REMOVABLE SETS FOR HÖLDER CONTINUOUS p(x)-HARMONIC FUNCTIONS

Removable Sets for Subharmonic Functions

It is a classical result that a closed exceptional polar set is removable for subharmonic functions which are bounded above. Gardiner has shown that in the case of a compact exceptional set the above boundedness condition can be relaxed by imposing certain smoothness and Hausdorff measure conditions on the set. We give related results for a closed exceptional set, by replacing the smoothness an...

improved regularity of harmonic map flows with hölder continuous energy ∗

For a smooth harmonic map flow u : M× [0, T ) → N with blow-up as t ↑ T , it has been asked ([6], [5], [7]) whether the weak limit u(T ) : M→ N is continuous. Recently, in [12], we showed that in general it need not be. Meanwhile, the energy function E(u(·)) : [0, T ) → R, being weakly positive, smooth and weakly decreasing, has a continuous extension to [0, T ]. Here we show that if this exten...

Sets of determination for harmonic functions in an NTA domain

Approximating Continuous Functions by Holomorphic and Harmonic Functions

If Q is a Widom domain in the plane (e.g., finitely connected) and / is any bounded harmonic function on Q which is not holomorphic, then we prove the algebra H°°(Sl)[f] contains all the uniformly continuous functions on ÎÎ . The basic tools are the solution of the 5 equation with L°° estimates and some estimates on the level sets of functions in BMOA.

Micro Tangent Sets of Continuous Functions

Motivated by the concept of tangent measures and by H. Fürstenberg’s definition of microsets of a compact set A we introduce micro tangent sets and central micro tangent sets of continuous functions. It turns out that the typical continuous function has a rich (universal) micro tangent set structure at many points. The Brownian motion, on the other hand, with probability one does not have graph...