Regularity for general functionals with double phase

  • Paolo Baroni
  • Maria Colombo
  • Giuseppe MingioneEmail author


We prove sharp regularity results for a general class of functionals of the type
$$\begin{aligned} w \mapsto \int F(x, w, Dw) \, dx, \end{aligned}$$
featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral
$$\begin{aligned} w \mapsto \int b(x,w)(|Dw|^p+a(x)|Dw|^q) \, dx,\quad 1<p < q, \quad a(x)\ge 0, \end{aligned}$$
with \(0<\nu \le b(\cdot )\le L \). This changes its ellipticity rate according to the geometry of the level set \(\{a(x)=0\}\) of the modulating coefficient \(a(\cdot )\). We also present new methods and proofs that are suitable to build regularity theorems for larger classes of non-autonomous functionals. Finally, we disclose some new interpolation type effects that, as we conjecture, should draw a general phenomenon in the setting of non-uniformly elliptic problems. Such effects naturally connect with the Lavrentiev phenomenon.

Mathematics Subject Classification

49N60 35D10 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Paolo Baroni
    • 1
  • Maria Colombo
    • 2
  • Giuseppe Mingione
    • 1
    Email author
  1. 1.Dipartimento SMFIUniversità di ParmaParmaItaly
  2. 2.Institute for Theoretical StudiesETH ZürichZurichSwitzerland

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