We prove the partial Hölder continuity for minimizers of quasiconvex functionals

\begin{equation*} \mathcal{F}({\bf u}) \colon =\int_{\Omega} f(x,{\bf u},D{\bf u})\,\textrm{d} x, \end{equation*}

$f$

$x$

${\bf u}$

We show local higher integrability of derivative of a suitable weak solution to the surface growth model, provided a scale-invariant quantity is locally bounded. If additionally our scale-invariant quantity is small, we prove local smoothness of solutions.

In this paper we prove a regularityresult for local minimizers of functionals of the Calculus of Variations of thetype

$$\int_{\Omega}f(x, Du)\ {\rm d}x$$

where Ω is a bounded open set in $\mathbb{R}^{n}$ , u∈ $W^{1,p}_{\rm loc}$ (Ω; $\mathbb{R}^{N}$ ), p> 1, n≥ 2 and N≥ 1.We use the technique of difference quotient without the usual assumption on the growth of the second derivatives of the function f. We apply this result to givea bound on the Hausdorff dimension of the singular set of minimizers.

An a priori Campanato type regularity condition is established for a class of W1X local minimisers $\overline{u}$ of the general variational integral $\int_{\Omega} F(\nabla u(x))\,{\rm d}x$ where $\Omega \subset \mathbb{R}^n$ is an open bounded domain, F is of class C2, F is strongly quasi-convex and satisfies the growth condition $F(\xi)\leq c(1+|\xi|^p)$ for a p > 1 and where the corresponding Banach spaces X are the Morrey-Campanato space $\mathcal{L}^{p,\mu} (\Omega,\mathbb{R}^{N\times n})$ , µ < n, Campanato space $\mathcal{L}^{p,n}(\Omega,\mathbb{R}^{N\times n})$ and the space of bounded mean oscillation $ {\rm BMO}\Omega,\mathbb{R}^{N\times n})$ . The admissible maps $u\colon \Omega \to \mathbb{R}^N$ are of Sobolev class W1,p , satisfying a Dirichlet boundary condition, and to help clarify the significance of the above result the sufficiency condition for W1BMO local minimisers is extended from Lipschitz maps to this admissible class.

We prove partial regularity with optimal Hölder exponent ofvector-valued minimizers u of the quasiconvex variational integral $\intF( x,u,Du) \,{\rm d}x$ under polynomial growth. We employ the indirectmethod of the bilinear form.


We prove a $C^{k,\alpha}$ partial regularity result for local minimizers of variationalintegrals of the type $I(u)=\int_\Omega f(D^{k}u(x)){\rm d}x$ , assumingthat the integrand f satisfies (p,q) growth conditions.


In this paper we prove that every weakand strong localminimizer $u\in{W^{1,2}(\Omega,\mathbb{R}^3)}$ of the functional $I(u)=\int_\Omega|Du|^2+f({\rm Adj}Du)+g({\rm det}Du),$ where $ u:\Omega\subset\mathbb{R}^3\to \mathbb{R}^3$ , f grows like $|{\rm Adj}Du|^p$ , g growslike $|{\rm det}Du|^q$ and1<q<p<2, is $C^{1,\alpha}$ on an opensubset $\Omega_0$ of Ω such that ${\it meas}(\Omega\setminus \Omega_0)=0$ . Suchfunctionals naturally arise from nonlinear elasticity problems. The keypoint in order to obtain the partial regularity result is toestablish an energy estimate of Caccioppoli type, which is based onan appropriate choice of the test functions. The limit case $p=q\le 2$ is also treated for weak local minimizers.

We prove a new energy or Caccioppoli type estimate for minimisers of the model functional ∫Ω|Du|2 + (det Du)2, where Ω ⊂ 2 and u: Ω → 2. We apply this to establish C∞ regularity for minimisers except on a closed set of measure zero. We also prove a maximum principle and use this to establish everywhere continuity of minimisers.