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Quasilinear divergence form parabolic equations in Reifenberg flat domains
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 |
References:
[1] |
E. Acerbi and N. Fusco, Regularity for minimizers of nonquadratic functionals: The case $1 |
[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 $1 |
[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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