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March  2016, 5(1): 147-184. doi: 10.3934/eect.2016.5.147

## 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

Received  March 2015 Revised  February 2016 Published  March 2016

In this paper we establish Hölder estimates for solutions to nonautonomous parabolic equations on non-smooth domains which are complemented with mixed boundary conditions. The corresponding elliptic operators are of divergence type, the coefficient matrix of which depends only measurably on time. These results are in the tradition of the classical book of Ladyshenskaya et al. [40], which also serves as the starting point for our investigations.
Citation: Hannes Meinlschmidt, Joachim Rehberg. Hölder-estimates for non-autonomous parabolic problems with rough data. Evolution Equations & Control Theory, 2016, 5 (1) : 147-184. doi: 10.3934/eect.2016.5.147
##### 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. Google Scholar [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. Google Scholar [3] H. Amann, Maximal regularity for nonautonomous evolution equations,, Adv. Nonlinear Stud., 4 (2004), 417. Google Scholar [4] H. Amann, Linear and Quasilinear Parabolic Problems,, Birkhäuser, (1995). doi: 10.1007/978-3-0348-9221-6. Google Scholar [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. Google Scholar [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. Google Scholar [7] C. Bennett and R. Sharpley, Interpolation of Operators,, Academic Press, (1988). Google Scholar [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. Google Scholar [9] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Springer, (2011). doi: 10.1007/978-0-387-70914-7. Google Scholar [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. Google Scholar [11] P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, North Holland, (1978). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in advanced mathematics, (1992). Google Scholar [19] I. Fonseca and G. Parry, Equilibrium configurations of defective crystals,, Arch. Rat. Mech. Anal., 120 (1992), 245. doi: 10.1007/BF00375027. Google Scholar [20] H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen,, Akademie-Verlag, (1974). Google Scholar [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. Google Scholar [23] E. Giusti, Metodi Diretti nel Calcolo Delle Variazioni,, Unione Matematica Italiana, (1994). Google Scholar [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. Google Scholar [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. Google Scholar [26] J. Griepentrog, Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces,, Adv. Differ. Equ., 12 (2007), 1031. Google Scholar [27] P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman, (1985). doi: 10.1137/1.9781611972030. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [38] A. Jonsson and H. Wallin, Function Spaces on Subsets of $\mathbbR^n$,, Harwood Academic Publishers, (1984). Google Scholar [39] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications,, Academic Press, (1980). doi: 10.1137/1.9780898719451. Google Scholar [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). Google Scholar [41] V. G. Maz'ya, Sobolev Spaces,, Second, (2011). doi: 10.1007/978-3-662-09922-3. Google Scholar [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. Google Scholar [43] J. Prüss, Maximal regularity for evolution equations in $L^p$-spaces,, Conf. Semin. Mat. Univ. Bari, 285 (2002), 1. Google Scholar [44] M. Struwe, On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems,, Manuscr. Math., 35 (1981), 125. doi: 10.1007/BF01168452. Google Scholar [45] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North Holland Publishing Company, (1978). Google Scholar

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. Google Scholar [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. Google Scholar [3] H. Amann, Maximal regularity for nonautonomous evolution equations,, Adv. Nonlinear Stud., 4 (2004), 417. Google Scholar [4] H. Amann, Linear and Quasilinear Parabolic Problems,, Birkhäuser, (1995). doi: 10.1007/978-3-0348-9221-6. Google Scholar [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. Google Scholar [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. Google Scholar [7] C. Bennett and R. Sharpley, Interpolation of Operators,, Academic Press, (1988). Google Scholar [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. Google Scholar [9] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Springer, (2011). doi: 10.1007/978-0-387-70914-7. Google Scholar [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. Google Scholar [11] P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, North Holland, (1978). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in advanced mathematics, (1992). Google Scholar [19] I. Fonseca and G. Parry, Equilibrium configurations of defective crystals,, Arch. Rat. Mech. Anal., 120 (1992), 245. doi: 10.1007/BF00375027. Google Scholar [20] H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen,, Akademie-Verlag, (1974). Google Scholar [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. Google Scholar [23] E. Giusti, Metodi Diretti nel Calcolo Delle Variazioni,, Unione Matematica Italiana, (1994). Google Scholar [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. Google Scholar [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. Google Scholar [26] J. Griepentrog, Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces,, Adv. Differ. Equ., 12 (2007), 1031. Google Scholar [27] P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman, (1985). doi: 10.1137/1.9781611972030. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [38] A. Jonsson and H. Wallin, Function Spaces on Subsets of $\mathbbR^n$,, Harwood Academic Publishers, (1984). Google Scholar [39] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications,, Academic Press, (1980). doi: 10.1137/1.9780898719451. Google Scholar [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). Google Scholar [41] V. G. Maz'ya, Sobolev Spaces,, Second, (2011). doi: 10.1007/978-3-662-09922-3. Google Scholar [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. Google Scholar [43] J. Prüss, Maximal regularity for evolution equations in $L^p$-spaces,, Conf. Semin. Mat. Univ. Bari, 285 (2002), 1. Google Scholar [44] M. Struwe, On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems,, Manuscr. Math., 35 (1981), 125. doi: 10.1007/BF01168452. Google Scholar [45] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North Holland Publishing Company, (1978). Google Scholar
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