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The energy conservation for weak solutions to the relativistic Nordström-Vlasov system
Hölder-estimates for non-autonomous parabolic problems with rough data
1. | Technische Universität Darmstadt, Fachbereich Mathematik, Dolivostr. 15, D-64293 Darmstadt, Germany |
2. | Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Germany |
References:
[1] |
H. Amann, Parabolic evolution equations and nonlinear boundary conditions,, J. Differ. Equations, 72 (1988), 201.
doi: 10.1016/0022-0396(88)90156-8. |
[2] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, in: Function spaces, 133 (1993), 9.
doi: 10.1007/978-3-663-11336-2_1. |
[3] |
H. Amann, Maximal regularity for nonautonomous evolution equations,, Adv. Nonlinear Stud., 4 (2004), 417.
|
[4] |
H. Amann, Linear and Quasilinear Parabolic Problems,, Birkhäuser, (1995).
doi: 10.1007/978-3-0348-9221-6. |
[5] |
W. Arendt, D. Dier, H. Laasri and E. M. Ouhabaz, Maximal regularity for evolution equations governed by non-autonomous forms,, Adv. Differential Equations, 19 (2014), 1043.
|
[6] |
P. Auscher, N. Badr, R. Haller-Dintelmann and J. Rehberg, The square root problem for second order, divergence form operators with mixed boundary conditions on $L^p$,, J. Evol. Eq., 15 (2015), 165.
doi: 10.1007/s00028-014-0255-1. |
[7] |
C. Bennett and R. Sharpley, Interpolation of Operators,, Academic Press, (1988).
|
[8] |
K. Brewster, D. Mitrea, I. Mitrea and M. Mitrea, Extending Sobolev functions with partially vanishing traces from locally $(\epsilon,\delta)$-domains and applications to mixed boundary problems,, J. Funct. Anal., 266 (2014), 4314.
doi: 10.1016/j.jfa.2014.02.001. |
[9] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Springer, (2011).
doi: 10.1007/978-0-387-70914-7. |
[10] |
E. Casas, J. C. de los Reyes and F. Tröltzsch, Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints,, SIAM J. Control Optim., 19 (2008), 616.
doi: 10.1137/07068240X. |
[11] |
P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, North Holland, (1978).
|
[12] |
R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5: Evolution Problems I,, Springer-Verlag, (1992).
doi: 10.1007/978-3-642-58090-1. |
[13] |
J. C. de los Reyes, P. Merino, J. Rehberg and F. Tröltzsch, Optimality conditions for state-constrained PDE control problems with time-dependent controls,, Control Cybernet., 37 (2008), 5.
|
[14] |
D. Dier, Non-autonomous maximal regularity for forms of bounded variation,, J. Math. Anal. Appl., 425 (2015), 33.
doi: 10.1016/j.jmaa.2014.12.006. |
[15] |
K. Disser, H.-C. Kaiser and J. Rehberg, Optimal Sobolev regularity for linear second-order divergence elliptic operators occuring in real-world problems,, SIAM J. Math. Anal., 47 (2015), 1719.
doi: 10.1137/140982969. |
[16] |
J. Elschner, J. Rehberg and G. Schmidt, Optimal regularity for elliptic transmission problems including $C^1$ interfaces,, Interfaces Free Bound., 9 (2007), 233.
doi: 10.4171/IFB/163. |
[17] |
A. F. M. ter Elst and J. Rehberg, Hölder estimates for second-order operators on domains with rough boundary,, Adv. Differential Equations, 20 (2015), 299.
|
[18] |
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in advanced mathematics, (1992).
|
[19] |
I. Fonseca and G. Parry, Equilibrium configurations of defective crystals,, Arch. Rat. Mech. Anal., 120 (1992), 245.
doi: 10.1007/BF00375027. |
[20] |
H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen,, Akademie-Verlag, (1974).
|
[21] |
Ch. Gallarati and M. Veraar, Maximal regularity for non-autonomous equations with measurable dependence on time,, , (). Google Scholar |
[22] |
M. Giaquinta and M. Struwe, An optimal regularity result for a class of quasilinear parabolic systems,, Manuscr. Math., 36 (1981), 223.
