<p>We prove the higher differentiability of integer order of locally bounded minimizers of integral functionals of the form <Equation ID="Equ82"> <EquationSource Format="TEX">$\begin{aligned} \mathcal {F}(u,\Omega ):= \,\sum _{i=1}^{n} \dfrac{1}{p_i}\displaystyle \int _\Omega \, a_i(x) |u_{x_i} |^{p_i} dx- \int _\Omega \omega (x)u(x) dx, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mi mathvariant="script">F</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <mspace width="0.166667em" /> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mi>i</mi> </msub> </mfrac> </mstyle> <msub> <mo>∫</mo> <mi mathvariant="normal">Ω</mi> </msub> <mspace width="0.166667em" /> <msub> <mi>a</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">|</mo> <msub> <mi>u</mi> <msub> <mi>x</mi> <mi>i</mi> </msub> </msub> <mo stretchy="false">|</mo> </mrow> <msub> <mi>p</mi> <mi>i</mi> </msub> </msup> <mi>d</mi> <mi>x</mi> <mo>-</mo> <msub> <mo>∫</mo> <mi mathvariant="normal">Ω</mi> </msub> <mi>ω</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where the exponents <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ p_i \ge 2 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and the coefficients <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ a_i(x) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> satisfy a suitable Sobolev regularity. The main novelty consists in dealing with non-autonomous, anisotropic functionals, which depend also on the solution.</p>

Regularity for minimizers of degenerate, non-autonomous, orthotropic integral functionals

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2026年3月6日

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Antonio Giuseppe Grimaldi,
Stefania Russo

摘要

We prove the higher differentiability of integer order of locally bounded minimizers of integral functionals of the form $\begin{aligned} \mathcal {F}(u,\Omega ):= \,\sum _{i=1}^{n} \dfrac{1}{p_i}\displaystyle \int _\Omega \, a_i(x) |u_{x_i} |^{p_i} dx- \int _\Omega \omega (x)u(x) dx, \end{aligned}$ $\begin{matrix} F (u, Ω) : = \sum_{i = 1}^{n} \frac{1}{p_{i}} \int_{Ω} a_{i} (x) {| u_{x_{i}} |}^{p_{i}} d x - \int_{Ω} ω (x) u (x) d x, \end{matrix}$ where the exponents $ p_i \ge 2 $ $p_{i} \geq 2$ and the coefficients $ a_i(x) $ $a_{i} (x)$ satisfy a suitable Sobolev regularity. The main novelty consists in dealing with non-autonomous, anisotropic functionals, which depend also on the solution.