Parabolic equations and the bounded slope condition
Annales de l'I.H.P. Analyse non linéaire, Volume 34 (2017) no. 2, pp. 355-379.

In this paper we establish the existence of Lipschitz-continuous solutions to the Cauchy Dirichlet problem of evolutionary partial differential equations

{tudivDf(Du)=0in ΩT,u=uoon PΩT.
The only assumptions needed are the convexity of the generating function f:RnR, and the classical bounded slope condition on the initial and the lateral boundary datum uoW1,(Ω). We emphasize that no growth conditions are assumed on f and that – an example which does not enter in the elliptic case – uo could be any Lipschitz initial and boundary datum, vanishing at the boundary ∂Ω, and the boundary may contain flat parts, for instance Ω could be a rectangle in Rn.

DOI: 10.1016/j.anihpc.2015.12.005
Classification: 35A01, 35K61, 35K86, 49J40
Keywords: Existence, Parabolic equations, Bounded slope condition, Lipschitz solutions
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Bögelein, Verena; Duzaar, Frank; Marcellini, Paolo; Signoriello, Stefano. Parabolic equations and the bounded slope condition. Annales de l'I.H.P. Analyse non linéaire, Volume 34 (2017) no. 2, pp. 355-379. doi : 10.1016/j.anihpc.2015.12.005. http://www.numdam.org/articles/10.1016/j.anihpc.2015.12.005/

[1] Akagi, G.; Stefanelli, U. Doubly nonlinear evolution equations as convex minimization, SIAM J. Math. Anal., Volume 46 (2014) no. 3, pp. 1922–1945 | DOI | MR | Zbl

[2] Ambrosio, L.; Gigli, N.; Savaré, G. Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008 | MR | Zbl

[3] Bögelein, V.; Duzaar, F.; Marcellini, P. Parabolic systems with p,q-growth: a variational approach, Arch. Ration. Mech. Anal., Volume 210 (2013) no. 1, pp. 219–267 | DOI | MR | Zbl

[4] Bögelein, V.; Duzaar, F.; Marcellini, P. Existence of evolutionary variational solutions via the calculus of variations, J. Differ. Equ., Volume 256 (2014) no. 12, pp. 3912–3942 | DOI | MR | Zbl

[5] Bögelein, V.; Duzaar, F.; Marcellini, P.; Signoriello, S. Nonlocal diffusion equations, J. Math. Anal. Appl., Volume 432 (2015) no. 1, pp. 398–428 | DOI | MR | Zbl

[6] Bögelein, V.; Duzaar, F.; Mingione, G. The boundary regularity of nonlinear parabolic systems I, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 27 (2010) no. 1, pp. 201–255 | Numdam | MR | Zbl

[7] Bögelein, V.; Duzaar, F.; Mingione, G. The regularity of general parabolic systems with degenerate diffusion, Mem. Am. Math. Soc., Volume 221 (2013) | MR | Zbl

[8] Bousquet, P. On the lower bounded slope condition, J. Convex Anal., Volume 14 (2007) no. 1, pp. 119–136 | MR | Zbl

[9] Bousquet, P. Boundary continuity of solutions to a basic problem in the calculus of variations, Adv. Calc. Var., Volume 3 (2010) no. 1, pp. 1–27 | DOI | MR | Zbl

[10] Cellina, A. On the bounded slope condition and the validity of the Euler Lagrange equation, SIAM J. Control Optim., Volume 40 (2001–2002) no. 4, pp. 1270–1279 | DOI | MR | Zbl

[11] Clarke, F. Continuity of solutions to a basic problem in the calculus of variations, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 4 (2005) no. 3, pp. 511–530 | Numdam | MR | Zbl

[12] Da Prato, G. Spazi L(p,ϑ)(Ω,δ) e loro proprieta, Ann. Mat. Pura Appl. (4), Volume 69 (1965), pp. 383–392 | DOI | MR | Zbl

[13] De Giorgi, E. Conjectures concerning some evolution problems. (Italian) A celebration of John F. Nash, Jr., Duke Math. J., Volume 81 (1996) no. 2, pp. 255–268 | MR | Zbl