doi: 10.1007/BF01170135. |
[23] |
E. Giusti, Metodi Diretti nel Calcolo Delle Variazioni,, Unione Matematica Italiana, (1994).
|
[24] |
J. A. Griepentrog, W. Höppner, H.-C. Kaiser and J. Rehberg, A bi-Lipschitz continuous, volume preserving map from the unit ball onto a cube,, Note Mat., 28 (2008), 177.
doi: 10.1285/i15900932v28n1p177. |
[25] |
J. A. Griepentrog, K. Gröger, H. C. Kaiser and J. Rehberg, Interpolation for function spaces related to mixed boundary value problems,, Math. Nachr., 241 (2002), 110.
doi: 10.1002/1522-2616(200207)241:1<110::AID-MANA110>3.0.CO;2-R. |
[26] |
J. Griepentrog, Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces,, Adv. Differ. Equ., 12 (2007), 1031.
|
[27] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman, (1985).
doi: 10.1137/1.9781611972030. |
[28] |
K. Gröger, A $W^{1,p}$-estimate for solutions to mixed boundary value problems for second order elliptic differential equations,, Math. Ann., 283 (1989), 679.
doi: 10.1007/BF01442860. |
[29] |
K. Gröger, $W^{1,p}$-estimates of solutions to evolution equations corresponding to nonsmooth second order elliptic differential operators,, Nonlinear Anal., 18 (1992), 569.
doi: 10.1016/0362-546X(92)90211-V. |
[30] |
B. H. Haak and E. M. Ouhabaz, Maximal regularity for non-autonomous evolution equations,, Math. Ann., 363 (2015), 1117.
doi: 10.1007/s00208-015-1199-7. |
[31] |
R. Haller-Dintelmann, H.-C. Kaiser and J. Rehberg, Elliptic model problems including mixed boundary conditions and material heterogeneities,, J. Math. Pures Appl., 89 (2008), 25.
doi: 10.1016/j.matpur.2007.09.001. |
[32] |
R. Haller-Dintelmann, C. Meyer, J. Rehberg and A. Schiela, Hölder continuity and optimal control for nonsmooth elliptic problems,, Appl. Math. Optim., 60 (2009), 397.
doi: 10.1007/s00245-009-9077-x. |
[33] |
R. Haller-Dintelmann and J. Rehberg, Maximal parabolic regularity for divergence operators including mixed boundary conditions,, J. Differ. Equations, 247 (2009), 1354.
doi: 10.1016/j.jde.2009.06.001. |
[34] |
R. Haller-Dintelmann and J. Rehberg, Coercivity for elliptic operators and positivity of solutions on Lipschitz domains,, Arch. Math., 95 (2010), 457.
doi: 10.1007/s00013-010-0184-3. |
[35] |
R. Haller-Dintelmann and J. Rehberg, Maximal parabolic regularity for divergence operators on distribution spaces,, in Parabolic problems: The Herbert Amann Festschrift, 80 (2011), 313.
doi: 10.1007/978-3-0348-0075-4_17. |
[36] |
M. Hieber and J. Rehberg, Quasilinear parabolic systems with mixed boundary conditions on nonsmooth domains,, SIAM J. Math. Anal., 40 (2008), 292.
doi: 10.1137/070683829. |
[37] |
M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints,, Springer Netherlands, (2009).
doi: 10.1007/978-1-4020-8839-1. |
[38] |
A. Jonsson and H. Wallin, Function Spaces on Subsets of $\mathbbR^n$,, Harwood Academic Publishers, (1984).
|
[39] |
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications,, Academic Press, (1980).
doi: 10.1137/1.9780898719451. |
[40] |
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type,, American Mathematical Society (AMS), (1968).
|
[41] |
V. G. Maz'ya, Sobolev Spaces,, Second, (2011).
doi: 10.1007/978-3-662-09922-3. |
[42] |
E. M. Ouhabaz, Maximal regularity for non-autonomous evolution equations governed by forms having less regularity,, Arch. Math. (Basel), 105 (2015), 79.
doi: 10.1007/s00013-015-0783-0. |
[43] |
J. Prüss, Maximal regularity for evolution equations in $L^p$-spaces,, Conf. Semin. Mat. Univ. Bari, 285 (2002), 1.
|
[44] |
M. Struwe, On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems,, Manuscr. Math., 35 (1981), 125.
doi: 10.1007/BF01168452. |
[45] |
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North Holland Publishing Company, (1978).
