On Hölder continuity of solutions for a class of nonlinear elliptic systems with $$p$$ p -growth via weighted integral techniques
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| 024 | 7 | 0 | |a 10.1007/s10231-014-0410-x |2 doi |
| 035 | |a (NATIONALLICENCE)springer-10.1007/s10231-014-0410-x | ||
| 245 | 0 | 0 | |a On Hölder continuity of solutions for a class of nonlinear elliptic systems with $$p$$ p -growth via weighted integral techniques |h [Elektronische Daten] |c [Miroslav Bulíček, Jens Frehse, Mark Steinhauer] |
| 520 | 3 | |a 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. | |
| 540 | |a Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2014 | ||
| 690 | 7 | |a Nonlinear elliptic systems |2 nationallicence | |
| 690 | 7 | |a Regularity |2 nationallicence | |
| 690 | 7 | |a Noether equation |2 nationallicence | |
| 690 | 7 | |a Hölder continuity |2 nationallicence | |
| 690 | 7 | |a Liouville theorem |2 nationallicence | |
| 700 | 1 | |a Bulíček |D Miroslav |u Faculty of Mathematics and Physics, Mathematical Institute, Charles University Sokolovská 83, 18675, Praha 8, Czech Republic |4 aut | |
| 700 | 1 | |a Frehse |D Jens |u Department of applied analysis, Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany |4 aut | |
| 700 | 1 | |a Steinhauer |D Mark |u Mathematical Institute, University of Koblenz-Landau, Campus Koblenz Universitstr. 1, 56070, Koblenz, Germany |4 aut | |
| 773 | 0 | |t Annali di Matematica Pura ed Applicata (1923 -) |d Springer Berlin Heidelberg |g 194/4(2015-08-01), 1025-1069 |x 0373-3114 |q 194:4<1025 |1 2015 |2 194 |o 10231 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s10231-014-0410-x |q text/html |z Onlinezugriff via DOI |
| 898 | |a BK010053 |b XK010053 |c XK010000 | ||
| 900 | 7 | |a Metadata rights reserved |b Springer special CC-BY-NC licence |2 nationallicence | |
| 908 | |D 1 |a research-article |2 jats | ||
| 949 | |B NATIONALLICENCE |F NATIONALLICENCE |b NL-springer | ||
| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s10231-014-0410-x |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Bulíček |D Miroslav |u Faculty of Mathematics and Physics, Mathematical Institute, Charles University Sokolovská 83, 18675, Praha 8, Czech Republic |4 aut | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Frehse |D Jens |u Department of applied analysis, Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany |4 aut | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Steinhauer |D Mark |u Mathematical Institute, University of Koblenz-Landau, Campus Koblenz Universitstr. 1, 56070, Koblenz, Germany |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Annali di Matematica Pura ed Applicata (1923 -) |d Springer Berlin Heidelberg |g 194/4(2015-08-01), 1025-1069 |x 0373-3114 |q 194:4<1025 |1 2015 |2 194 |o 10231 | ||