<p>We establish that locally bounded relaxed minimizers of degenerate elliptic symmetric gradient functionals on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textrm{BD}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>BD</mtext> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> have weak gradients in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textrm{L}_{\textrm{loc}}^{1}(\Omega ;\mathbb {R}^{n\times n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mtext>L</mtext> <mrow> <mtext>loc</mtext> </mrow> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>;</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. This is achieved for the sharp ellipticity range that is presently known to yield <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textrm{W}_{\textrm{loc}}^{1,1}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mtext>W</mtext> <mrow> <mtext>loc</mtext> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msubsup> </math></EquationSource> </InlineEquation>-regularity in the full gradient case on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\textrm{BV}(\Omega ;\mathbb {R}^{n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>BV</mtext> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>;</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. As a consequence, we also obtain the first Sobolev regularity results for minimizers of the area-type functional on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\textrm{BD}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>BD</mtext> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

Gradient Integrability for Bounded $\textrm{BD}$-Minimizers

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2026年5月11日

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Lisa Beck,
Ferdinand Eitler,
Franz Gmeineder

摘要

We establish that locally bounded relaxed minimizers of degenerate elliptic symmetric gradient functionals on $\textrm{BD}(\Omega )$ $BD (Ω)$ have weak gradients in $\textrm{L}_{\textrm{loc}}^{1}(\Omega ;\mathbb {R}^{n\times n})$ $L_{loc}^{1} (Ω; R^{n \times n})$ . This is achieved for the sharp ellipticity range that is presently known to yield $\textrm{W}_{\textrm{loc}}^{1,1}$ $W_{loc}^{1, 1}$ -regularity in the full gradient case on $\textrm{BV}(\Omega ;\mathbb {R}^{n})$ $BV (Ω; R^{n})$ . As a consequence, we also obtain the first Sobolev regularity results for minimizers of the area-type functional on $\textrm{BD}(\Omega )$ $BD (Ω)$ .

Gradient Integrability for Bounded \(\textrm{BD}\)-Minimizers

摘要