|
show all references
References:
[1] |
H. Amann, Parabolic evolution equations and nonlinear boundary conditions,, J. Differ. Equations, 72 (1988), 201.
doi: 10.1016/0022-0396(88)90156-8. |
[2] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, in: Function spaces, 133 (1993), 9.
doi: 10.1007/978-3-663-11336-2_1. |
[3] |
H. Amann, Maximal regularity for nonautonomous evolution equations,, Adv. Nonlinear Stud., 4 (2004), 417.
|
[4] |
H. Amann, Linear and Quasilinear Parabolic Problems,, Birkhäuser, (1995).
doi: 10.1007/978-3-0348-9221-6. |
[5] |
W. Arendt, D. Dier, H. Laasri and E. M. Ouhabaz, Maximal regularity for evolution equations governed by non-autonomous forms,, Adv. Differential Equations, 19 (2014), 1043.
|
[6] |
P. Auscher, N. Badr, R. Haller-Dintelmann and J. Rehberg, The square root problem for second order, divergence form operators with mixed boundary conditions on $L^p$,, J. Evol. Eq., 15 (2015), 165.
doi: 10.1007/s00028-014-0255-1. |
[7] |
C. Bennett and R. Sharpley, Interpolation of Operators,, Academic Press, (1988).
|
[8] |
K. Brewster, D. Mitrea, I. Mitrea and M. Mitrea, Extending Sobolev functions with partially vanishing traces from locally $(\epsilon,\delta)$-domains and applications to mixed boundary problems,, J. Funct. Anal., 266 (2014), 4314.
doi: 10.1016/j.jfa.2014.02.001. |
[9] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Springer, (2011).
doi: 10.1007/978-0-387-70914-7. |
[10] |
E. Casas, J. C. de los Reyes and F. Tröltzsch, Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints,, SIAM J. Control Optim., 19 (2008), 616.
doi: 10.1137/07068240X. |
[11] |
P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, North Holland, (1978).
|
[12] |
R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5: Evolution Problems I,, Springer-Verlag, (1992).
doi: 10.1007/978-3-642-58090-1. |
[13] |
J. C. de los Reyes, P. Merino, J. Rehberg and F. Tröltzsch, Optimality conditions for state-constrained PDE control problems with time-dependent controls,, Control Cybernet., 37 (2008), 5.
|
[14] |
D. Dier, Non-autonomous maximal regularity for forms of bounded variation,, J. Math. Anal. Appl., 425 (2015), 33.
doi: 10.1016/j.jmaa.2014.12.006. |
[15] |
K. Disser, H.-C. Kaiser and J. Rehberg, Optimal Sobolev regularity for linear second-order divergence elliptic operators occuring in real-world problems,, SIAM J. Math. Anal., 47 (2015), 1719.
doi: 10.1137/140982969. |
[16] |
J. Elschner, J. Rehberg and G. Schmidt, Optimal regularity for elliptic transmission problems including $C^1$ interfaces,, Interfaces Free Bound., 9 (2007), 233.
doi: 10.4171/IFB/163. |
[17] |
A. F. M. ter Elst and J. Rehberg, Hölder estimates for second-order operators on domains with rough boundary,, Adv. Differential Equations, 20 (2015), 299.
|
[18] |
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in advanced mathematics, (1992).
|
[19] |
I. Fonseca and G. Parry, Equilibrium configurations of defective crystals,, Arch. Rat. Mech. Anal., 120 (1992), 245.
doi: 10.1007/BF00375027. |
[20] |
H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen,, Akademie-Verlag, (1974).
|
[21] |
Ch. Gallarati and M. Veraar, Maximal regularity for non-autonomous equations with measurable dependence on time,, , (). Google Scholar |
[22] |
M. Giaquinta and M. Struwe, An optimal regularity result for a class of quasilinear parabolic systems,, Manuscr. Math., 36 (1981), 223.
doi: 10.1007/BF01170135. |
[23] |
E. Giusti, Metodi Diretti nel Calcolo Delle Variazioni,, Unione Matematica Italiana, (1994).
|
[24] |
J. A. Griepentrog, W. Höppner, H.-C. Kaiser and J. Rehberg, A bi-Lipschitz continuous, volume preserving map from the unit ball onto a cube,, Note Mat., 28 (2008), 177.
