Local Lipschitz regularity for degenerate elliptic systems

By Frank Duzaar, Giuseppe Mingione. 2009

105 citations recorded by the source corpus.

We start presenting an $L^{\infty}$-gradient bound for solutions to non-homogeneous $p$-Laplacean type systems and equations, via suitable non-linear potentials of the right hand side. Such a bound implies a Lorentz space characterization of Lipschitz regularity of solutions which surprisingly turns out to be independent of $p$, and that reveals to be the same classical one for the standard Laplacean operator. In turn, the a priori estimates derived imply the existence of locally Lipschitz regular solutions to certain degenerate systems with critical growth of the type arising when considering geometric analysis problems, as recently emphasized by Rivi\`ere

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8a745515731e6ef013d67a762a54a537b23fac19

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In Applied Mathematics

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Citations in the corpus

Gradient estimates via linear and nonlinear potentials (2010, background): “Here we see that we may reduce to the nondegenerate case s > 0 via an approximation that the reader may for instance find in [12,10].”
A nonlinear Stein theorem (2014, background): “γ(·) is a constant function) as derived in [3,6], but the known techniques do not apparently extend to treat the case when coefficients are involved, at least in the case they are not differentiable, as here.”
Global Lipschitz Regularity for a Class of Quasilinear Elliptic Equations (2010, background): “Duzaar were about to submit a new paper [ 19 ] dealing with local estimates for the gradient of local solutions to nonlinear elliptic equations and systems (from a more general class than that considered here).”
Potential estimates and gradient boundedness for nonlinear parabolic systems (2012, methodology): “Possible extensions and refinements The techniques demonstrated in the previous sections allow for a rather large number of refinements and extensions, especially when keeping in mind the results valid in the elliptic case [14].”
Linear Potentials in Nonlinear Potential Theory (2013, background): “6 strikingly extends the known results available in the literature where the condition μ ∈ L(n, 1) is found to be a sufficient one for gradient boundedness of solutions [4,12].”