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

Published online by Cambridge University Press:  12 May 2007

Pietro Celada ,
Giovanni Cupini  and
Marcello Guidorzi
Pietro Celada
Affiliation:
Dipartimento di Matematica – Università degli Studi di Parma, Parco Area delle Scienze 53/A, 43100 Parma, Italy; celada@math.unipr.it
Giovanni Cupini
Affiliation:
Dipartimento di Matematica “U. Dini” – Università degli Studi di Firenze, V. le Morgagni 67/A, 50134 Firenze Italy; cupini@math.unifi.it
Marcello Guidorzi
Affiliation:
Dipartimento di Matematica – Università degli Studi di Ferrara, Via Machiavelli 35, 44100 Ferrara, Italy; guidorzi@dm.unife.it
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Abstract

We show that local minimizers of functionals of the form $\int_{\Omega} \left[f(Du(x)) + g(x\,,u(x))\right]\,{\rm d}x$,  $u \in u_0 + W_0^{1,p}(\Omega)$, 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.

Keywords

Nonstandard growthexistenceLipschitz continuity
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 2 , April 2007 , pp. 343 - 358
Copyright
© EDP Sciences, SMAI, 2007

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References

M. Bildhauer, Convex variational problems. Linear, nearly linear and anisotropic growth conditions, Springer-Verlag, Berlin and New York. Lect. Notes Math. 1818 (2003).
Celada, P., Existence and regularity of minimizers of non convex functionals depending on u and $\nabla u$ . J. Math. Anal. Appl. 230 (1999) 30–56. CrossRef
Celada, P. and Perrotta, S., Minimizing non convex, multiple integrals: a density result. Proc. Roy. Soc. Edinburgh 130A (2000) 721–741. CrossRef
Celada, P. and Perrotta, S., On the minimum problem for nonconvex, multiple integrals of product type. Calc. Var. Partial Differential Equations 12 (2001) 371–398. CrossRef
Celada, P., Cupini, G. and Guidorzi, M., A sharp attainment result for nonconvex variational problems. Calc. Var. Partial Differential Equations 20 (2004) 301–328. CrossRef
Cellina, A., On minima of a functional of the gradient: necessary conditions. Nonlinear Anal. 20 (1993) 337–341. CrossRef
Cellina, A., On minima of a functional of the gradient: sufficient conditions. Nonlinear Anal. 20 (1993) 343–347. CrossRef
G. Cupini and A.P. Migliorini, Hölder continuity for local minimizers of a nonconvex variational problem, J. Convex Anal. 10 (2003) 389–408.
Cupini, G., Guidorzi, M. and Mascolo, E., Regularity of minimizers of vectorial integrals with p-q growth. Nonlinear Anal. 54 (2003) 591–616. CrossRef
G. Dal Maso, An introduction to $\Gamma$ -convergence, Birkhäuser, Boston. Progr. Nonlinear Differential Equations Appl. 8 (1993).
De Blasi, F.S. and Pianigiani, G., On the Dirichlet problem for Hamilton-Jacobi equations. A Baire category approach. Nonlinear Differential Equations Appl. 6 (1999) 13–34. CrossRef
Esposito, L., Leonetti, F. and Mingione, G., Regularity results for minimizers of irregular integrals with $(p,q)$ growth. Forum Math. 14 (2002) 245–272. CrossRef
Esposito, L., Leonetti, F. and Mingione, G., Sharp regularity for functionals with $(p,q)$ growth. J. Differential Equations 204 (2004) 5–55. CrossRef
Fonseca, I. and Fusco, N., Regularity results for anisotropic image segmentation models. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 244 (1997) 463–499.
I. Fonseca, N. Fusco and P. Marcellini, An existence result for a nonconvex variational problem via regularity. ESAIM: COCV. 7 (2002) 69–95.
Friesecke, G., 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. CrossRef
Giaquinta, M. and Giusti, E., On the regularity of the minima of variational integrals. Acta Math. 148 (1982) 31–46. CrossRef
Giaquinta, M. and Giusti, E., Differentiability of minima of non-differentiable functionals. Invent. Math. 72 (1983) 285–298. CrossRef
Giaquinta, M. and Modica, G., Remarks on the regularity of the minimizers of certain degenerate functionals. Manuscripta Math. 57 (1986) 55–99. CrossRef
E. Giusti, Direct methods in the calculus of variations, World Scientific Publishing Co., Inc., River Edge, NJ (2003).
Manfredi, J.J., Regularity for minima of functionals with p-growth. J. Differential Equations 76 (1988) 203–212. CrossRef
Marcellini, P., Alcune osservazioni sull'esistenza del minimo di integrali del calcolo delle variazioni senza ipotesi di convessità. Rend. Mat. 13 (1980) 271–281.
Marcellini, P., 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. CrossRef
Marcellini, P., Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions. Arch. Rational Mech. Anal. 105 (1989) 267–284. CrossRef
Marcellini, P., Regularity for elliptic equations with general growth conditions. J. Differential Equations 105 (1993) 296–333. CrossRef
Sychev, M.A., Characterization of homogeneous scalar variational problems solvable for all boundary data. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000) 611–631.
Zagatti, S., Minimization of functionals of the gradient by Baire's theorem. SIAM J. Control Optim. 38 (2000) 384–399. CrossRef