Calculus of Variations and Geometric Measure Theory
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R. De Arcangelis - C. Trombetti

On the Lavrentieff Phenomenon for Some Classes of Dirichlet Minimum Problems

created on 26 Dec 2001


Published Paper

Inserted: 26 dec 2001

Journal: Journal of Convex Analysis
Volume: 7
Number: 2
Pages: 271-297
Year: 2000


Starting from the results of DE ARCANGELIS R., TROMBETTI C.: {\it On the Relaxation of Some Classes of Dirichlet Minimum Problems}; Comm. Partial Differential Equations *24*, (1999), 975-1006, the Lavrentieff phenomenon for the functional $F(\Omega,\varphi_O,\cdot)\colon u\in BV(\Omega)\mapsto \int_\Omega f(\nabla u)dx + \int_\Omega f^\infty({dD^su\over d
+\int_{\partial\Omega} f^\infty((\varphi_0-\gamma_\Omega (u))*n*) d{\cal H}^{n-1}$ between $BV(\Omega)$ and $BV(\Omega)\cap C^1(\Omega)$ is studied, where $f\colon *R*^n\to [0,+\infty[$ is convex, $f^\infty$ is its recession function, $\varphi_0\in L^1(\partial\Omega)$, and $\gamma_\Omega$ is the trace operator on $\partial\Omega$. The occurrence of the phenomenon is first discussed by means of an example, and then completely characterized. Sufficient conditions implying the absence of the phenomenon are also proved, and some relaxation properties of $F(\Omega,\varphi_O,\cdot)$ are also established.

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