Local regularity results for minima of functionals of the calculus of variation (Q1961256)

The authors consider local minima \(v\in W^1_{p,\text{loc}}(\Omega)\) of the variational integral \[ I(v)= \int_\Omega f(x,v,\nabla v) dx \] where \(\Omega\subset\mathbb{R}^n\) is open and \(f\) denotes a Carathéodory function satisfying the growth condition \[ a|\xi|^p\leq f(x,u,\xi)\leq b|\xi|^p+ \varphi_0(x), \] where \(p>1\) and \(\varphi_0\in L^r_{\text{loc}}(\Omega)\) for some \(r>1\). A typical result is the following one: Suppose \(1<p<n\) and \(1< r<{n\over p}\). Then \(v\) is in the space \(L^s_{\text{loc}}(\Omega)\) for \(s\) defined through \({1\over s}={1\over p\cdot r}-{1\over n}\).