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We prove partial regularity for minimizers of the functional ${\int }_{\Omega }f(x,u\left(x\right),Du\left(x\right))dx$ where the integrand f(x,u,ξ) is quasiconvex with subquadratic growth: $\left|f(x,u,\xi )\right|\le L(1+|\xi {|}^p)$, p < 2. We also obtain the same results for ω-minimizers.

We examine the p-harmonic equation div |grad u|. grad u = mu, where mu is a bounded Radon measure. We determine a range of p's for which solutions to the equation verify an a priori estimate. For such p's we also prove a higher integrability result.


We prove a $C^{k,\alpha }$ partial regularity result for local minimizers of variational integrals of the type $I\left(u\right)={\int }_{\Omega }f\left(D^{k}u\left(x\right)\right)\mathrm{d}x$, assuming that the integrand satisfies growth conditions.


In this paper we prove that every weak and strong local minimizer $u\in W^{1,2}(\Omega ,{ℝ}^3)$ of the functional $I\left(u\right)={\int }_{\Omega }{\left|Du\right|}^2+f\left(\mathrm{Adj}Du\right)+g\left(\det Du\right),$ where $u:\Omega \subset {ℝ}^3\to {ℝ}^3$, grows like ${\left|\mathrm{Adj}Du\right|}^p$, grows like ${\left|\det Du\right|}^q$ and , is $C^{1,\alpha }$ on an open subset ${\Omega }_0$ of such that $\mathrm{𝑚𝑒𝑎𝑠}(\Omega \setminus {\Omega }_0)=0$. Such functionals naturally arise from nonlinear elasticity problems. The key point in order to obtain the partial regularity result is to establish an energy estimate of Caccioppoli type, which is based on an appropriate choice of the test functions. The limit case $p=q\le 2$ is also treated for weak local minimizers.

In this paper we prove a regularity result for very weak solutions of equations of the type $-divA(x,u,Du)=B(x,u,Du)$, where $A$, $B$ grow in the gradient like $t^{p-1}$ and $B(x,u,Du)$ is not in divergence form. Namely we prove that a very weak solution $u\in W^{1,r}$ of our equation belongs to $W^{1,p}$. We also prove global higher integrability for a very weak solution for the Dirichlet problem $$\left\{\begin{array}{cc}-divA(x,u,Du)=B(x,u,Du)\hfill & \text{in}\Omega ,u-u_o\in W^{1,r}(\Omega ,{ℝ}^m).\hfill \end{array}\right.$$

Regularity results for minimal configurations of variational problems involving both bulk and surface energies and subject to a volume constraint are established. The bulk energies are convex functions with -power growth, but are otherwise not subjected to any further structure conditions. For a minimal configuration (), Hölder continuity of the function is proved as well as partial regularity of the boundary of the minimal set . Moreover, full regularity of the boundary of the minimal set is obtained...

We prove an existence and uniqueness theorem for the Dirichlet problem for the equation $\text{div}\left(a\left(x\right)\nabla u\right)=\text{div}f$ in an open cube $\mathrm{\Omega }\subset {\mathbb{R}}^N$, when $f$ belongs to some $L^p\left(\mathrm{\Omega }\right)$, with $p$ close to 2. Here we assume that the coefficient $a$ belongs to the space BMO($\mathrm{\Omega }$) of functions of bounded mean oscillation and verifies the condition $a\left(x\right)\ge {\lambda }_0>0$ for a.e. $x\in \mathrm{\Omega }$.