doi: 10.1285/i15900932v28n1p177. |
[25] |
J. A. Griepentrog, K. Gröger, H. C. Kaiser and J. Rehberg, Interpolation for function spaces related to mixed boundary value problems,, Math. Nachr., 241 (2002), 110.
doi: 10.1002/1522-2616(200207)241:1<110::AID-MANA110>3.0.CO;2-R. |
[26] |
J. Griepentrog, Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces,, Adv. Differ. Equ., 12 (2007), 1031.
|
[27] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman, (1985).
doi: 10.1137/1.9781611972030. |
[28] |
K. Gröger, A $W^{1,p}$-estimate for solutions to mixed boundary value problems for second order elliptic differential equations,, Math. Ann., 283 (1989), 679.
doi: 10.1007/BF01442860. |
[29] |
K. Gröger, $W^{1,p}$-estimates of solutions to evolution equations corresponding to nonsmooth second order elliptic differential operators,, Nonlinear Anal., 18 (1992), 569.
doi: 10.1016/0362-546X(92)90211-V. |
[30] |
B. H. Haak and E. M. Ouhabaz, Maximal regularity for non-autonomous evolution equations,, Math. Ann., 363 (2015), 1117.
doi: 10.1007/s00208-015-1199-7. |
[31] |
R. Haller-Dintelmann, H.-C. Kaiser and J. Rehberg, Elliptic model problems including mixed boundary conditions and material heterogeneities,, J. Math. Pures Appl., 89 (2008), 25.
doi: 10.1016/j.matpur.2007.09.001. |
[32] |
R. Haller-Dintelmann, C. Meyer, J. Rehberg and A. Schiela, Hölder continuity and optimal control for nonsmooth elliptic problems,, Appl. Math. Optim., 60 (2009), 397.
doi: 10.1007/s00245-009-9077-x. |
[33] |
R. Haller-Dintelmann and J. Rehberg, Maximal parabolic regularity for divergence operators including mixed boundary conditions,, J. Differ. Equations, 247 (2009), 1354.
doi: 10.1016/j.jde.2009.06.001. |
[34] |
R. Haller-Dintelmann and J. Rehberg, Coercivity for elliptic operators and positivity of solutions on Lipschitz domains,, Arch. Math., 95 (2010), 457.
doi: 10.1007/s00013-010-0184-3. |
[35] |
R. Haller-Dintelmann and J. Rehberg, Maximal parabolic regularity for divergence operators on distribution spaces,, in Parabolic problems: The Herbert Amann Festschrift, 80 (2011), 313.
doi: 10.1007/978-3-0348-0075-4_17. |
[36] |
M. Hieber and J. Rehberg, Quasilinear parabolic systems with mixed boundary conditions on nonsmooth domains,, SIAM J. Math. Anal., 40 (2008), 292.
doi: 10.1137/070683829. |
[37] |
M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints,, Springer Netherlands, (2009).
doi: 10.1007/978-1-4020-8839-1. |
[38] |
A. Jonsson and H. Wallin, Function Spaces on Subsets of $\mathbbR^n$,, Harwood Academic Publishers, (1984).
|
[39] |
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications,, Academic Press, (1980).
doi: 10.1137/1.9780898719451. |
[40] |
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type,, American Mathematical Society (AMS), (1968).
|
[41] |
V. G. Maz'ya, Sobolev Spaces,, Second, (2011).
doi: 10.1007/978-3-662-09922-3. |
[42] |
E. M. Ouhabaz, Maximal regularity for non-autonomous evolution equations governed by forms having less regularity,, Arch. Math. (Basel), 105 (2015), 79.
doi: 10.1007/s00013-015-0783-0. |
[43] |
J. Prüss, Maximal regularity for evolution equations in $L^p$-spaces,, Conf. Semin. Mat. Univ. Bari, 285 (2002), 1.
|
[44] |
M. Struwe, On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems,, Manuscr. Math., 35 (1981), 125.
doi: 10.1007/BF01168452. |
[45] |
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North Holland Publishing Company, (1978).
|
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