Regularity and weak regularity of solutions of quasi-linear elliptic systems

1994 ◽  
pp. 163-185
Author(s):  
A. I. Koshelev ◽  
S. I. Chelkak
Keyword(s):  
Elliptic Systems ◽  
Regularity Of Solutions ◽  
Weak Regularity

scholarly journals Existence via regularity of solutions for elliptic systems and saddle points of functionals of the calculus of variations

2017 ◽  
Vol 6 (2) ◽  
pp. 99-120 ◽  
Author(s):  
Lucio Boccardo ◽  
Luigi Orsina
Keyword(s):  
Calculus Of Variations ◽  
Weak Solutions ◽  
Saddle Points ◽  
Elliptic Systems ◽  
Regularity Of Solutions ◽  
Integral Functionals ◽  
Content Type ◽  
The Core ◽  
Regularity Of Weak Solutions

AbstractThe core of this paper concerns the existence (via regularity) of weak solutions in ${W_{0}^{1,2}}$ of a class of elliptic systems such as$\left\{\begin{aligned} \displaystyle-\operatorname{div}((A+\varphi)\nabla u)&% \displaystyle=f,\\ \displaystyle-\operatorname{div}(M(x)\nabla\varphi)&\displaystyle=\frac{1}{2}% \lvert\nabla u\rvert^{2},\end{aligned}\right.$deriving from saddle points of integral functionals of the type$J(v,\psi)=\frac{1}{2}\int_{\Omega}(A+\psi_{+})\lvert\nabla v\rvert^{2}-\frac{1% }{2}\int_{\Omega}M(x)\nabla\psi\nabla\psi-\int_{\Omega}fv.$

scholarly journals Partial regularity result for non-autonomous elliptic systems with general growth

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Teresa Isernia ◽  
Chiara Leone ◽  
Anna Verde
Keyword(s):  
Weak Solutions ◽  
Partial Regularity ◽  
Elliptic Systems ◽  
Regularity Result ◽  
Crucial Point ◽  
Weak Regularity ◽  
General Growth ◽  
Regularity Properties ◽  
Controllable Growth ◽  
Partial Regularity Result

<p style='text-indent:20px;'>In this paper we prove a partial Hölder regularity result for weak solutions <inline-formula><tex-math id="M1">\begin{document}$ u:\Omega\to \mathbb{R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ N\geq 2 $\end{document}</tex-math></inline-formula>, to non-autonomous elliptic systems with general growth of the type:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} -{\rm{div}} a(x, u, Du) = b(x, u, Du) \quad \;{\rm{ in }}\; \Omega. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>The crucial point is that the operator <inline-formula><tex-math id="M3">\begin{document}$ a $\end{document}</tex-math></inline-formula> satisfies very weak regularity properties and a general growth, while the inhomogeneity <inline-formula><tex-math id="M4">\begin{document}$ b $\end{document}</tex-math></inline-formula> has a controllable growth.</p>

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