a ball. Although his result was rediscovered
independently by Duffin 23 years later, it still does not seem to be widely
known. In this paper we return to Malmheden's theorem, give an alternative
proof of the result that allows generalization to polyharmonic functions and,
also, discuss applications of his theorem to geometric properties of harmonic
measures in balls in Euclidean spaces
Two-dimensional shapes and lemniscates
A shape in the plane is an equivalence class of sufficiently smooth Jordan
curves, where two curves are equivalent if one can be obtained from the other
by a translation and a scaling. The fingerprint of a shape is an equivalence of
orientation preserving diffeomorphisms of the unit circle, where two
diffeomorphisms are equivalent if they differ by right composition with an
automorphism of the unit disk. The fingerprint is obtained by composing Riemann
maps onto the interior and exterior of a representative of a shape in a
suitable way. In this paper, we show that there is a one-to-one correspondence
between shapes defined by polynomial lemniscates of degree n and nth roots of
Blaschke products of degree n. The facts that lemniscates approximate all
Jordan curves in the Hausdorff metric and roots of Blaschke products
approximate all orientation preserving diffeomorphisms of the circle in the
C^1-norm suggest that lemniscates and roots of Blaschke products are natural
objects to study in the theory of shapes and their fingerprints
Laplacian Growth, Elliptic Growth, and Singularities of the Schwarz Potential
The Schwarz function has played an elegant role in understanding and in
generating new examples of exact solutions to the Laplacian growth (or "Hele-
Shaw") problem in the plane. The guiding principle in this connection is the
fact that "non-physical" singularities in the "oil domain" of the Schwarz
function are stationary, and the "physical" singularities obey simple dynamics.
We give an elementary proof that the same holds in any number of dimensions for
the Schwarz potential, introduced by D. Khavinson and H. S. Shapiro [17]
(1989). A generalization is also given for the so-called "elliptic growth"
problem by defining a generalized Schwarz potential. New exact solutions are
constructed, and we solve inverse problems of describing the driving
singularities of a given flow. We demonstrate, by example, how \mathbb{C}^n -
techniques can be used to locate the singularity set of the Schwarz potential.
One of our methods is to prolong available local extension theorems by
constructing "globalizing families". We make three conjectures in potential
theory relating to our investigation
Beurling's theorem for the Bergman space
A celebrated theorem in operator theory is A. Beurling's description of the invariant subspaces in in terms of inner functions [Acta Math. {\bf81} (1949), 239--255; MR0027954 (10,381e)]. To do the same thing for the Bergman space has been deemed virtually impossible by many analysts, in view of the fact that the lattice of invariant subspaces is so large, and that the invariant subspaces may have weird properties as viewed from the perspective. The size of the lattice can be appreciated from the known fact that essentially every operator on separable Hilbert space can be realized as the compression of the Bergman shift on , where and are invariant subspaces, . But a Beurling-type theorem is precisely what the present paper delivers. Given an invariant subspace in , consider the subspace , where stands for multiplication by . This makes sense because is a closed subspace of . In Beurling's case, is one-dimensional and spanned by an inner function. In the setting, the dimension of may be arbitrarily large, even infinite. However, with the correct analogous definition of inner functions in , all vectors of unit norm in\break are -inner. Following Halmos, the subspace is called the wandering subspace of . Given an invariant subspace, a natural question is: which collections of elements generate it? In particular, one can ask for the least number of elements in a set of generators. It is known that the dimension of the wandering subspace represents a lower bound for the least number of generators. In the paper, it is shown that any orthonormal basis in the wandering subspace (which then consists of -inner functions) generates as an invariant subspace. This settles the issue of the minimal number of generators. Let be the orthogonal projection , and let be the operator such that is the orthogonal projection . Then, for , . If we do the same for , we get and, inserting it into the original relation for , we get . As we go on repeating this process, we get . The point with this decomposition is that, apart from the last term, each term is of the form to some power times an element of , so that is in\break , the invariant subspace generated by . If the operators happened to be uniformly bounded, as they are in the case of , would tend to in the weak topology, and would be in the weak closure of , which by standard functional analysis coincides with . However, for the Bergman space, it seems unlikely that the are uniformly bounded for all possible invariant subspaces , although no immediate counterexample comes to mind. For this reason, the authors try Abel summation instead, and consider for the operators given by ; it is easy\break to show that the series converges in norm. To prove that M=\break [M \ominus TM], it suffices to check that (a) for some constant independent of , and (b) in the topology of uniform convergence on compacts in the unit disk. This is so because then converges weakly to as . Part (b) is easy; the trick is to obtain (a). The authors show at an early stage that for , M=(M\ominus TM)+\break (T-\lambda)M, and that the sum is direct (in the Banach space sense). They write for the skewed projection operator associated with this decomposition. The operator has a convergent Taylor series expansion , which very much resembles the expression for . In fact, one sees that . Now one observes that , because the element of which one has to add to to obtain vanishes at the point . It follows that normally in as . We come to the hard part: obtaining the uniform boundedness of the operators . What is needed is a concrete representation formula for the norm in which makes it possible to compare the norms of and . Here, the factorization theory due to the reviewer [J. Reine Angew. Math. 422 (1991), 45--68; MR1133317 (93c:30053)], and subsequent improvements by Duren-Khavinson-Shapiro-Sundberg, come to assistance. Let be an -inner function, and let , the invariant subspace generated by . The norm of can then be expressed in terms of the quotient . That formula suggests the norm identity (1) , which turns out to be valid for general invariant subspaces , where is the biharmonic Green function for the unit disk (suitably normalized), which is known to be positive. One first checks that the formula (1) holds for all in . Second, a related integral formula for the norm in , in terms of integrals along concentric circles, is established for all . Then an intricate argument, involving Green's formula and rather subtle analysis of signs of functions, shows that for , we have at least a inequality in (1). To get the identity (1) for general , it is necessary to show first that . Formula (1), which we know to hold for and with for general , applies to with equality, and if we notice that , we get (2) . An almost radial monotonicity property of in the variable then shows that the last term on the right-hand side of (2) has a as bounded by twice the value one gets when is set equal . Moreover, the first term on the right-hand side of (2) has a as which equals what one gets when is plugged in. Thus, . It follows that weakly, which completes the proof. In the paper, it is even shown that in norm as . Incidentally, the proof also answers affirmatively one of the conjectures raised by the reviewer concerning polynomial approximation in certain weighted Bergman spaces [in Linear and complex analysis. Problem book 3. Part II, 114, Lecture Notes in Math., 1574, Springer, Berlin, 1994]. The above-mentioned theorem is new also in the case when the wandering subspace is one-dimensional. The theorem seems to represent a breakthrough in our understanding of the invariant subspaces of the Bergman space. What is desirable and remains to be developed is a better understanding of the wandering subspaces