Joseph F. Grotowski

Kernel density estimation via diffusion

We present a new adaptive kernel density estimator based on linear diffusion processes. The proposed estimator builds on existing ideas for adaptive smoothing by incorporating information from a pilot density estimate. In addition, we propose a new plug-in bandwidth selection method that is free from the arbitrary normal reference rules used by existing methods. We present simulation examples in which the proposed approach outperforms existing methods in terms of accuracy and reliability.

PARTIAL REGULARITY FOR ALMOST MINIMIZERS OF QUASI-CONVEX INTEGRALS ∗

We consider almost minimizers of variational integrals whose integrands are quasi-convex. Under suitable growth conditions on the integrand and on the function determining the almost minimality, we establish almost everywhere regularity for almost minimizers and obtain results on the regularity of the gradient away from the singular set. We give examples of problems from the calculus of variations whose solutions can be viewed as such almost minimizers.

BOUNDARY REGULARITY FOR QUASILINEAR ELLIPTIC SYSTEMS

ABSTRACT We consider boundary regularity for solutions of certain systems of second-order nonlinear elliptic equations, and obtain a general criterion for a weak solution to be regular in the neighbourhood of a given boundary point. Combined with existing results on interior partial regularity, this result yields an upper bound on the Hausdorff dimension of the singular set at the boundary.

Harmonic map heat flow for axially symmetric data

We examine the harmonic map heat flow problem for maps between the three-dimensional ball and the two-sphere. We give blow-up results for certain initial data. We establish convergence results for suitable axially symmetric initial data, and discuss generalizations to higher dimensions.

Existence and regularity for higher-dimensional H-systems

for a map u from B to Rn+1. (Obviously for (1.1) to make sense classically, we look for u ∈ C2(B,Rn+1). As we discuss in Section 2, it also makes sense to look for a weak solution u ∈W 1,n(B,Rn+1) to (1.1) under suitable restrictions on H .) Here we use the summation convention, and the cross product w1×·· ·×wn : Rn+1⊕·· ·⊕ Rn+1 →Rn+1 is defined by the property thatw ·w1×·· ·×wn = detW for all vectors w ∈Rn+1, whereW is the (n+1)×(n+1) matrix whose first row is (w1, . . . ,wn+1) and whose j th row is (w1 j−1, . . . ,w n+1 j−1) for 2 ≤ j ≤ n+1. Equation (1.1) has a natural geometric property; namely, if u fulfills certain additional conditions, then it represents a hypersurface in Rn+1 whose mean curvature at the point u(x), for x ∈ B, is given by H ◦u(x). Specifically, a map u :B→ Rn+1 is called conformal if

Finite time blow-up for the harmonic map heat flow

SummaryWe consider the harmonic map heat flow from the three-dimensional ball to the two-sphere. We establish the existence of regular initial data leading to blow-up in finite time.

Finite time blow-up for the Yang-Mills heat flow in higher dimensions

Abstract. We consider the

On variational models for quasi-static Bingham fluids

L^2

Concentrated boundary data and axially symmetric harmonic maps

-gradient flow associated with the Yang-Mills functional, the so-called Yang-Mills heat flow. In the setting of a trivial principal SO(n)-bundle over

Optimal regularity results via A-harmonic approximation

{\mathbb R}^n