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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

Maria Alessandra Ragusa 1, and  Atsushi Tachikawa 2,

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 $L^\infty\cap VMO$.
Keywords: local minimizers, variational integrals., regularity results, Nonlinear elliptic systems.
Mathematics Subject Classification: Primary: 49N60, 35J50, 35R05; Secondary: 46E30, 35B6.
Citation: Maria Alessandra Ragusa, Atsushi Tachikawa. Estimates of the derivatives of minimizers of a special class of variational integrals. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1411-1425. doi: 10.3934/dcds.2011.31.1411
References:
[1]

E. Acerbi and N. Fusco, Regularity for minimizers of nonquadratic functionals: The case $1140 (1989), 115-135.  Google Scholar

[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.  Google Scholar

[3]

L. Caffarelli, Elliptic second order equations, Rend. Sem. Mat. Fis. Milano, 58 (1988), 253-284. doi: 10.1007/BF02925245.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 253-266. doi: 10.2307/2373037.  Google Scholar

[11]

M. Fuchs, Everywhere regularity theorems for mapping which minimize $p$-energy, Comment. Math. Univ. Carolin., 28 (1987), 673-677.  Google Scholar

[12]

M. Fuchs, $p$-harmonic obstacle problems. I. Partial regularity theory, Ann. Mat. Pura Appl. (4), 156 (1990), 127-158. doi: 10.1007/BF01766976.  Google Scholar

[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.  Google Scholar

[14]

M. Giaquinta, "Introduction to Regularity Theory for Nonlinear Elliptic Systems," Lectures in Mathematics, ETH Zürich, Birkhäuser Verlag, Basel-Boston-Berlin, 1993,  Google Scholar

[15]

M. Giaquinta and E. Giusti, Partial regularity for the solution to nonlinear parabolic systems, Ann. Mat. Pura Appl., 47 (1973), 253-266.  Google Scholar

[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.  Google Scholar

[17]

M. Giaquinta and E. Giusti, Differentiability of minima of non-differentiable functionals, Inv. Math., 72 (1983), 285-298. doi: 10.1007/BF01389324.  Google Scholar

[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.  Google Scholar

[19]

M. Giaquinta and P. A. Ivert, Partial regularity for minima of variational integrals, Ark. Mat., 25 (1987), 221-229. doi: 10.1007/BF02384445.  Google Scholar

[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.  Google Scholar

[21]

M. Giaquinta and G. Modica, Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. H. Poincaré, Analyse nonlinéare, 3 (1986), 185-208.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

E. Giusti, "Direct Method in the Calculus of Variations," World Scientific, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

J. Kinnunen and S. Zhou, A local estimate for nonlinear equations with discontinuous coefficients, Comm. Partial Differential Equations, 24 (1999), 2043-2068.  Google Scholar

[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.  Google Scholar

[30]

J. J. Manfredi, Regularity for minima of functionals with $p$-growth, J. Differential Equations, 76 (1988), 203-212.  Google Scholar

[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.  Google Scholar

[32]

C. B. Morrey Jr., Partial regularity results for nonlinear elliptic systems,, Journ. Math. and Mech., 17 (): 649.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.  Google Scholar

[39]

K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems, Acta Math., 138 (1977), 219-240. doi: 10.1007/BF02392316.  Google Scholar

show all references

References:
[1]

E. Acerbi and N. Fusco, Regularity for minimizers of nonquadratic functionals: The case $1140 (1989), 115-135.  Google Scholar

[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.  Google Scholar

[3]

L. Caffarelli, Elliptic second order equations, Rend. Sem. Mat. Fis. Milano, 58 (1988), 253-284. doi: 10.1007/BF02925245.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 253-266. doi: 10.2307/2373037.  Google Scholar

[11]

M. Fuchs, Everywhere regularity theorems for mapping which minimize $p$-energy, Comment. Math. Univ. Carolin., 28 (1987), 673-677.  Google Scholar

[12]

M. Fuchs, $p$-harmonic obstacle problems. I. Partial regularity theory, Ann. Mat. Pura Appl. (4), 156 (1990), 127-158. doi: 10.1007/BF01766976.  Google Scholar

[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.  Google Scholar

[14]

M. Giaquinta, "Introduction to Regularity Theory for Nonlinear Elliptic Systems," Lectures in Mathematics, ETH Zürich, Birkhäuser Verlag, Basel-Boston-Berlin, 1993,  Google Scholar

[15]

M. Giaquinta and E. Giusti, Partial regularity for the solution to nonlinear parabolic systems, Ann. Mat. Pura Appl., 47 (1973), 253-266.  Google Scholar

[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.  Google Scholar

[17]

M. Giaquinta and E. Giusti, Differentiability of minima of non-differentiable functionals, Inv. Math., 72 (1983), 285-298. doi: 10.1007/BF01389324.  Google Scholar

[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.  Google Scholar

[19]

M. Giaquinta and P. A. Ivert, Partial regularity for minima of variational integrals, Ark. Mat., 25 (1987), 221-229. doi: 10.1007/BF02384445.  Google Scholar

[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.  Google Scholar

[21]

M. Giaquinta and G. Modica, Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. H. Poincaré, Analyse nonlinéare, 3 (1986), 185-208.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

E. Giusti, "Direct Method in the Calculus of Variations," World Scientific, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

J. Kinnunen and S. Zhou, A local estimate for nonlinear equations with discontinuous coefficients, Comm. Partial Differential Equations, 24 (1999), 2043-2068.  Google Scholar

[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.  Google Scholar

[30]

J. J. Manfredi, Regularity for minima of functionals with $p$-growth, J. Differential Equations, 76 (1988), 203-212.  Google Scholar

[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.  Google Scholar

[32]

C. B. Morrey Jr., Partial regularity results for nonlinear elliptic systems,, Journ. Math. and Mech., 17 (): 649.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.  Google Scholar

[39]

K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems, Acta Math., 138 (1977), 219-240. doi: 10.1007/BF02392316.  Google Scholar

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