scholarly journals Higher integrability for variational integrals with non-standard growth

Author(s):  
Mathias Schäffner

AbstractWe consider autonomous integral functionals of the form $$\begin{aligned} {\mathcal {F}}[u]:=\int _\varOmega f(D u)\,dx \quad \text{ where } u:\varOmega \rightarrow {\mathbb {R}}^N, N\ge 1, \end{aligned}$$ F [ u ] : = ∫ Ω f ( D u ) d x where u : Ω → R N , N ≥ 1 , where the convex integrand f satisfies controlled (p, q)-growth conditions. We establish higher gradient integrability and partial regularity for minimizers of $${\mathcal {F}}$$ F assuming $$\frac{q}{p}<1+\frac{2}{n-1}$$ q p < 1 + 2 n - 1 , $$n\ge 3$$ n ≥ 3 . This improves earlier results valid under the more restrictive assumption $$\frac{q}{p}<1+\frac{2}{n}$$ q p < 1 + 2 n .

2019 ◽  
Vol 12 (1) ◽  
pp. 85-110 ◽  
Author(s):  
Raffaella Giova ◽  
Antonia Passarelli di Napoli

AbstractWe prove the higher differentiability and the higher integrability of the a priori bounded local minimizers of integral functionals of the form\mathcal{F}(v,\Omega)=\int_{\Omega}f(x,Dv(x))\,{\mathrm{d}}x,with convex integrand satisfyingp-growth conditions with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to thex-variable belongs to a suitable Sobolev space. The a priori boundedness of the minimizers allows us to obtain the higher differentiability under a Sobolev assumption which is independent on the dimensionnand that, in the case{p\leq n-2}, improves previous known results. We also deal with solutions of elliptic systems with discontinuous coefficients under the so-called Uhlenbeck structure. In this case, it is well known that the solutions are locally bounded and therefore we obtain analogous regularity results without the a priori boundedness assumption.


2018 ◽  
Vol 177 ◽  
pp. 543-552 ◽  
Author(s):  
Petteri Harjulehto ◽  
Peter Hästö ◽  
Arttu Karppinen

1998 ◽  
Vol 3 (1-2) ◽  
pp. 41-64 ◽  
Author(s):  
Martin Fuchs ◽  
Li Gongbao

We consider the obstacle problem{minimize????????I(u)=?OG(?u)dx??among functions??u:O?Rsuch?that???????u|?O=0??and??u=F??a.e.for a given functionF?C2(O¯),F|?O<0and a bounded Lipschitz domainOinRn. The growth properties of the convex integrandGare described in terms of aN-functionA:[0,8)?[0,8)withlimt?8¯A(t)t-2<8. Ifn=3, we prove, under certain assumptions onG,C1,8-partial regularity for the solution to the above obstacle problem. For the special case whereA(t)=tln(1+t)we obtainC1,a-partial regularity whenn=4. One of the main features of the paper is that we do not require any power growth ofG.


Sign in / Sign up

Export Citation Format

Share Document