Existence and regularity of minimizers of nonconvex integrals with $p-q$ growth

Celada, Pietro; Cupini, Giovanni; Guidorzi, Marcello

ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 2, pp. 343-358.

We show that local minimizers of functionals of the form ${\int }_{\Omega }\left[f\left(Du\right(x\left)\right)+g(x,u(x\left)\right)\right]\mathrm{d}x$, $u\in u_0+W_0^{1,p}\left(\Omega \right)$, are locally Lipschitz continuous provided $f$ is a convex function with $p-q$ growth satisfying a condition of qualified convexity at infinity and $g$ is Lipschitz continuous in $u$. As a consequence of this, we obtain an existence result for a related nonconvex functional.

[1] M. Bildhauer, Convex variational problems. Linear, nearly linear and anisotropic growth conditions, Springer-Verlag, Berlin and New York. Lect. Notes Math. 1818 (2003). | MR | Zbl

[2] P. Celada, Existence and regularity of minimizers of non convex functionals depending on $u$ and $\nabla u$. J. Math. Anal. Appl. 230 (1999) 30-56. | Zbl

[3] P. Celada and S. Perrotta, Minimizing non convex, multiple integrals: a density result. Proc. Roy. Soc. Edinburgh 130A (2000) 721-741. | Zbl

[4] P. Celada and S. Perrotta, On the minimum problem for nonconvex, multiple integrals of product type. Calc. Var. Partial Differential Equations 12 (2001) 371-398. | Zbl

[5] P. Celada, G. Cupini and M. Guidorzi, A sharp attainment result for nonconvex variational problems. Calc. Var. Partial Differential Equations 20 (2004) 301-328. | Zbl

[6] A. Cellina, On minima of a functional of the gradient: necessary conditions. Nonlinear Anal. 20 (1993) 337-341. | Zbl

[7] A. Cellina, On minima of a functional of the gradient: sufficient conditions. Nonlinear Anal. 20 (1993) 343-347. | Zbl

[8] G. Cupini and A.P. Migliorini, Hölder continuity for local minimizers of a nonconvex variational problem, J. Convex Anal. 10 (2003) 389-408. | Zbl

[9] G. Cupini, M. Guidorzi and E. Mascolo, Regularity of minimizers of vectorial integrals with $p$-$q$ growth. Nonlinear Anal. 54 (2003) 591-616. | Zbl

[10] G. Dal Maso, An introduction to $\Gamma$-convergence, Birkhäuser, Boston. Progr. Nonlinear Differential Equations Appl. 8 (1993). | MR | Zbl

[11] F.S. De Blasi and G. Pianigiani, On the Dirichlet problem for Hamilton-Jacobi equations. A Baire category approach. Nonlinear Differential Equations Appl. 6 (1999) 13-34. | Zbl

[12] L. Esposito, F. Leonetti and G. Mingione, Regularity results for minimizers of irregular integrals with $(p,q)$ growth. Forum Math. 14 (2002) 245-272. | Zbl

[13] L. Esposito, F. Leonetti and G. Mingione, Sharp regularity for functionals with $(p,q)$ growth. J. Differential Equations 204 (2004) 5-55. | Zbl

[14] I. Fonseca and N. Fusco, Regularity results for anisotropic image segmentation models. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 244 (1997) 463-499. | EuDML | Numdam | Zbl

[15] I. Fonseca, N. Fusco and P. Marcellini, An existence result for a nonconvex variational problem via regularity. ESAIM: COCV. 7 (2002) 69-95. | EuDML | Numdam | Zbl

[16] G. Friesecke, A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 437-471. | Zbl

[17] M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals. Acta Math. 148 (1982) 31-46. | Zbl

[18] M. Giaquinta and E. Giusti, Differentiability of minima of non-differentiable functionals. Invent. Math. 72 (1983) 285-298. | EuDML | Zbl

[19] M. Giaquinta and G. Modica, Remarks on the regularity of the minimizers of certain degenerate functionals. Manuscripta Math. 57 (1986) 55-99. | EuDML | Zbl

[20] E. Giusti, Direct methods in the calculus of variations, World Scientific Publishing Co., Inc., River Edge, NJ (2003). | MR | Zbl

[21] J.J. Manfredi, Regularity for minima of functionals with $p$-growth. J. Differential Equations 76 (1988) 203-212. | Zbl

[22] P. Marcellini, Alcune osservazioni sull'esistenza del minimo di integrali del calcolo delle variazioni senza ipotesi di convessità. Rend. Mat. 13 (1980) 271-281. | Zbl

[23] P. Marcellini, A relation between existence of minima for non convex integrals and uniqueness for non strictly convex integrals of the Calculus of Variations, in Mathematical theories of optimization (S. Margherita Ligure (1981)), J.P. Cecconi and T. Zolezzi Eds., Springer, Berlin. Lect. Notes Math. 979 (1983) 216-231. | Zbl

[24] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions. Arch. Rational Mech. Anal. 105 (1989) 267-284. | Zbl

[25] P. Marcellini, Regularity for elliptic equations with general growth conditions. J. Differential Equations 105 (1993) 296-333. | Zbl

[26] M.A. Sychev, Characterization of homogeneous scalar variational problems solvable for all boundary data. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000) 611-631. | Zbl

[27] S. Zagatti, Minimization of functionals of the gradient by Baire's theorem. SIAM J. Control Optim. 38 (2000) 384-399. | Zbl