Regularity results up to the boundary for minimizers of \(p(x)\)-energy with \(p(x)>1\)
Given a bounded domain \(\Omega\subset\mathbb{R}^m\) with \(C^1\)-smooth boundary, the authors obtain partial regularity in \(\overline{\Omega}\) for the minimizers \(u: \Omega\to \mathbb{R}^n\) of the \(p(x)\)-growth type functional \[ \int_\Omega \left( A^{\alpha\beta}_{ij}(x,u)D_\alpha u^i(x)D_\beta u^j(x)\right)^{p(x)/2}dx \] with a Hölder continuous \(p(x)>1.\) It is assumed that \(A^{\alpha\beta}_{ij}(x,u)\) are bounded Hölder continuous functions on \(\Omega\times \mathbb{R}^n,\) \(A^{\alpha\beta}_{ij}(x,u)=A^{\beta\alpha}_{ji}(x,u)\) and \[ \lambda |\xi|^2\leq A^{\alpha\beta}_{ij}(x,u) \xi^i_\alpha\xi^j_\beta\leq \Lambda |\xi|^2. \] In the particular case when the coefficients satisfy the splitting condition \[ A^{\alpha\beta}_{ij}(x,u)=g^{\alpha\beta}(x) G_{ij}(x,u), \] it is shown that the minimizers have no singular points on \(\partial\Omega\).
scientific article; zbMATH DE number 6680077
scientific article