## Regularity for some scalar variational problems under general growth conditions.(English)Zbl 0901.49030

The author continues his study of regularity for minima of integral functionals of the form $F(v) = \int_\Omega f(Dv) dx,$ where $$\Omega$$ is an open set of the Euclidean $$n$$-space with $$n \geq 2$$, and $$f$$ is a scalar valued function. It is well known that if $$f$$ behaves likes $$f_p$$ defined by $$f^p(\xi) = | \xi| ^p$$, then minimizers of $$F$$ (subject to suitable boundary conditions) are at least locally Lipschitz. Here, the interest is in a more general class of functions $$f$$. Specifically, the function $$f$$ is assumed to be twice differentiable and the eigenvalues of the second derivative matrix $$\partial_2 f/\partial \xi^2$$ are assumed to lie between two positive, increasing functions $$g_1(| \xi|)$$ and $$g_2(| \xi|)$$. In addition, the functions $$g_1$$ and $$g_2$$ are related, first, by the obvious inequality $$g_1 \leq g_2$$, and by the restrictions $g_2(t)t^2 = O \left(\int_0^{t} (g_1(s))^{1/2}) ds)^\alpha\right)$ and $g_2(| \xi|)| \xi| ^2 = O \left( f(\xi)^\beta\right)$ as $$t$$ and $$| \xi|$$ go to $$\infty$$ for exponents $$\alpha$$ and $$\beta$$ satisfying the inequalities $2 \leq \sigma < \frac {2n}{n-2},\qquad 1 \leq\beta < \frac 2n \frac{\alpha}{\alpha-2}$ using the convention that $$a/0 =\infty$$ if $$a > 0$$. The main result is that the $$L^\infty(B_\rho)$$ norm of the gradient of a minimizer can be estimated in terms of the $$L^1(B_r)$$ norm of that gradient, where $$B_\rho$$ and $$B_r$$ are concentric balls with $$\rho < r$$. After discussing a number of examples of functions $$f$$ which satisfy these conditions, the author proves the gradient estimate via an approximation argument and a variant of the Moser iteration scheme.

### MSC:

 49N60 Regularity of solutions in optimal control
Full Text:

