Manifold Constrained Non-uniformly Elliptic Problems

Abstract

We consider the problem of minimizing variational integrals defined on nonlinear Sobolev spaces of competitors taking values into the sphere. The main novelty is that the underlying energy features a nonuniformly elliptic integrand exhibiting different polynomial growth conditions and no homogeneity. We develop a few intrinsic methods aimed at proving partial regularity of minima and providing techniques for treating larger classes of similar constrained non-uniformly elliptic variational problems. In order to give estimates for the singular sets we use a general family of Hausdorff type measures following the local geometry of the integrand. A suitable comparison is provided with respect to the naturally associated capacities.

Removable sets in non-uniformly elliptic problems

“…holding for all x ∈ Ω, s, t ∈ [0, ∞), and ε ∈ (0, 1), see [3,15]. We shall often deal with the vector field…”

“…which turn out to be useful in dealing with the regularity theory for double/multi phase functionals, see [3,11,12,15,16]. In the following, we will also consider the double phase integrand…”

“…Naturally associated to those capacities is the concept of intristic Hausdorff measures, introduced in [15], see also [46,51].…”

Regularity for solutions of fully nonlinear elliptic equations with nonhomogeneous degeneracy

“…The degeneracy term appearing in (1.1) is modelled upon the Double-Phase energy, which first appeared in [24][25][26] in the study of the Lavrentiev phenomenon and Homogeneization theory. It received lots of attention also from the viewpoint of regularity theory, look at [1,2,11,13] for a rather comprehensive account on the regularity of local minimizers of the variational integral W 1,p (Ω) ∋ w → min Ω |Dw| p + a(x)|Dw| q dx, a ∈ C 0,α (Ω), see also [10] for the obstacle problem and some potential theoretic considerations, [19] for the manifold constrained case, [12,18] for nonlinear Calderón-Zygmund-type results and [20] for the regularity features of viscosity solutions of the fractional Double-Phase operator |w(x) − w(y)| p−2 (w(x) − w(y)) |x − y| n+sp + a(x, y) |w(x) − w(y)| q−2 (w(x) − w(y)) |x − y| n+tq dy.…”

Removable sets in elliptic equations with Musielak–Orlicz growth

“…Naturally associated to these capacities is the concept of intrinsic Hausdorff measures, introduced in [17], see also [50,55].…”