The quasi-minimizer of integral functionals with m(x) growth conditions

2000 ◽  
Vol 39 (7) ◽  
pp. 807-816 ◽  
Author(s):  
Xianling Fan ◽  
Dun Zhao
Keyword(s):  
Growth Conditions ◽  
Integral Functionals

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

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
Mathias Schäffner
Keyword(s):  
Partial Regularity ◽  
Growth Conditions ◽  
Integral Functionals ◽  
Restrictive Assumption ◽  
Higher Integrability ◽  
Variational Integrals

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 .

scholarly journals Growth conditions and regularity, an optimal local boundedness result

2020 ◽  
pp. 2050029
Author(s):  
Jonas Hirsch ◽  
Mathias Schäffner
Keyword(s):  
Growth Conditions ◽  
Integral Functionals ◽  
Local Minimizers ◽  
Local Boundedness ◽  
Scalar Integral ◽  
Boundedness Result

We prove local boundedness of local minimizers of scalar integral functionals [Formula: see text], [Formula: see text] where the integrand satisfies [Formula: see text]-growth of the form [Formula: see text] under the optimal relation [Formula: see text].

Regularity results for a priori bounded minimizers of non-autonomous functionals with discontinuous coefficients

2019 ◽  
Vol 12 (1) ◽  
pp. 85-110 ◽  
Author(s):  
Raffaella Giova ◽  
Antonia Passarelli di Napoli
Keyword(s):  
Sobolev Space ◽  
A Priori ◽  
Elliptic Systems ◽  
Discontinuous Coefficients ◽  
Growth Conditions ◽  
Integral Functionals ◽  
Local Minimizers ◽  
Higher Integrability ◽  
Regularity Results ◽  
Higher Differentiability

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.

Sign in / Sign up

Export Citation Format

Share Document