Regularity theory in the calculus of variations and elliptic systems

Thesis

In this thesis we will investigate the regularity of critical points and minimisers of integral functionals of the form \[ \mathcal{F}(w) = F (x, w, \nabla w) dx, \] where the mappings w we consider will be vector-valued. This will be done from several different perspectives, with the goal of understanding when solutions are locally regular (of class C^{1,α} ) near a given point $x_0 \in \Omega$. This is typically done through a so-called “ε-regularity result,” which characterises the points x_0 which are regular in the above sense, by a smallness condition for a suitable excess quantity. We will investigate in three different aspects such regularity issues.

In the first part, we will consider the regularity for critical points of functionals satisfying a Legendre-Hadamard ellipticity condition. It is known by the works of Müller & Šverák [MŠ03] that critical points may be highly irregular (in particular nowhere C^1 ), however we show that regularity does hold if we assume an a-priori smallness condition of the gradient in BMO . Results will be obtained up to the boundary, and global consequences will also be explored.

In the second part, we will consider the existence and partial regularity theory for minimisers of non-autonomous quasiconvex integrands, subject to a general growth condition. By this we mean the growth of the integrand F is governed by an N -function, which we assume satisfies a ∆ 2 -condition. We will obtain a general existence theorem which will involve selecting a regularised minimising sequence along which we establish lower-semicontinuity, along with a partial regularity result. For the regularity theory we will additionally assume a non-degeneracy condition from below in the form of a ∇ 2 -condition, and we will discuss how this can be partially relaxed.

Finally in the third part, which is joint work with Lukas Koch (MPI Leipzig), we will consider relaxed minimisers for convex integrands satisfying a (p, q)-growth condition of the form \[ |z|^p ≲ F (x, z) ≲ 1 + |z|^q . \] We will establish interior improved differentiability results in the Besov scales $B^{s,p}_{\infty}$, by means of a novel second order difference quotient technique. By using known ε-regularity results for minimisers in this setting, we obtain improved dimension estimates for the singular set.

Authors: Irving, C

• DOI: 10.5287/ora-mv1vjpm7n
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: elliptic PDEs; quasiconvexity; general growth; calculus of variations
• Subjects: Mathematics