| 总结: | <p>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.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
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