Clarke and Limiting Subdifferentials of Integral Functionals

Abstract

<jats:p> Some comparisons (equality, strong density or complete characterizations) are made between the Clarke and limiting subdifferentials for the class of integral functionals defined on reflexive Lebesgue spaces <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">L_p</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , without any locally Lipschitzian assumptions on these functionals. Let <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">I_f</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> be the integral functional associated with a normal integrand <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">f:\Omega\times E\to {\mathbb{R}}\cup\{\pm\infty\}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo>×</mml:mo> <mml:mi>E</mml:mi> <mml:mo>→</mml:mo> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mo>∪</mml:mo> <mml:mo stretchy="false">{</mml:mo> <mml:mo>±</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo stretchy="false">}</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , where <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">E</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is a finite dimensional space and <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\Omega</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is endowed with an atomless <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\sigma</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -finite measure. At a given point <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">x</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> such that almost everywhere the Clarke subdifferential of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">f_{\omega}</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>ω</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> at <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">x(\omega)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> coincide with the closed convex hull of the limiting subdifferential and such that the limiting subdifferential multifunction at <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">x</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\omega\rightrightarrows \partial^{L}f_{\omega}(x(\omega))</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>⇉</mml:mo> <mml:msup> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi>L</mml:mi> </mml:msup> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>ω</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , admits a <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">q=p(p-1)^{-1}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> integrable selection, we show that the limiting subdifferential of the integral functional is strongly dense in the Clarke subdifferential. This property allows us to give an exact computation of the Clarke subdifferential by a legitimate disintegration formula. As a consequence, when for almost every <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\omega</jats:tex-math> <mml:math xmlns:mml="http://w