Research Group:
Speaker:
Cristiana De Filippis
Institution:
University of Oxford
Schedule:
Friday, April 24, 2020 - 16:30
Location:
Online (sign in to get the link)
Abstract:
Variational integrals of $p$-Laplacean type like
\[
W^{1,p}_{\operatorname{loc}}(\Omega,\mathbb{R}^{N})\ni w\mapsto \int_{\Omega}F(x,w,Dw)\ \,{\rm d}x,
\]
where
\[
F(x,v,z)\sim {\lvert {z}\rvert}^{p}\qquad 1<p<\infty
\]
play a prominent role in the regularity theory for elliptic or parabolic problems. However, there are several models in use in materials science or in continuum mechanics such as
\[
\mathcal{A}(w,\Omega):=\int_{\Omega}\left[{\lvert {Dw}\rvert}^{p}+\sum_{i=1}^{n}{\lvert {D_{i}w}\rvert}^{p_{i}}\right] \ \,{\rm d}x
\]
or
\[
\mathcal{P}(w,\Omega):=\int_{\Omega}\left[{\lvert {Dw}\rvert}^{p}+a(x){\lvert {Dw}\rvert}^{q}\right] \ \,{\rm d}x
\]
which cannot be included in the $p$-Laplacean framework, but certainly fall into the realm of functionals with $(p,q)$-growth:
\[
W^{1,p}_{\operatorname{loc}}(\Omega,\mathbb{R}^{N})\ni w\mapsto \int_{\Omega}G(x,w,Dw) \ \,{\rm d}x,
\]
with
\[
{\lvert {z}\rvert}^{p}\lesssim G(x,v,z)\lesssim 1+{\lvert {z}\rvert}^{q}\qquad 1<p\le q<\infty,
\]
which exhibit a structure flexible enough to cover models $\mathcal{A}(\cdot)$-$\mathcal{P}(\cdot)$ and several other examples. In this talk I will review the basic literature available for $(p,q)$ variational problems and present some new results both in the elliptic and in the parabolic setting [1,2,3].
References
References
- C. De Filippis, Gradient bounds for solutions to irregular parabolic equations with $(p,q)$-growth. Preprint (2020), submitted.
- C. De Filippis, L. Koch, J. Kristensen, Higher differentiability for minimizers of variational integrals under controlled growth hypothesis. Preprint (2020).
- C. De Filippis, G. Mingione, On the regularity of minima of non-autonomous functionals. Journal of Geometric Analysis 30:1584-1626, (2020).