[14] Giusti, E. Direct Methods in the Calculus of Variations, World Scientific, Singapore and River Edge and NJ, 2003 | DOI | MR | Zbl

[15] Haar, A. Über das Plateausche Problem, Math. Ann., Volume 97 (1927) no. 1, pp. 124–158 | DOI | JFM | MR

[16] Hartman, P.; Nirenberg, L. On spherical image maps whose Jacobians do not change sign, Am. J. Math., Volume 81 (1959), pp. 901–920 | DOI | MR | Zbl

[17] Hartman, P.; Stampacchia, G. On some non-linear elliptic differential-functional equations, Acta Math., Volume 115 (1966), pp. 271–310 | DOI | MR | Zbl

[18] Ilmanen, T. Elliptic regularization and partial regularity for motion by mean curvature, Mem. Am. Math. Soc., Volume 108 (1994) no. 520 | MR | Zbl

[19] Kinnunen, J.; Lindqvist, P. Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation, Ann. Mat. Pura Appl. (4), Volume 185 (2006) no. 3, pp. 411–435 | DOI | MR | Zbl

[20] Lichnewsky, A.; Temam, R.M. Pseudosolutions of the time-dependent minimal surface problem, J. Differ. Equ., Volume 30 (1978) no. 3, pp. 340–364 | DOI | MR | Zbl

[21] Lieberman, G.M. Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996 | DOI | MR | Zbl

[22] Mariconda, C.; Treu, G. Existence and Lipschitz regularity for minima, Proc. Am. Math. Soc., Volume 130 (2002) no. 2, pp. 395–404 | DOI | MR | Zbl

[23] Mariconda, C.; Treu, G. Lipschitz regularity for minima without strict convexity of the Lagrangian, J. Differ. Equ., Volume 243 (2007) no. 2, pp. 388–413 | DOI | MR | Zbl

[24] Mariconda, C.; Treu, G. Hölder regularity for a classical problem of the calculus of variations, Adv. Calc. Var., Volume 2 (2009) no. 4, pp. 311–320 | DOI | MR | Zbl

[25] Mascolo, E.; Schianchi, E. Existence theorems for nonconvex problems, J. Math. Pures Appl. (9), Volume 62 (1983) no. 3, pp. 349–359 | MR | Zbl

[26] Mielke, A.; Stefanelli, U. Weighted energy-dissipation functionals for gradient flows, ESAIM Control Optim. Calc. Var., Volume 17 (2011) no. 1, pp. 52–85 | DOI | Numdam | MR | Zbl

[27] Miranda, M. Un teorema di esistenza e unicità per il problema dell'area minima in n variabili, Ann. Sc. Norm. Super. Pisa, Volume 19 (1965) no. 3, pp. 233–249 | Numdam | MR | Zbl

[28] Moser, J. A Harnack inequality for parabolic differential equations, Commun. Pure Appl. Math., Volume 17 (1964), pp. 101–134 | DOI | MR | Zbl

[29] Moser, J. Correction to: ”A Harnack inequality for parabolic differential equations”, Commun. Pure Appl. Math., Volume 20 (1967), pp. 231–236 | DOI | MR | Zbl

[30] Serra, E.; Tilli, P. Nonlinear wave equations as limits of convex minimization problems: proof of a conjecture by De Giorgi, Ann. Math. (2), Volume 175 (2012) no. 3, pp. 1551–1574 | DOI | MR | Zbl

[31] Simon, J. Compact sets in the space Lp(0,T;B) , Ann. Mat. Pura Appl. (4), Volume 146 (1987), pp. 65–96 | MR | Zbl

[32] Stampacchia, G. On some regular multiple integral problems in the calculus of variations, Commun. Pure Appl. Math., Volume 16 (1963), pp. 383–421 | DOI | MR | Zbl

[33] Wieser, W. Parabolic Q-minima and minimal solutions to variational flow, Manuscr. Math., Volume 59 (1987) no. 1, pp. 63–107 | DOI | MR | Zbl

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