caa a22 4500 605496323 CHVBK 20210128100535.0 cr unu---uuuuu 210128e20150801xx s 000 0 eng 10.1007/s10231-014-0410-x doi (NATIONALLICENCE)springer-10.1007/s10231-014-0410-x On Hölder continuity of solutions for a class of nonlinear elliptic systems with $$p$$ p -growth via weighted integral techniques [Elektronische Daten] [Miroslav Bulíček, Jens Frehse, Mark Steinhauer] We consider weak solutions of nonlinear elliptic systems in a $$W^{1,p}$$ W 1 , p -setting which arise as Euler-Lagrange equations of certain variational integrals with pollution term, and we also consider minimizers of a variational problem. The solutions are assumed to be stationary in the sense that the differential of the variational integral vanishes with respect to variations of the independent and the dependent variables. We impose new structural conditions on the nonlinearities which yield $$\fancyscript{C}^{\alpha }$$ C α -regularity and $$\fancyscript{C}^{\alpha }$$ C α -estimates for the solutions. These structure conditions cover variational integrals like $$\int F(\nabla u)\,\mathrm{d}x $$ ∫ F ( ∇ u ) d x with potential $$F(\nabla u):=\tilde{F} (Q_1(\nabla u),\ldots , Q_N(\nabla u))$$ F ( ∇ u ) : = F ~ ( Q 1 ( ∇ u ) , ... , Q N ( ∇ u ) ) and positive definite quadratic forms $$Q_i$$ Q i in $$\nabla u$$ ∇ u defined as $$Q_i(\nabla u)=\sum \nolimits _{\alpha \beta } a_i^{\alpha \beta } \nabla u^\alpha \cdot \nabla u^\beta $$ Q i ( ∇ u ) = ∑ α β a i α β ∇ u α · ∇ u β . A simple example consists in $${\tilde{F}}(\xi _1,\xi _2):= |\xi _1|^{\frac{p}{2}} + |\xi _2|^{\frac{p}{2}}$$ F ~ ( ξ 1 , ξ 2 ) : = | ξ 1 | p 2 + | ξ 2 | p 2 or $$\tilde{F}(\xi _1,\xi _2):= |\xi _1|^{\frac{p}{4}}|\xi _2|^{\frac{p}{4}}.$$ F ~ ( ξ 1 , ξ 2 ) : = | ξ 1 | p 4 | ξ 2 | p 4 . Since the quadratic forms $$Q_i$$ Q i need not to be linearly dependent, our result covers a class of nondiagonal, possibly nonmonotone elliptic systems. As a by-product, we also prove a kind of Liouville theorem. As a new analytical tool, we use a weighted integral technique with singular weights in an $$L^p$$ L p -setting for the proof and establish a weighted hole-filling inequality in a setting where Green-function techniques are not available. Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2014 Nonlinear elliptic systems nationallicence Regularity nationallicence Noether equation nationallicence Hölder continuity nationallicence Liouville theorem nationallicence Bulíček Miroslav Faculty of Mathematics and Physics, Mathematical Institute, Charles University Sokolovská 83, 18675, Praha 8, Czech Republic aut Frehse Jens Department of applied analysis, Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany aut Steinhauer Mark Mathematical Institute, University of Koblenz-Landau, Campus Koblenz Universitstr. 1, 56070, Koblenz, Germany aut Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/4(2015-08-01), 1025-1069 0373-3114 194:4<1025 2015 194 10231 https://doi.org/10.1007/s10231-014-0410-x text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10231-014-0410-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Bulíček Miroslav Faculty of Mathematics and Physics, Mathematical Institute, Charles University Sokolovská 83, 18675, Praha 8, Czech Republic aut NATIONALLICENCE 700 1- Frehse Jens Department of applied analysis, Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany aut NATIONALLICENCE 700 1- Steinhauer Mark Mathematical Institute, University of Koblenz-Landau, Campus Koblenz Universitstr. 1, 56070, Koblenz, Germany aut NATIONALLICENCE 773 0- Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/4(2015-08-01), 1025-1069 0373-3114 194:4<1025 2015 194 10231