# American Institute of Mathematical Sciences

December  2011, 31(4): 1411-1425. doi: 10.3934/dcds.2011.31.1411

## Estimates of the derivatives of minimizers of a special class of variational integrals

 1 Dipartimento di Matematica, Università di Catania, Università di Catania, Viale A. Doria, 6, 95125 Catania, Italy 2 Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba, 278-8510, Japan

Received  June 2009 Revised  September 2010 Published  September 2011

The note concerns on some estimates in Morrey Spaces for the derivatives of local minimizers of variational integrals of the form $$\int_\Omega F (x,u,Du) dx$$ where the integrand has the following special form $$F(x,u,Du)\, =\, A(x,u, g^{\alpha\beta}(x) h_{ij}(u) \frac{\partial u^i}{\partial x^\alpha} \frac{\partial u^i }{\partial x^\beta}),$$ where $(g^{\alpha\beta})$ and $(h_{ij})$ symmetric positive definite matrices. We are not assuming the continuity of $A$ and $g$ with respect to $x$. We suppose that $A(\cdot, u,t)/(1+t)$ and $g(\cdot)$ are in the class $L^\infty\cap VMO$.
Citation: Maria Alessandra Ragusa, Atsushi Tachikawa. Estimates of the derivatives of minimizers of a special class of variational integrals. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1411-1425. doi: 10.3934/dcds.2011.31.1411
 [1] E. Acerbi and N. Fusco, Regularity for minimizers of nonquadratic functionals: The case $1140 (1989), 115-135. [2] E. Acerbi and G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J., 136 (2007), 285-320. doi: 10.1215/S0012-7094-07-13623-8. [3] L. Caffarelli, Elliptic second order equations, Rend. Sem. Mat. Fis. Milano, 58 (1988), 253-284. doi: 10.1007/BF02925245. [4] L. Caffarelli, Interior a priori estimates for solutions of fully non linear equations, Ann. of Math., 130 (1989), 189-213. doi: 10.2307/1971480. [5] S. Campanato, Equazioni ellittiche del$II$ordine e spazi$\mathcalL^{2,\lambda},$Ann. Mat. Pura Appl., 69 (1965), 321-382. doi: 10.1007/BF02414377. [6] S. Campanato, A maximum principle for non-linear elliptic systems: Boundary fundamental estimates, Adv. Math., 66 (1987), 291-317. doi: 10.1016/0001-8708(87)90037-5. [7] S. Campanato, Elliptic systems with non-linearity$q$greater or equal$2. $Regularity of the solution of the Dirichlet problem, Ann. Mat. Pura Appl., 147 (1987), 117-150. doi: 10.1007/BF01762414. [8] F. Chiarenza, M. Frasca and P. Longo, Interior$W^{2,p}$estimates for non divergence elliptic equations with discontinuous coefficients, Ric. di Mat., XL 1, (1991), 149-168. [9] E. Di Benedetto,$C^{1+\alpha}$local regularity of weak solutions of degenerate elliptic vequations, Nonlinear Anal., 7 (1983), 827-850. doi: 10.1016/0362-546X(83)90061-5. [10] J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 253-266. doi: 10.2307/2373037. [11] M. Fuchs, Everywhere regularity theorems for mapping which minimize$p$-energy, Comment. Math. Univ. Carolin., 28 (1987), 673-677. [12] M. Fuchs,$p$-harmonic obstacle problems. I. Partial regularity theory, Ann. Mat. Pura Appl. (4), 156 (1990), 127-158. doi: 10.1007/BF01766976. [13] N. Fusco and J. Hutchinson, Partial regularity for minimisers of certain functionals having nonquadratic growth, Ann. Mat. Pura Appl., 155 (1989), 1-24. doi: 10.1007/BF01765932. [14] M. Giaquinta, "Introduction to Regularity Theory for Nonlinear Elliptic Systems," Lectures in Mathematics, ETH Zürich, Birkhäuser Verlag, Basel-Boston-Berlin, 1993, [15] M. Giaquinta and E. Giusti, Partial regularity for the solution to nonlinear parabolic systems, Ann. Mat. Pura Appl., 47 (1973), 253-266. [16] M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals, Acta Math., 148 (1982), 31-46. doi: 10.1007/BF02392725. [17] M. Giaquinta and E. Giusti, Differentiability of minima of non-differentiable functionals, Inv. Math., 72 (1983), 285-298. doi: 10.1007/BF01389324. [18] M. Giaquinta and E. Giusti, The singular set of the minima of certain quadratic functionals, Ann. Sc. Norm. Sup. Pisa, 9 (1984), 45-55. [19] M. Giaquinta and P. A. Ivert, Partial regularity for minima of variational integrals, Ark. Mat., 25 (1987), 221-229. doi: 10.1007/BF02384445. [20] M. Giaquinta and G. Modica, Regularity results for some classes of higher order non linear elliptic systems, J. Reine Angew. Math., 311/312 (1979), 145-169. [21] M. Giaquinta