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.

A C1,α$C^{1,\alpha}$ partial regularity result for integral functionals with p⁢(x)$p(x)$-growth condition

2016 ◽  
Vol 9 (4) ◽  
pp. 395-407 ◽  
Author(s):  
Flavia Giannetti
Keyword(s):  
Growth Condition ◽  
Partial Regularity ◽  
Regularity Result ◽  
Integral Functionals ◽  
Local Minimizers ◽  
Zygmund Class ◽  
Partial Regularity Result

AbstractWe establish${C^{1,\alpha}}$partial regularity for the local minimizers of integral functionals of the type$\mathcal{F}(u;\Omega):=\int_{\Omega}(1+|Du|^{2})^{\frac{p(x)}{2}}\,dx,$where the gradient of the exponent function${p(\,\cdot\,)\geq 2}$belongs to a suitable Orlicz–Zygmund class.

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