# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021181
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## Existence of minimizers for one-dimensional vectorial non-semicontinuous functionals with second order lagrangian

 SISSA, Via Bonomea 265 - 34136 Trieste, Italy

* Corresponding author: Sandro Zagatti

Received  June 2021 Revised  October 2021 Early access November 2021

We study the minimum problem for functionals of the form
 $$$\mathcal{F}(u) = \int_{I} f(x, u(x), u^ \prime(x), u^ {\prime\prime}(x))\,dx,$$$
where the integrand
 $f:I\times \mathbb{R}^m\times \mathbb{R}^m\times \mathbb{R}^m \to \mathbb{R}$
is not convex in the last variable. We provide an existence result assuming that the lower convex envelope
 $\overline{f} = \overline{f}(x,p,q,\xi)$
of
 $f$
with respect to
 $\xi$
is regular and enjoys a special dependence with respect to the i-th single components
 $p_i, q_i, \xi_i$
of the vector variables
 $p,q,\xi$
. More precisely, we assume that it is monotone in
 $p_i$
and that it satisfies suitable affinity properties with respect to
 $\xi_i$
on the set
 $\{f> \overline{f}\}$
and with respect to
 $q_i$
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.
Citation: Sandro Zagatti. Existence of minimizers for one-dimensional vectorial non-semicontinuous functionals with second order lagrangian. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021181
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##### References:
 [1] Ana Cristina Barroso, José Matias. Necessary and sufficient conditions for existence of solutions of a variational problem involving the curl. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 97-114. doi: 10.3934/dcds.2005.12.97 [2] M. Soledad Aronna. Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1233-1258. doi: 10.3934/dcdss.2018070 [3] Hongwei Lou. Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1445-1464. doi: 10.3934/dcdsb.2010.14.1445 [4] Yong Wang, Wanquan Liu, Guanglu Zhou. An efficient algorithm for non-convex sparse optimization. Journal of Industrial & Management Optimization, 2019, 15 (4) : 2009-2021. doi: 10.3934/jimo.2018134 [5] Xiaoni Chi, Zhongping Wan, Zijun Hao. Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1111-1125. doi: 10.3934/jimo.2015.11.1111 [6] RazIye Mert, A. Zafer. A necessary and sufficient condition for oscillation of second order sublinear delay dynamic equations. Conference Publications, 2011, 2011 (Special) : 1061-1067. doi: 10.3934/proc.2011.2011.1061 [7] Qilin Wang, Liu He, Shengjie Li. Higher-order weak radial epiderivatives and non-convex set-valued optimization problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 465-480. doi: 10.3934/jimo.2018051 [8] Nurullah Yilmaz, Ahmet Sahiner. On a new smoothing technique for non-smooth, non-convex optimization. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 317-330. doi: 10.3934/naco.2020004 [9] Jingang Xiong, Jiguang Bao. The obstacle problem for Monge-Ampère type equations in non-convex domains. Communications on Pure & Applied Analysis, 2011, 10 (1) : 59-68. doi: 10.3934/cpaa.2011.10.59 [10] Yoon Mo Jung, Taeuk Jeong, Sangwoon Yun. Non-convex TV denoising corrupted by impulse noise. Inverse Problems & Imaging, 2017, 11 (4) : 689-702. doi: 10.3934/ipi.2017032 [11] Tong Li, Jeungeun Park. Stability of traveling waves of models for image processing with non-convex nonlinearity. Communications on Pure & Applied Analysis, 2018, 17 (3) : 959-985. doi: 10.3934/cpaa.2018047 [12] Kaizhi Wang, Yong Li. Existence and monotonicity property of minimizers of a nonconvex variational problem with a second-order Lagrangian. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 687-699. doi: 10.3934/dcds.2009.25.687 [13] Lihua Li, Yan Gao, Hongjie Wang. Second order sufficient optimality conditions for hybrid control problems with state jump. Journal of Industrial & Management Optimization, 2015, 11 (1) : 329-343. doi: 10.3934/jimo.2015.11.329 [14] C. M. Elliott, B. Gawron, S. Maier-Paape, E. S. Van Vleck. Discrete dynamics for convex and non-convex smoothing functionals in PDE based image restoration. Communications on Pure & Applied Analysis, 2006, 5 (1) : 181-200. doi: 10.3934/cpaa.2006.5.181 [15] Alberto Cabada, João Fialho, Feliz Minhós. Non ordered lower and upper solutions to fourth order problems with functional boundary conditions. Conference Publications, 2011, 2011 (Special) : 209-218. doi: 10.3934/proc.2011.2011.209 [16] Piernicola Bettiol, Nathalie Khalil. Necessary optimality conditions for average cost minimization problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2093-2124. doi: 10.3934/dcdsb.2019086 [17] Tong Li, Hui Yin. Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux. Communications on Pure & Applied Analysis, 2014, 13 (2) : 835-858. doi: 10.3934/cpaa.2014.13.835 [18] Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076 [19] Asadollah Aghajani. Regularity of extremal solutions of semilinear elliptic problems with non-convex nonlinearities on general domains. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3521-3530. doi: 10.3934/dcds.2017150 [20] . Adimurthi, Siddhartha Mishra, G.D. Veerappa Gowda. Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes. Networks & Heterogeneous Media, 2007, 2 (1) : 127-157. doi: 10.3934/nhm.2007.2.127

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