and G. Modica, Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. H. Poincaré, Analyse nonlinéare, 3 (1986), 185-208. [22] M. Giaquinta and G. Modica, Remarks on the regularity of the minimizers of certain degenerate functionals, Manuscripta Math., 57 (1986), 55-99. doi: 10.1007/BF01172492. [23] E. Giusti, Regolarita' parziale delle soluzioni deboli di una classe di sistemi ellittici quasi lineari di ordine arbitrario, Ann. Sc. Norm. Sup. Pisa, 23 (1969), 115-141. [24] E. Giusti, "Direct Method in the Calculus of Variations," World Scientific, 2003. [25] E. Giusti and M. Miranda, Sulla regolarita' delle soluzioni deboli di una classe di sistemi ellittici quasilineari, Arch. Rat. Mech. Anal., 31 (1968), 173-184. doi: 10.1007/BF00282679. [26] R. Hardt and F.-H. Lin, Mappings minimizing the$L^p$norm of the gradient, Comm. Pure Appl. Math., 40 (1987), 555-588. doi: 10.1002/cpa.3160400503. [27] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14 (1961), 415-476. doi: 10.1002/cpa.3160140317. [28] J. Kinnunen and S. Zhou, A local estimate for nonlinear equations with discontinuous coefficients, Comm. Partial Differential Equations, 24 (1999), 2043-2068. [29] J. Kristensen and G. Mingione, The singular set of minima of integral functionals, Arch. Ration. Mech. Anal., 180 (2006), 331-398. doi: 10.1007/s00205-005-0402-5. [30] J. J. Manfredi, Regularity for minima of functionals with$p$-growth, J. Differential Equations, 76 (1988), 203-212. [31] G. Mingione, Singularities of minima: A walk on the wild side of the calculus of variations, J. Global Optim., 40 (2008), 209-223. doi: 10.1007/s10898-007-9226-1. [32] C. B. Morrey Jr., Partial regularity results for nonlinear elliptic systems,, Journ. Math. and Mech., 17 (): 649. [33] M. A. Ragusa and A. Tachikawa, "Interior Estimates in Campanato Spaces Related to Quadratic Functionals," Proceedings of Research Institute of Mathematical Sciences, Kyoto, (2004), 54-65. [34] M. A. Ragusa and A. Tachikawa, Regularity of the minimizers of some variational integrals with discontinuity, Z. Anal. Anwend., 27 (2008), 469-482. doi: 10.4171/ZAA/1366. [35] D. Sarason, On functions of vanishing mean oscillation, Trans. Amer. Math. Soc., 207 (1975), 391-405. doi: 10.1090/S0002-9947-1975-0377518-3. [36] L. M. Sibner and R. B. Sibner, A non-linear Hodge de Rham theorem, Acta Math., 125 (1970), 57-73. doi: 10.1007/BF02392330. [37] P. Tolksdorf, Everywhere-regularity for some quasilinear systems with a lack of ellipticity, Ann. Mat. Pura Appl., 134 (1983), 241-266. doi: 10.1007/BF01773507. [38] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150. [39] K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems, Acta Math., 138 (1977), 219-240. doi: 10.1007/BF02392316. show all references ##### References:  [1] E. Acerbi and N. Fusco, Regularity for minimizers of nonquadratic functionals: The case$1140 (1989), 115-135. [2] E. Acerbi and G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J., 136 (2007), 285-320. doi: 10.1215/S0012-7094-07-13623-8. [3] L. Caffarelli, Elliptic second order equations, Rend. Sem. Mat. Fis. Milano, 58 (1988), 253-284. doi: 10.1007/BF02925245. [4] L. Caffarelli, Interior a priori estimates for solutions of fully non linear equations, Ann. of Math., 130 (1989), 189-213. doi: 10.2307/1971480. [5] S. Campanato, Equazioni ellittiche del $II$ ordine e spazi $\mathcalL^{2,\lambda},$ Ann. Mat. Pura Appl., 69 (1965), 321-382. doi: 10.1007/BF02414377. [6] S. Campanato, A maximum principle for non-linear elliptic systems: Boundary fundamental estimates, Adv. Math., 66 (1987), 291-317. doi: 10.1016/0001-8708(87)90037-5. [7] S. Campanato, Elliptic systems with non-linearity $q$ greater or equal $2.