Given <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p\in [1,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we provide sufficient and necessary conditions on the non-negative measurable kernels <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(\rho _t)_{t\in (0,1)}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mi>t</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> ensuring convergence of the associated Bourgain–Brezis–Mironescu (BBM) energies <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(\mathscr {F}_{t,p})_{t\in (0,1)}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">F</mi> <mrow> <mi>t</mi> <mo>,</mo> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> to a variant of the p-Dirichlet energy on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {R}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$t\rightarrow 0^+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> both in the pointwise and in the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-sense. We also devise sufficient conditions on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(\rho _t)_{t\in (0,1)}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mi>t</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> yielding local compactness in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^p(\mathbb {R}^N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of sequences with bounded BBM energy. Moreover, we give sufficient conditions on <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$(\rho _t)_{t\in (0,1)}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mi>t</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> implying pointwise and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-convergence and equicoercivity of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$({\mathscr {F}}_{t,p})_{t\in (0,1)}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">F</mi> <mrow> <mi>t</mi> <mo>,</mo> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> when the limit p-energy is of non-local type. Finally, we apply our results to provide asymptotic formulas in the pointwise and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-sense for heat content-type energies both in the local and non-local settings.

Sharp Conditions for the BBM Formula and Asymptotics of Heat Content-Type Energies

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2026年1月7日

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Luca Gennaioli,
Giorgio Stefani

摘要

Given $p\in [1,\infty )$ $p \in [1, \infty)$ , we provide sufficient and necessary conditions on the non-negative measurable kernels $(\rho _t)_{t\in (0,1)}$ ${(ρ_{t})}_{t \in (0, 1)}$ ensuring convergence of the associated Bourgain–Brezis–Mironescu (BBM) energies $(\mathscr {F}_{t,p})_{t\in (0,1)}$ ${(F_{t, p})}_{t \in (0, 1)}$ to a variant of the p-Dirichlet energy on $\mathbb {R}^N$ $R^{N}$ as $t\rightarrow 0^+$ $t \to 0^{+}$ both in the pointwise and in the $\Gamma $ $Γ$ -sense. We also devise sufficient conditions on $(\rho _t)_{t\in (0,1)}$ ${(ρ_{t})}_{t \in (0, 1)}$ yielding local compactness in $L^p(\mathbb {R}^N)$ $L^{p} (R^{N})$ of sequences with bounded BBM energy. Moreover, we give sufficient conditions on $(\rho _t)_{t\in (0,1)}$ ${(ρ_{t})}_{t \in (0, 1)}$ implying pointwise and $\Gamma $ $Γ$ -convergence and equicoercivity of $({\mathscr {F}}_{t,p})_{t\in (0,1)}$ ${(F_{t, p})}_{t \in (0, 1)}$ when the limit p-energy is of non-local type. Finally, we apply our results to provide asymptotic formulas in the pointwise and $\Gamma $ $Γ$ -sense for heat content-type energies both in the local and non-local settings.