Mathematics

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

1. C. De Filippis, Gradient bounds for solutions to irregular parabolic equations with $(p,q)$-growth. Preprint (2020), submitted.
2. C. De Filippis, L. Koch, J. Kristensen, Higher differentiability for minimizers of variational integrals under controlled growth hypothesis. Preprint (2020).
3. C. De Filippis, G. Mingione, On the regularity of minima of non-autonomous functionals. Journal of Geometric Analysis 30:1584-1626, (2020).