$Regularity of the solution of the Dirichlet problem, Ann. Mat. Pura Appl., 147 (1987), 117-150. doi: 10.1007/BF01762414. [8] F. Chiarenza, M. Frasca and P. Longo, Interior $W^{2,p}$ estimates for non divergence elliptic equations with discontinuous coefficients, Ric. di Mat., XL 1, (1991), 149-168. [9] E. Di Benedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic vequations, Nonlinear Anal., 7 (1983), 827-850. doi: 10.1016/0362-546X(83)90061-5. [10] J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 253-266. doi: 10.2307/2373037. [11] M. Fuchs, Everywhere regularity theorems for mapping which minimize $p$-energy, Comment. Math. Univ. Carolin., 28 (1987), 673-677. [12] M. Fuchs, $p$-harmonic obstacle problems. I. Partial regularity theory, Ann. Mat. Pura Appl. (4), 156 (1990), 127-158. doi: 10.1007/BF01766976. [13] N. Fusco and J. Hutchinson, Partial regularity for minimisers of certain functionals having nonquadratic growth, Ann. Mat. Pura Appl., 155 (1989), 1-24. doi: 10.1007/BF01765932. [14] M. Giaquinta, "Introduction to Regularity Theory for Nonlinear Elliptic Systems," Lectures in Mathematics, ETH Zürich, Birkhäuser Verlag, Basel-Boston-Berlin, 1993, [15] M. Giaquinta and E. Giusti, Partial regularity for the solution to nonlinear parabolic systems, Ann. Mat. Pura Appl., 47 (1973), 253-266. [16] M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals, Acta Math., 148 (1982), 31-46. doi: 10.1007/BF02392725. [17] M. Giaquinta and E. Giusti, Differentiability of minima of non-differentiable functionals, Inv. Math., 72 (1983), 285-298. doi: 10.1007/BF01389324. [18] M. Giaquinta and E. Giusti, The singular set of the minima of certain quadratic functionals, Ann. Sc. Norm. Sup. Pisa, 9 (1984), 45-55. [19] M. Giaquinta and P. A. Ivert, Partial regularity for minima of variational integrals, Ark. Mat., 25 (1987), 221-229. doi: 10.1007/BF02384445. [20] M. Giaquinta and G. Modica, Regularity results for some classes of higher order non linear elliptic systems, J. Reine Angew. Math., 311/312 (1979), 145-169. [21] M. Giaquinta and G. Modica, Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. H. Poincaré, Analyse nonlinéare, 3 (1986), 185-208. [22] M. Giaquinta and G. Modica, Remarks on the regularity of the minimizers of certain degenerate functionals, Manuscripta Math., 57 (1986), 55-99. doi: 10.1007/BF01172492. [23] E. Giusti, Regolarita' parziale delle soluzioni deboli di una classe di sistemi ellittici quasi lineari di ordine arbitrario, Ann. Sc. Norm. Sup. Pisa, 23 (1969), 115-141. [24] E. Giusti, "Direct Method in the Calculus of Variations," World Scientific, 2003. [25] E. Giusti and M. Miranda, Sulla regolarita' delle soluzioni deboli di una classe di sistemi ellittici quasilineari, Arch. Rat. Mech. Anal., 31 (1968), 173-184. doi: 10.1007/BF00282679. [26] R. Hardt and F.-H. Lin, Mappings minimizing the $L^p$ norm of the gradient, Comm. Pure Appl. Math., 40 (1987), 555-588. doi: 10.1002/cpa.3160400503. [27] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14 (1961), 415-476. doi: 10.1002/cpa.3160140317. [28] J. Kinnunen and S. Zhou, A local estimate for nonlinear equations with discontinuous coefficients, Comm. Partial Differential Equations, 24 (1999), 2043-2068. [29] J. Kristensen and G. Mingione, The singular set of minima of integral functionals, Arch. Ration. Mech. Anal., 180 (2006), 331-398. doi: 10.1007/s00205-005-0402-5. [30] J. J. Manfredi, Regularity for minima of functionals with $p$-growth, J. Differential Equations, 76 (1988), 203-212. [31] G. Mingione, Singularities of minima: A walk on the wild side of the calculus of variations, J. Global Optim., 40 (2008), 209-223. doi: 10.1007/s10898-007-9226-1. [32] C. B. Morrey Jr., Partial regularity results for nonlinear elliptic systems,, Journ. Math. and Mech., 17 (): 649. [33] M. A. Ragusa and A. Tachikawa, "Interior Estimates in Campanato Spaces Related to Quadratic Functionals," Proceedings of Research Institute of Mathematical Sciences, Kyoto, (2004), 54-65. [34] M. A. Ragusa and A. Tachikawa, Regularity of the minimizers of some variational integrals with discontinuity, Z. Anal. Anwend., 27 (2008), 469-482. doi: 10.4171/ZAA/1366. [35] D. Sarason, On functions of vanishing mean oscillation, Trans. Amer. Math. Soc., 207 (1975), 391-405. doi: 10.1090/S0002-9947-1975-0377518-3. [36] L. M. Sibner and R. B. Sibner, A non-linear Hodge de Rham theorem, Acta Math., 125 (1970), 57-73. doi: 10.1007/BF02392330. [37] P. Tolksdorf, Everywhere-regularity for some quasilinear systems with a lack of ellipticity, Ann. Mat. Pura Appl., 134 (1983), 241-266. doi: 10.1007/BF01773507. [38] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150. [39] K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems, Acta Math., 138 (1977), 219-240. doi: 10.1007/BF02392316.
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