Regularity for general functionals with double phase
TL;DR: In this article, sharp regularity results for a general class of functionals with non-standard growth conditions and non-uniform ellipticity properties were proved for the double phase integral model.
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Abstract: 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.
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Citations
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