A relaxation result for autonomous integral functionals with discontinuous non-coercive integrand

Carlo Mariconda; Giulia Treu

Displaying similar documents to “A relaxation result for autonomous integral functionals with discontinuous non-coercive integrand”

Structure of approximate solutions of variational problems with extended-valued convex integrands

Alexander J. Zaslavski (2009)

ESAIM: Control, Optimisation and Calculus of Variations

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In this work we study the structure of approximate solutions of autonomous variational problems with a lower semicontinuous strictly convex integrand $f$ : $R^{n}$ $\times$ $R^{n}$ $\to$ $R^{1}$ $\cup$ ${\infty}$ , where $R^{n}$ is the $n$ -dimensional euclidean space. We obtain a full description of the structure of the approximate solutions which is independent of the length of the interval, for all sufficiently large intervals.

A note on propagation of singularities of semiconcave functions of two variables

Luděk Zajíček (2010)

Commentationes Mathematicae Universitatis Carolinae

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P. Albano and P. Cannarsa proved in 1999 that, under some applicable conditions, singularities of semiconcave functions in $ℝ^{n}$ propagate along Lipschitz arcs. Further regularity properties of these arcs were proved by P. Cannarsa and Y. Yu in 2009. We prove that, for $n = 2$ , these arcs are very regular: they can be found in the form (in a suitable Cartesian coordinate system) $ψ (x) = (x, y_{1} (x) - y_{2} (x))$ , $x \in [0, α]$ , where $y_{1}$ , $y_{2}$ are convex and Lipschitz on $[0, α]$ . In other words: singularities propagate along arcs with finite turn. ...

Vector integral equations with discontinuous right-hand side

Filippo Cammaroto, Paolo Cubiotti (1999)

Commentationes Mathematicae Universitatis Carolinae

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We deal with the integral equation $u (t) = f (\int_{I} g (t, z) u (z) d z)$ , with $t \in I = [0, 1]$ , $f : 𝐑^{n} \to 𝐑^{n}$ and $g : I \times I \to [0, + \infty [$ . We prove an existence theorem for solutions $u \in L^{\infty} (I, 𝐑^{n})$ where the function $f$ is not assumed to be continuous, extending a result previously obtained for the case $n = 1$ .

Local Lipschitz continuity of the stop operator

Wolfgang Desch (1998)

Applications of Mathematics

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On a closed convex set $Z$ in $ℝ^{N}$ with sufficiently smooth ( $𝒲^{2, \infty}$ ) boundary, the stop operator is locally Lipschitz continuous from $𝐖^{1, 1} ([0, T], ℝ^{N}) \times Z$ into $𝐖^{1, 1} ([0, T], ℝ^{N})$ . The smoothness of the boundary is essential: A counterexample shows that $𝒞^{1}$ -smoothness is not sufficient.

A remark on the local Lipschitz continuity of vector hysteresis operators

Pavel Krejčí (2001)

Applications of Mathematics

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It is known that the vector stop operator with a convex closed characteristic $Z$ of class $C^{1}$ is locally Lipschitz continuous in the space of absolutely continuous functions if the unit outward normal mapping $n$ is Lipschitz continuous on the boundary $\partial Z$ of $Z$ . We prove that in the regular case, this condition is also necessary.