We provide necessary and sufficient conditions on the density <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$W:\mathbb {R}^d\times \mathbb {R}^d\rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <mo>:</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> in order to ensure the sequential weak<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$^*$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mrow /> <mo>∗</mo> </mmultiscripts> </math></EquationSource> </InlineEquation> lower semicontinuity of the functional <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$J: W^{1,\infty }(I;\mathbb {R}^d)\rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>J</mi> <mo>:</mo> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>∞</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>I</mi> <mo>;</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, defined as <Equation ID="Equ14"> <EquationSource Format="TEX">$\begin{aligned} J(u):=\underset{I\times I}{\mathrm {ess\,sup}}\,W(u'(x), u'(y)), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>J</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <munder> <mrow> <mi mathvariant="normal">ess</mi> <mspace width="0.166667em" /> <mi mathvariant="normal">sup</mi> </mrow> <mrow> <mi>I</mi> <mo>×</mo> <mi>I</mi> </mrow> </munder> <mspace width="0.166667em" /> <mi>W</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>u</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msup> <mi>u</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>when I is an open and bounded interval of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation>. We also show that, when <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$d=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the lower semicontinuous envelope of I in general can be obtained by replacing W by its separately level convex envelope.

A Note on the Relaxation of One-Dimensional Nonlocal Supremal Functionals in the Sobolev Setting

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发表日期：

2025年4月10日

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Andrea Torricelli

摘要

We provide necessary and sufficient conditions on the density $W:\mathbb {R}^d\times \mathbb {R}^d\rightarrow \mathbb {R}$ $W : R^{d} \times R^{d} \to R$ in order to ensure the sequential weak $^*$ $^{*}$ lower semicontinuity of the functional $J: W^{1,\infty }(I;\mathbb {R}^d)\rightarrow \mathbb {R}$ $J : W^{1, \infty} (I; R^{d}) \to R$ , defined as $\begin{aligned} J(u):=\underset{I\times I}{\mathrm {ess\,sup}}\,W(u'(x), u'(y)), \end{aligned}$ $\begin{matrix} J (u) : = \underset{I \times I}{ess \sup} W (u^{'} (x), u^{'} (y)), \end{matrix}$ when I is an open and bounded interval of $\mathbb {R}$ $R$ . We also show that, when $d=1$ $d = 1$ , the lower semicontinuous envelope of I in general can be obtained by replacing W by its separately level convex envelope.