Relaxation in BV of integrals with superlinear growth

Parth Soneji

ESAIM: Control, Optimisation and Calculus of Variations (2014)

  • Volume: 20, Issue: 4, page 1078-1122
  • ISSN: 1292-8119

Abstract

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We study properties of the functional ℱ loc ( u , Ω ) : = inf ( u j ) lim inf j → ∞ ∫ Ω f ( ∇ u j ) x ( u j ) ⊂ W loc 1 , r Ω , u j u in Ω , , F loc ( u,Ω ) : = inf ( u j ) lim inf j → ∞ ∫ Ω f ( ∇ u j ) d x , whereu ∈ BV(Ω;RN), and f:RN × n → R is continuous and satisfies 0 ≤ f(ξ) ≤ L(1 + | ξ | r). For r ∈ [1,2), assuming f has linear growth in certain rank-one directions, we combine a result of [A. Braides and A. Coscia, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 737–756] with a new technique involving mollification to prove an upper bound for Floc. Then, for r ∈ [ 1 , n n - 1 ) r ∈ [ 1 , n n − 1 ) , we prove that Flocsatisfies the lower bound loc ( u , Ω ) ≥ ∫ Ω f ( ∇ u ( x ) ) x + ∫ Ω D s u | D s u | | D s u | , F loc ( u,Ω ) ≥ ∫ Ω f ( ∇ u ( x ) ) d x + ∫ Ω f ∞ D s u | D s u | | D s u | , providedf is quasiconvex, and the recession function f∞(defined as f ∞ ( ξ ) : = lim ¯ t → ∞ f ( t ξ ) / t f∞(ξ):=limt→∞f(tξ)/t) is assumed to be finite in certain rank-one directions. The proof of this result involves adapting work by [Kristensen,Calc. Var. Partial Differ. Eqs. 7 (1998) 249–261], and [Ambrosio and Dal Maso, J. Funct. Anal. 109 (1992) 76–97], and applying a non-standard blow-up technique that exploits fine properties of BV maps. It also makes use of the fact that Floc has a measure representation, which is proved in the appendix using a method of [Fonseca and Malý, Annal. Inst. Henri Poincaré Anal. Non Linéaire 14 (1997) 309–338].

How to cite

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Soneji, Parth. "Relaxation in BV of integrals with superlinear growth." ESAIM: Control, Optimisation and Calculus of Variations 20.4 (2014): 1078-1122. <http://eudml.org/doc/272909>.

@article{Soneji2014,
abstract = {We study properties of the functional\begin\{eqnarray\} \mathcal \{F\}\_\{\{\rm loc\}\}(u,\Omega ):= \inf \_\{(u\_\{j\})\}\bigg \lbrace \liminf \_\{j\rightarrow \infty \}\int \_\{\Omega \}f(\nabla u\_\{j\})x\, \left| \!\!\begin\{array\}\{rl\} & (u\_\{j\})\subset W\_\{\{\rm loc\}\}^\{1,r\}\left(\Omega , \right) \\ & u\_\{j\}u\,\,\textrm \{in \}\left(\Omega , \right) \end\{array\} \right. \bigg \rbrace , \end\{eqnarray\}F loc ( u,Ω ) : = inf ( u j ) lim inf j → ∞ ∫ Ω f ( ∇ u j ) d x , whereu ∈ BV(Ω;RN), and f:RN × n → R is continuous and satisfies 0 ≤ f(ξ) ≤ L(1 + | ξ | r). For r ∈ [1,2), assuming f has linear growth in certain rank-one directions, we combine a result of [A. Braides and A. Coscia, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 737–756] with a new technique involving mollification to prove an upper bound for Floc. Then, for $r\in [1,\frac\{n\}\{n-1\})$ r ∈ [ 1 , n n − 1 ) , we prove that Flocsatisfies the lower bound\begin\{equation*\} \_\{\{\rm loc\}\}(u,\Omega ) \ge \int \_\{\Omega \} f(\nabla u (x))x + \int \_\{\Omega \}\bigg (\frac\{D^\{s\}u\}\{|D^\{s\}u|\}\bigg )\,|D^\{s\}u|, \end\{equation*\}F loc ( u,Ω ) ≥ ∫ Ω f ( ∇ u ( x ) ) d x + ∫ Ω f ∞ D s u | D s u | | D s u | , providedf is quasiconvex, and the recession function f∞(defined as $ f^\{\infty \}(\xi ):= \overline\{\lim \}_\{t\rightarrow \infty \}f(t\xi )/t$f∞(ξ):=limt→∞f(tξ)/t) is assumed to be finite in certain rank-one directions. The proof of this result involves adapting work by [Kristensen,Calc. Var. Partial Differ. Eqs. 7 (1998) 249–261], and [Ambrosio and Dal Maso, J. Funct. Anal. 109 (1992) 76–97], and applying a non-standard blow-up technique that exploits fine properties of BV maps. It also makes use of the fact that Floc has a measure representation, which is proved in the appendix using a method of [Fonseca and Malý, Annal. Inst. Henri Poincaré Anal. Non Linéaire 14 (1997) 309–338].},
author = {Soneji, Parth},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {quasiconvexity; lower semicontinuity; relaxation; BV; lower semi-continuity; variation integral; bounded variation; quasicovexity},
language = {eng},
number = {4},
pages = {1078-1122},
publisher = {EDP-Sciences},
title = {Relaxation in BV of integrals with superlinear growth},
url = {http://eudml.org/doc/272909},
volume = {20},
year = {2014},
}

