Quasiconvexity, growth conditions and partial regularity

1988 ◽  
pp. 211-237 ◽  
Author(s):  
Mariano Giaquinta
Keyword(s):  
Partial Regularity ◽  
Growth Conditions

scholarly journals Variational inequalities for energy functionals with nonstandard growth conditions

1998 ◽  
Vol 3 (1-2) ◽  
pp. 41-64 ◽  
Author(s):  
Martin Fuchs ◽  
Li Gongbao
Keyword(s):  
Variational Inequalities ◽  
Obstacle Problem ◽  
Partial Regularity ◽  
Growth Conditions ◽  
Energy Functionals ◽  
Nonstandard Growth ◽  
Bounded Lipschitz Domain ◽  
Growth Properties ◽  
Nonstandard Growth Conditions ◽  
Special Case

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.

Partial regularity results for non-autonomous functionals with $$\varPhi $$-growth conditions

2017 ◽  
Vol 196 (6) ◽  
pp. 2147-2165 ◽  
Author(s):  
Flavia Giannetti ◽  
Antonia Passarelli di Napoli ◽  
Atsushi Tachikawa
Keyword(s):  
Partial Regularity ◽  
Growth Conditions ◽  
Regularity Results

Partial regularity of strong local minimizers of quasiconvex integrals with (p, q)-growth

2009 ◽  
Vol 139 (3) ◽  
pp. 595-621 ◽  
Author(s):  
Sabine Schemm ◽  
Thomas Schmidt
Keyword(s):  
Calculus Of Variations ◽  
Partial Regularity ◽  
Growth Conditions ◽  
Local Minimizers ◽  
Image Position

We consider strictly quasiconvex integralsin the multi-dimensional calculus of variations. For the C2-integrand f : ℝNn → ℝ we impose (p, q)-growth conditionswith γ, Γ > 0 and 1 < p ≤ q < min {p + 1/n, p(2n − 1)/(2n − 2)}. Under these assumptions we prove partial C1, αloc-regularity for strong local minimizers of F and the associated relaxed functional F.

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 Partial Regularity for Nonlinear Subelliptic Systems with Dini Continuous Coefficients in Heisenberg Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Jialin Wang ◽  
Pingzhou Hong ◽  
Dongni Liao ◽  
Zefeng Yu
Keyword(s):  
Weak Solution ◽  
Heisenberg Group ◽  
Partial Regularity ◽  
Harmonic Approximation ◽  
Growth Conditions ◽  
Heisenberg Groups ◽  
Hölder Exponent ◽  
Hölder Continuous ◽  
Horizontal Gradients ◽  
Controllable Growth

This paper is concerned with partial regularity to nonlinear subelliptic systems with Dini continuous coefficients under quadratic controllable growth conditions in the Heisenberg groupℍn. Based on a generalization of the technique of𝒜-harmonic approximation introduced by Duzaar and Steffen, partial regularity to the sub-elliptic system is established in the Heisenberg group. Our result is optimal in the sense that in the case of Hölder continuous coefficients we establish the optimal Hölder exponent for the horizontal gradients of the weak solution on its regular set.

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