Hölder-estimates for non-autonomous parabolic problems with rough data

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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

Hannes Meinlschmidt ^1, and Joachim Rehberg ^2,

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

Abstract
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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.

Keywords: second order parabolic equations, Hölder continuity of the solution., non-smooth data, Linear.

Mathematics Subject Classification: 35B65, 35K10, 35K1.

Citation: Hannes Meinlschmidt, Joachim Rehberg. Hölder-estimates for non-autonomous parabolic problems with rough data. Evolution Equations and 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), 201-269. doi: 10.1016/0022-0396(88)90156-8.
[2]	H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in: Function spaces, differential operators and nonlinear analysis, (eds. H.-J. Schmeisser et al.), Teubner, Stuttgart, 133 (1993), 9-126. doi: 10.1007/978-3-663-11336-2_1.
[3]	H. Amann, Maximal regularity for nonautonomous evolution equations, Adv. Nonlinear Stud., 4 (2004), 417-430.
[4]	H. Amann, Linear and Quasilinear Parabolic Problems, Birkhäuser, Basel-Boston-Berlin, 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), 1043-1066.
[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), 165-208. doi: 10.1007/s00028-014-0255-1.
[7]	C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston etc., 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), 4314-4421. doi: 10.1016/j.jfa.2014.02.001.
[9]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 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), 616-643. doi: 10.1137/07068240X.
[11]	P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam-New York-Oxford, 1978.
[12]	R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5: Evolution Problems I, Springer-Verlag, Berlin, 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), 5-38.
[14]	D. Dier, Non-autonomous maximal regularity for forms of bounded variation, J. Math. Anal. Appl., 425 (2015), 33-54. 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), 1719-1746. 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), 233-252. 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), 299-360.
[18]	L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in advanced mathematics, CRC Press, Boca Raton-New York-London-Tokyo, 1992.
[19]	I. Fonseca and G. Parry, Equilibrium configurations of defective crystals, Arch. Rat. Mech. Anal., 120 (1992), 245-283. doi: 10.1007/BF00375027.
[20]	H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974.
[21]	Ch. Gallarati and M. Veraar, Maximal regularity for non-autonomous equations with measurable dependence on time,, , ().
[22]	M. Giaquinta and M. Struwe, An optimal regularity result for a class of quasilinear parabolic systems, Manuscr. Math., 36 (1981), 223-239. doi: 10.1007/BF01170135.
[23]	E. Giusti, Metodi Diretti nel Calcolo Delle Variazioni, Unione Matematica Italiana, Bologna, 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), 177-193. 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), 110-120. 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), 1031-1078.
[27]	P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 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), 679-687. 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), 569-577. 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), 1117-1145. 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), 25-48. 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), 397-428. 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), 1354-1396. 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), 457-468. 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, (eds. J. Escher, P. Guidotti, M. Hieber, P. Mucha, J. Prüss, Y. Shibata, G. Simonett, C. Walker, W. Zajaczkowski), Springer, Basel, 80 (2011), 313-341. 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), 292-305. 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, Chur-London-Paris-Utrecht-New York, 1984.
[39]	D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 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), Providence, RI, 1968.
[41]	V. G. Maz'ya, Sobolev Spaces, Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften, 342. Springer, Heidelberg, 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), 79-91. 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), 1-39.
[44]	M. Struwe, On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems, Manuscr. Math., 35 (1981), 125-145. doi: 10.1007/BF01168452.
[45]	H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North Holland Publishing Company, Amsterdam-New York-Oxford, 1978.

show all references

References:

[1]	H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differ. Equations, 72 (1988), 201-269. doi: 10.1016/0022-0396(88)90156-8.
[2]	H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in: Function spaces, differential operators and nonlinear analysis, (eds. H.-J. Schmeisser et al.), Teubner, Stuttgart, 133 (1993), 9-126. doi: 10.1007/978-3-663-11336-2_1.
[3]	H. Amann, Maximal regularity for nonautonomous evolution equations, Adv. Nonlinear Stud., 4 (2004), 417-430.
[4]	H. Amann, Linear and Quasilinear Parabolic Problems, Birkhäuser, Basel-Boston-Berlin, 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), 1043-1066.
[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), 165-208. doi: 10.1007/s00028-014-0255-1.
[7]	C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston etc., 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), 4314-4421. doi: 10.1016/j.jfa.2014.02.001.
[9]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 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), 616-643. doi: 10.1137/07068240X.
[11]	P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam-New York-Oxford, 1978.
[12]	R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5: Evolution Problems I, Springer-Verlag, Berlin, 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), 5-38.
[14]	D. Dier, Non-autonomous maximal regularity for forms of bounded variation, J. Math. Anal. Appl., 425 (2015), 33-54. 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), 1719-1746. 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), 233-252. 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), 299-360.
[18]	L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in advanced mathematics, CRC Press, Boca Raton-New York-London-Tokyo, 1992.
[19]	I. Fonseca and G. Parry, Equilibrium configurations of defective crystals, Arch. Rat. Mech. Anal., 120 (1992), 245-283. doi: 10.1007/BF00375027.
[20]	H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974.
[21]	Ch. Gallarati and M. Veraar, Maximal regularity for non-autonomous equations with measurable dependence on time,, , ().
[22]	M. Giaquinta and M. Struwe, An optimal regularity result for a class of quasilinear parabolic systems, Manuscr. Math., 36 (1981), 223-239. doi: 10.1007/BF01170135.
[23]	E. Giusti, Metodi Diretti nel Calcolo Delle Variazioni, Unione Matematica Italiana, Bologna, 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), 177-193. 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), 110-120. 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), 1031-1078.
[27]	P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 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), 679-687. 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), 569-577. 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), 1117-1145. 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), 25-48. 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), 397-428. 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), 1354-1396. 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), 457-468. 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, (eds. J. Escher, P. Guidotti, M. Hieber, P. Mucha, J. Prüss, Y. Shibata, G. Simonett, C. Walker, W. Zajaczkowski), Springer, Basel, 80 (2011), 313-341. 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), 292-305. 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, Chur-London-Paris-Utrecht-New York, 1984.
[39]	D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 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), Providence, RI, 1968.
[41]	V. G. Maz'ya, Sobolev Spaces, Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften, 342. Springer, Heidelberg, 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), 79-91. 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), 1-39.
[44]	M. Struwe, On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems, Manuscr. Math., 35 (1981), 125-145. doi: 10.1007/BF01168452.
[45]	H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North Holland Publishing Company, Amsterdam-New York-Oxford, 1978.

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