TY - JOUR
AU - Soneji, Parth
TI - Relaxation in BV of integrals with superlinear growth
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2014
PB - EDP-Sciences
VL - 20
IS - 4
SP - 1078
EP - 1122
AB - We study properties of the functional\begin{eqnarray} \mathcal {F}_{{\rm loc}}(u,\Omega ):= \inf _{(u_{j})}\bigg \lbrace \liminf _{j\rightarrow \infty }\int _{\Omega }f(\nabla u_{j})x\, \left| \!\!\begin{array}{rl} & (u_{j})\subset W_{{\rm loc}}^{1,r}\left(\Omega , \right) \\ & u_{j}u\,\,\textrm {in }\left(\Omega , \right) \end{array} \right. \bigg \rbrace , \end{eqnarray}F loc ( u,Ω ) : = inf ( u j ) lim inf j → ∞ ∫ Ω f ( ∇ u j ) d x , whereu ∈ BV(Ω;RN), and f:RN × n → R is continuous and satisfies 0 ≤ f(ξ) ≤ L(1 + | ξ | r). For r ∈ [1,2), assuming f has linear growth in certain rank-one directions, we combine a result of [A. Braides and A. Coscia, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 737–756] with a new technique involving mollification to prove an upper bound for Floc. Then, for $r\in [1,\frac{n}{n-1})$ r ∈ [ 1 , n n − 1 ) , we prove that Flocsatisfies the lower bound\begin{equation*} _{{\rm loc}}(u,\Omega ) \ge \int _{\Omega } f(\nabla u (x))x + \int _{\Omega }\bigg (\frac{D^{s}u}{|D^{s}u|}\bigg )\,|D^{s}u|, \end{equation*}F loc ( u,Ω ) ≥ ∫ Ω f ( ∇ u ( x ) ) d x + ∫ Ω f ∞ D s u | D s u | | D s u | , providedf is quasiconvex, and the recession function f∞(defined as $ f^{\infty }(\xi ):= \overline{\lim }_{t\rightarrow \infty }f(t\xi )/t$f∞(ξ):=limt→∞f(tξ)/t) is assumed to be finite in certain rank-one directions. The proof of this result involves adapting work by [Kristensen,Calc. Var. Partial Differ. Eqs. 7 (1998) 249–261], and [Ambrosio and Dal Maso, J. Funct. Anal. 109 (1992) 76–97], and applying a non-standard blow-up technique that exploits fine properties of BV maps. It also makes use of the fact that Floc has a measure representation, which is proved in the appendix using a method of [Fonseca and Malý, Annal. Inst. Henri Poincaré Anal. Non Linéaire 14 (1997) 309–338].
LA - eng
KW - quasiconvexity; lower semicontinuity; relaxation; BV; lower semi-continuity; variation integral; bounded variation; quasicovexity
UR - http://eudml.org/doc/272909
ER -

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