Minimizers of non convex scalar functionals and viscosity solutions of Hamilton-Jacobi equations

Zagatti, Sandro

doi:10.1007/s00526-007-0124-7

Cited by 12 publications

(9 citation statements)

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“…In section 7 we give a stronger result which holds true whenever the monotonicity expressed in item (i) above is strict: in such case we see that any minimizer of the relaxed functional minimizes the non-convex one too. The last section 8 is devoted to the comparison with integro extremality method which we have introduced in [7,8,9] and which is the key tool adopted by the authors in [6]. We stress that our result subsumes the existence theorem provided in the quoted paper, which is restricted to the scalar case m = 1.…”

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confidence: 74%

“…Relation with Integro-Extremality method. In previous section we have used a refined version of the integro-extremization method that we have used, for example, in [8] and in [9]. We refer to [10] for a more accurate discussion.…”

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confidence: 99%

“…In addition, it is easy to prove that S is compact in the strong L 1 (I, R)-topology (see for example [8] for details) and then there exists v, v ∈ S such that…”

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Existence of minimizers for one-dimensional vectorial non-semicontinuous functionals with second order lagrangian

Zagatti¹

2022

DCDS

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We study the minimum problem for functionals of the form<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \mathcal{F}(u) = \int_{I} f(x, u(x), u^ \prime(x), u^ {\prime\prime}(x))\,dx, \end{equation} $\end{document} </tex-math></disp-formula>where the integrand <inline-formula><tex-math id="M1">\begin{document}$ f:I\times \mathbb{R}^m\times \mathbb{R}^m\times \mathbb{R}^m \to \mathbb{R} $\end{document}</tex-math></inline-formula> is not convex in the last variable. We provide an existence result assuming that the lower convex envelope <inline-formula><tex-math id="M2">\begin{document}$ \overline{f} = \overline{f}(x,p,q,\xi) $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M3">\begin{document}$ f $\end{document}</tex-math></inline-formula> with respect to <inline-formula><tex-math id="M4">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> is regular and enjoys a special dependence with respect to the i-th single components <inline-formula><tex-math id="M5">\begin{document}$ p_i, q_i, \xi_i $\end{document}</tex-math></inline-formula> of the vector variables <inline-formula><tex-math id="M6">\begin{document}$ p,q,\xi $\end{document}</tex-math></inline-formula>. More precisely, we assume that it is monotone in <inline-formula><tex-math id="M7">\begin{document}$ p_i $\end{document}</tex-math></inline-formula> and that it satisfies suitable affinity properties with respect to <inline-formula><tex-math id="M8">\begin{document}$ \xi_i $\end{document}</tex-math></inline-formula> on the set <inline-formula><tex-math id="M9">\begin{document}$ \{f> \overline{f}\} $\end{document}</tex-math></inline-formula> and with respect to <inline-formula><tex-math id="M10">\begin{document}$ q_i $\end{document}</tex-math></inline-formula> on the whole domain. We adopt refined versions of the integro-extremality method, extending analogous results already obtained for functionals with first order lagrangians. In addition we show that our hypotheses are nearly optimal, providing in such a way an almost necessary and sufficient condition for the solvability of this class of variational problems.

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Existence of minimizers for one-dimensional vectorial non-semicontinuous functionals with second order lagrangian

Zagatti¹

2022

DCDS

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“…In such situation we may apply the integro-extremality method introduced for scalar functional, along the lines that the reader can find, for example, in [19]. In this case we do not need strong approximate continuity (iii), but classical differentiability almost everywhere, as it happens in the scalar case (see [18,19]).…”

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confidence: 99%

On the minimum problem for non-quasiconvex vectorial functionals

Zagatti

2022

Nonlinear Differ. Equ. Appl.

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We consider functionals of the form $$\begin{aligned} {\mathcal {F}}(u) = \int _{\Omega } f(x, u(x), D u(x))\,dx, \quad u\in u_0 + W_0^{1,r}(\Omega ,{\mathbb {R}}^m), \end{aligned}$$ F ( u ) = ∫ Ω f ( x , u ( x ) , D u ( x ) ) d x , u ∈ u 0 + W 0 1 , r ( Ω , R m ) , where the integrand $$f:\Omega \times {\mathbb {R}}^m\times {\mathbb {M}}^{m\times n} \rightarrow {\mathbb {R}}$$ f : Ω × R m × M m × n → R is assumed to be non-quasiconvex in the last variable and $$u_0 \in W^{1,r}(\Omega ,{\mathbb {R}}^m)$$ u 0 ∈ W 1 , r ( Ω , R m ) is an arbitrary boundary value. We study the minimum problem by the introduction of the lower quasiconvex envelope $${\overline{f}}$$ f ¯ of f and of the relaxed functional $$\begin{aligned} \overline{{\mathcal {F}}}(u) = \int _{\Omega } {\overline{f}}(x, u(x), D u(x))\,dx, \quad u\in u_0 + W_0^{1,r}(\Omega ,{\mathbb {R}}^m), \end{aligned}$$ F ¯ ( u ) = ∫ Ω f ¯ ( x , u ( x ) , D u ( x ) ) d x , u ∈ u 0 + W 0 1 , r ( Ω , R m ) , imposing standard differentiability and growth properties on $${\overline{f}}$$ f ¯ . In addition we assume a suitable structural condition on $${\overline{f}}$$ f ¯ and a special regularity on the minimizers of $$\overline{{\mathcal {F}}}$$ F ¯ , showing that under such assumptions $${\mathcal {F}}$$ F attains its infimum. Futhermore, we study the minimum problem for a class of functionals with separate dependence on the gradients of competing maps by the use of integro-extremality method, proving an existence result inspired by analogous ones obtained in the scalar case ($$m=1$$ m = 1 ). This last argument does not require the special regularity assumption mentioned above but the usual notion of classical differentiability (almost everywhere).

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“…in Ω , which turns out to be a viscosity solution of P ϕ and of P * * ϕ . The method that we use is inspired by the results contained in [15], [16], [17] and [18], which are mainly devoted to the minimization of nonsemicontinuous functionals of the Calculus of Variations.…”

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confidence: 99%