# American Institute of Mathematical Sciences

June  2012, 32(6): 2089-2099. doi: 10.3934/dcds.2012.32.2089

## A direct proof of the Tonelli's partial regularity result

Received  April 2011 Revised  July 2011 Published  February 2012

The aim of this work is to give a simple proof of the Tonelli's partial regularity result which states that any absolutely continuous solution to the variational problem $$\min\left\{\int_a^b L(t,u(t),\dot u(t))dt: u\in{\bf W}_0^{1,1}(a,b)\right\}$$ has extended-values continuous derivative if the Lagrangian function $L(t,u,\xi)$ is strictly convex in $\xi$ and Lipschitz continuous in $u$, locally uniformly in $\xi$ (but not in $t$). Our assumption is weaker than the one used in [2, 4, 5, 6, 13] since we do not require the Lipschitz continuity of $L$ in $u$ to be locally uniform in $t$, and it is optimal as shown by the example in [12].
Citation: Alessandro Ferriero. A direct proof of the Tonelli's partial regularity result. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2089-2099. doi: 10.3934/dcds.2012.32.2089
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##### References:
 [1] Tuhin Ghosh, Karthik Iyer. Cloaking for a quasi-linear elliptic partial differential equation. Inverse Problems & Imaging, 2018, 12 (2) : 461-491. doi: 10.3934/ipi.2018020 [2] Vasily Denisov and Andrey Muravnik. On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations. Electronic Research Announcements, 2003, 9: 88-93. [3] Vitali Liskevich, Igor I. Skrypnik. Pointwise estimates for solutions of singular quasi-linear parabolic equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1029-1042. doi: 10.3934/dcdss.2013.6.1029 [4] Priyanjana M. N. Dharmawardane. Decay property of regularity-loss type for quasi-linear hyperbolic systems of viscoelasticity. Conference Publications, 2013, 2013 (special) : 197-206. doi: 10.3934/proc.2013.2013.197 [5] Simone Creo, Valerio Regis Durante. Convergence and density results for parabolic quasi-linear Venttsel' problems in fractal domains. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 65-90. doi: 10.3934/dcdss.2019005 [6] Lu Yang, Meihua Yang, Peter E. Kloeden. Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2635-2651. doi: 10.3934/dcdsb.2012.17.2635 [7] Guangying Lv, Mingxin Wang. Existence, uniqueness and stability of traveling wave fronts of discrete quasi-linear equations with delay. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 415-433. doi: 10.3934/dcdsb.2010.13.415 [8] Kunio Hidano, Dongbing Zha. Remarks on a system of quasi-linear wave equations in 3D satisfying the weak null condition. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1735-1767. doi: 10.3934/cpaa.2019082 [9] Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345 [10] Timur Akhunov. Local well-posedness of quasi-linear systems generalizing KdV. Communications on Pure & Applied Analysis, 2013, 12 (2) : 899-921. doi: 10.3934/cpaa.2013.12.899 [11] Boris Buffoni, Laurent Landry. Multiplicity of homoclinic orbits in quasi-linear autonomous Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 75-116. doi: 10.3934/dcds.2010.27.75 [12] Teemu Tyni, Valery Serov. Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line. Inverse Problems & Imaging, 2019, 13 (1) : 159-175. doi: 10.3934/ipi.2019009 [13] Li Ma, Lin Zhao. Regularity for positive weak solutions to semi-linear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (3) : 631-643. doi: 10.3934/cpaa.2008.7.631 [14] Yingte Sun, Xiaoping Yuan. Quasi-periodic solution of quasi-linear fifth-order KdV equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6241-6285. doi: 10.3934/dcds.2018268 [15] Yusuke Murase, Risei Kano, Nobuyuki Kenmochi. Elliptic Quasi-variational inequalities and applications. Conference Publications, 2009, 2009 (Special) : 583-591. doi: 10.3934/proc.2009.2009.583 [16] Jean Ginibre, Giorgio Velo. Modified wave operators without loss of regularity for some long range Hartree equations. II. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1357-1376. doi: 10.3934/cpaa.2015.14.1357 [17] Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81 [18] Wen-Rong Dai. Formation of singularities to quasi-linear hyperbolic systems with initial data of small BV norm. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3501-3524. doi: 10.3934/dcds.2012.32.3501 [19] Massimo Lanza de Cristoforis, aolo Musolino. A quasi-linear heat transmission problem in a periodic two-phase dilute composite. A functional analytic approach. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2509-2542. doi: 10.3934/cpaa.2014.13.2509 [20] Yongqin Liu, Shuichi Kawashima. Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1113-1139. doi: 10.3934/dcds.2011.29.1113

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