2 results
Harnack inequality for the nonlocal equations with general growth
- Part of
-
- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 153 / Issue 5 / October 2023
- Published online by Cambridge University Press:
- 23 August 2022, pp. 1479-1502
- Print publication:
- October 2023
-
- Article
-
- You have access
- HTML
- Export citation
-
We consider a class of generalized nonlocal $p$
-Laplacian equations. We find some proper structural conditions to establish a version of nonlocal Harnack inequalities of weak solutions to such nonlocal problems by using the expansion of positivity and energy estimates.
Partial regularity for minimizers of discontinuous quasiconvex integrals with general growth
- Part of
-
- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 152 / Issue 5 / October 2022
- Published online by Cambridge University Press:
- 01 December 2021, pp. 1191-1232
- Print publication:
- October 2022
-
- Article
-
- You have access
- Open access
- HTML
- Export citation
-
We prove the partial Hölder continuity for minimizers of quasiconvex functionals
where
\begin{equation*} \mathcal{F}({\bf u}) \colon =\int_{\Omega} f(x,{\bf u},D{\bf u})\,\textrm{d} x, \end{equation*}
$f$ satisfies a uniform VMO condition with respect to the 
$x$-variable and is continuous with respect to 
${\bf u}$. The growth condition with respect to the gradient variable is assumed a general one.
