We prove the local Lipschitz regularity of the minimizers of functionals of the form <Equation ID="Equ58"> <EquationSource Format="TEX">$ \mathcal {I}(u)=\int _\Omega f(\nabla u(x))+g(x)u(x)\,dx\qquad u\in \phi +W^{1,1}_0(\Omega ) $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">I</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mi mathvariant="normal">Ω</mi> </msub> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <mi>d</mi> <mi>x</mi> <mspace width="2em" /> <mi>u</mi> <mo>∈</mo> <mi>ϕ</mi> <mo>+</mo> <msubsup> <mi>W</mi> <mn>0</mn> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </Equation>where g is bounded and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation> satisfies the Lower Bounded Slope Condition. The function f is assumed to be convex but not uniformly convex everywhere. As a byproduct, we also prove the existence of a locally Lipschitz minimizer for a class of functionals of the type above but allowing the function f to be nonconvex.

Local Lipschitz continuity of the minimizers of nonuniformly convex functionals under the Lower Bounded Slope Condition

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2026年4月24日

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Flavia Giannetti,
Giulia Treu

摘要

We prove the local Lipschitz regularity of the minimizers of functionals of the form $ \mathcal {I}(u)=\int _\Omega f(\nabla u(x))+g(x)u(x)\,dx\qquad u\in \phi +W^{1,1}_0(\Omega ) $ $I (u) = \int_{Ω} f (\nabla u (x)) + g (x) u (x) d x u \in ϕ + W_{0}^{1, 1} (Ω)$ where g is bounded and $\phi $ $ϕ$ satisfies the Lower Bounded Slope Condition. The function f is assumed to be convex but not uniformly convex everywhere. As a byproduct, we also prove the existence of a locally Lipschitz minimizer for a class of functionals of the type above but allowing the function f to be nonconvex.