### References:

  Morrey, C. B.,Multiple Integrals in the Calculus of Variations, Grundlehren der Mathematischen Wissenschaften, Springer Verlag, Berlin, Germany, Vol. 130, 1966. · Zbl 0142.38701  Ball, J. M.,Convexity Conditions and Existence Theorems in Nonlinear Elasticity, Archive for Rational Mechanics and Analysis, Vol. 63, pp. 337–403, 1977. · Zbl 0368.73040  Dacorogna, B.,Direct Methods in the Calculus of Variations, Applied Mathematical Sciences, Springer Verlag, Berlin, Germany, Vol. 78, 1989. · Zbl 0703.49001  Ladyzhenskaya, O., andUral’tseva, N.,Linear and Quasilinear Elliptic Equations, Mathematics in Science and Engineering, Academic Press, San Diego, California, Vol. 46, 1968.  Giaquinta, M.,Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematical Studies, Princeton University Press, Princeton, New Jersey, Vol. 105, 1983. · Zbl 0516.49003  Marcellini, P.,Regularity of Minimizers of Integrals of the Calculus of Variations with Nonstandard Growth Conditions, Archive for Rational Mechanics and Analysis, Vol. 105, pp. 267–284, 1989. · Zbl 0667.49032  Marcellini, P.,Regularity and Existence of Solutions of Elliptic Equations with (p,q)-Growth Conditions, Journal of Differential Equations, Vol. 90, pp. 1–30, 1991. · Zbl 0724.35043  Giaquinta, M.,Growth Conditions and Regularity: A Counterexample, Manuscripta Mathematica, Vol. 59, pp. 245–248, 1987. · Zbl 0638.49005  Marcellini, P.,Un Example de Solution Discontinue d’un Problème Variationnel dans le Cas Scalaire, Report 11, Istituto Matematico U. Dini, Università di Firenze, 1987.  Boccardo, L., Marcellini, P., andSbordone, C.,L Regularity for Variational Problems with Sharp Nonstandard Growth Conditions, Bollettino della Unione Matematica Italiana, Vol. 4A, pp. 219–225, 1990. · Zbl 0711.49058  Marcellini, P.,Regularity for Elliptic Equations with General Growth Conditions, Journal of Differential Equations, Vol. 105, pp. 296–333, 1993. · Zbl 0812.35042  Lieberman, G. M.,Gradient Estimates for a Class of Elliptic Systems, Annali di Matematica Pura ed Applicata, Vol. 164, pp. 103–120, 1993. · Zbl 0819.35019  Marcellini, P.,Everywhere Regularity for a Class of Elliptic Systems without Growth Conditions, Annali della Scuola Normale Superiore di Pisa (to appear). · Zbl 0922.35031  Chipot, M., andEvans, L. C.,Linearization at Infinity and Lipschitz Estimates for Certain Problems in the Calculus of Variations, Proceedings of the Royal Society of Edinburgh, Vol. 102A, pp. 291–303, 1986. · Zbl 0602.49029  Di Benedetto, E.,C 1+{$$\alpha$$} Local Regularity of Weak Solutions of Degenerate Elliptic Equations, Nonlinear Analysis: Theory, Methods and Applications, Vol. 7, pp. 827–850, 1983. · Zbl 0539.35027  Evans, L. C.,A New Proof of Local C 1,{$$\alpha$$} Regularity for Solutions of Certain Degenerate Elliptic PDE, Journal of Differential Equations, Vol. 45, pp. 356–373, 1982. · Zbl 0508.35036  Lewis, J.,Regularity of Derivatives of Solutions to Certain Degenerate Elliptic Equations, Indiana University Mathematics Journal, Vol. 32, pp. 849–858, 1983. · Zbl 0554.35048  Manfredi, J. J.,Regularity for Minima of Functionals with p-Growth, Journal of Differential Equations, Vol. 76, pp. 203–212, 1988. · Zbl 0674.35008  Ivanov, A. V.,Quasilinear Degenerate and Nonuniformly Elliptic and Parabolic Equations of Second Order, Proceedings of the Steklov Institute of Mathematics, Moscow, Russia, Vol. 160, 1982; American Mathematical Society, Providence, Rhode Island, 1984. · Zbl 0517.35001  Ladyzhenskaya, O., andUral’tseva, N.,Local Estimates for Gradients of Solutions of Nonuniformly Elliptic and Parabolic Equations, Communications on Pure and Applied Mathematics, Vol. 23, pp. 677–703, 1970. · Zbl 0193.07202  Lieberman, G. M.,On the Regularity of the Minimizer of a Functional with Exponential Growth, Commentationes Mathematicae Universitatis Carolinae, Vol. 33, pp. 45–49, 1992. · Zbl 0776.49026  Serrin, J.,Gradient Estimates for Solutions of Nonlinear Elliptic and Parabolic Equations, Contributions to Nonlinear Functional Analysis, Edited by E. H. Zarantonello, Academic Press, San Diego, California, pp. 565–601, 1971.  Simon, L.,Interior Gradients Bounds for Nonuniformly Elliptic Equations, Indiana University Mathematics Journal, Vol. 25, pp. 821–855, 1976. · Zbl 0346.35016  Bhattacharya, T., andLeonetti, F.,W 2,2 Regularity for Weak Solutions of Elliptic Systems with Nonstandard Growth, Journal of Mathematical Analysis and Applications, Vol. 176, pp. 224–234, 1993. · Zbl 0809.35008  Giusti, E.,Metodi Diretti nel Calcolo delle Variazioni, Unione Matematica Italiana, Bologna, Italia, 1994.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.