Homogenization of Periodic Finsler Metrics

Abstract

<jats:p> We prove an homogenization result in W¹,¹ and in BV for a sequence <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">(F_\varepsilon)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> of functionals of the form <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">F_\varepsilon(u) = \int_0^1 f(u/\varepsilon,u')\,dt</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mn>0</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mi mathvariant="normal">/</mml:mi> <mml:mi>ε</mml:mi> <mml:mo separator="true">,</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mo mathvariant="normal" lspace="0em" rspace="0em">′</mml:mo> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> where ε is a positive parameter which tends to zero, f : ℝⁿ × ℝⁿ → [0, +∞) is [0, 1)ⁿ-periodic in the first variable, convex in the second variable and satisfies a suitable growth condition of order one. Under the additional assumption that f(x, ·) is positively 1-homogeneous, we show how our result is equivalent to the analogous homogenization result (dealt with by Acerbi and Buttazzo) in which growth conditions of order p > 1 are considered. </jats:p>