Search

We extend the celebrated theorem of Kellogg for conformal mappings to the minimizers of Dirichlet energy. Namely we prove that a diffeomorphic minimizer of Dirichlet energy of Sobolev mappings between doubly connected domains D and $$\Omega $$ having $${\mathscr {C}}^{n,\alpha }$$ boundary is $${\mathscr {C}}^{n,\alpha }$$ up to the boundary, provided $${{\,\mathrm{Mod}\,}}(D)\geqslant {{\,\mathrm{Mod}\,}}(\Omega )$$ . If $${{\,\mathrm{Mod}\,}}(D)< {{\,\mathrm{Mod}\,}}(\Omega )$$ and $$n=1$$ we obtain that the diffeomorphic minimizer has $${\mathscr {C}}^{1,\alpha '}$$ extension up to the boundary, for $$\alpha '=\alpha /(2+\alpha )$$ . It is crucial that, every diffeomorphic minimizer of Dirichlet energy has a very special Hopf differential and this fact is used to prove that every diffeomorphic minimizer of Dirichlet energy can be locally lifted to a certain minimal surface near an arbitrary point inside and at the boundary. This is a complementary result of an existence results proved by Iwaniec et al. (Invent Math 186(3):667–707, 2011).

The aim of this paper is to prove the following version of well-known Kellogg’s theorem. Let , , and is a conformal (one-to-one, onto) mapping. Then, extends to the homeomorphism from to ; moreover, , and the inverse mapping .

We give some extensions of classical results of Kellogg and Warschawski to a class of quasiconformal (q.c.) mappings. Among the other results we prove that a q.c. mappingf, between two planar domains with smoothC1,αboundaries, together with its inverse mappingf−1, isC1,αup to the boundary if and only if the Beltrami coefficient μfis uniformly α Hölder continuous (0 < α < 1).

The Kellogg property says that the set of irregular boundary points has capacity zero, i.e. given a bounded open set $\Omega$ there is a set $E \subset \partial\Omega$ with capacity zero such that for all $p$-harmonic functions $u$ in $\Omega$ with continuous boundary values in Sobolev sense, $u$ attains its boundary values at all boundary points in $\partial\Omega \setminus E$. In this paper, we show a weak Kellogg property for quasiminimizers: a quasiminimizer with continuous boundary values in Sobolev sense takes its boundary values at quasievery boundary point. The exceptional set may however depend on the quasiminimizer. To obtain this result we use the potential theory of quasisuperminimizers and prove a weak Kellogg property for quasisuperminimizers. This is done in complete doubling metric spaces supporting a Poincare inequality.

CONTENTS §1. The Kellogg conception and its development §2. Totally positive integral operators §3. The basic theorem. Oscillatory properties of a spectrum §4. The Kellogg property and the constant sign property §5. Characteristics of total positiveness of a kernel References

In this note, we shall supplement Kellogg's theory of interface problems, and provide complete eigenfunctions for the Sturm-Liouville system concerned. The complete eigenfunctions of interface problems are essential not only for theoretical research but also for numerical methods.

In 1975, R. B. Kellogg gave a condition guaranteeing uniqueness for the fixed point whose existence is insured by the Schauder Theorem. In this note, we indicate how to extend Kellogg's result to the class of k-set-contractions. R. B. Kellogg [1] established the following uniqueness criterion for the Schauder Fixed Point Theorem: THEOREM. Let D be a bounded convex open subset of a real Banach space X, and let F: D -* D be a compact continuous map which is continuously Frechet differentiable on D. Suppose that (a) for each x E D, 1 is not an eigenvalue of F'(x), and (b) for each x E aD, x # F(x). Then F has a unique fixed point. In this note we indicate how to extend Kellogg's theorem to a larger class of maps than the compact ones. In order to describe the class of maps with which we work, let X be a real Banach space. If B C X is bounded, we define the set measure of noncompactness of B, a(B), by a(B) = inf{c > 0: B has a finite cover by sets whose diameters do not exceed e}. Clearly, B is precompact iff a(B) = 0. A function F whose domain is a subset of X is called a k-set-contraction, or an a-Lipschitz operator, if there is a nonnegative constant k such that a(F(B)) 1 and X E a(F)} is a finite set each of whose elements is an eigenvalue of F of finite multiplicity. See [4]. In 1970, the same author extended the degree theory for Banach spaces to a Received by the editors June 27, 1977 and, in revised form, September 7, 1977. AMS (MOS) subject classifications (1970). Primary 47H10, 54H25, 55C20, 55C25.