Calculus of Variations and Geometric Measure Theory

G. Cupini - F. Leonetti - E. Mascolo

Existence of weak solutions for elliptic systems with $p,q$-growth

created by cupini on 22 Dec 2014


Accepted Paper

Inserted: 22 dec 2014
Last Updated: 22 dec 2014

Journal: Ann. Acad. Sci. Fenn. Math.
Year: 2015


We consider a non-linear system of $m$ equations in divergence form and a boundary condition: \[ \left\{\begin{array}{cl} \displaystyle \sum_{i=1}^n\frac{\partial }{\partial x_i}\left(A_i^\alpha(x,Du(x))\right)=0, \quad 1\le \alpha\le m, &\text{in $ \Omega$} \\ u=\tilde{u} &\text{on $ \partial \Omega$.} \end{array}\right. \] The functions $A_i^\alpha(x,z)$ are H\"older continuous with respect to $x$ and \[
^p-c_1\le \sum_{\alpha=1}^m\sum_{i=1}^n A_i^{\alpha}(x,z)z_i^{\alpha}\le c_2 (1+
)^{q},\qquad 2\le p\le q. \] We prove the existence of a weak solution $u$ in $(\tilde{u}+W_0^{1,p}(\Omega;\mathbb{R}^m))\cap W_{\rm loc}^{1,q}(\Omega;\mathbb{R}^m)$, provided $p$ and $q$ are close enough and under suitable summability assumptions on the boundary datum $\tilde{u}$.

Keywords: existence, regularity, weak solution, elliptic system, growth