Elliptic boundary value problems in spaces of continuous functions

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February 2016, 9(1): 43-52. doi: 10.3934/dcdss.2016.9.43

Elliptic boundary value problems in spaces of continuous functions

Hugo Beirão da Veiga ^1,

1.	Dipartimento di Matematica Applicata, Università di Pisa, Via Buonarroti 1/C, 56127 Pisa

Received September 2014 Revised February 2015 Published December 2015

Abstract
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In these notes we consider second order linear elliptic boundary value problems in the framework of different spaces on continuous functions. We appeal to a general formulation which contains some interesting particular cases as, for instance, a new class of functional spaces, called here Hölog spaces and denoted by the symbol $\, C^{0,\,\lambda}_\alpha(\overline{\Omega}) \,,$ $\,0 \leq\,\lambda<\,1\,,$ and $\,\alpha \in\,\mathbb{R}\,.$ One has the following inclusions $$ C^{0,\,\lambda+\,\epsilon}(\overline{\Omega})\subset \,C^{0,\,\lambda}_\alpha(\overline{\Omega})\subset \,C^{0,\,\lambda}(\overline{\Omega}) \subset \,C^{0,\,\lambda,}_{-\alpha}(\overline{\Omega}) \subset\,C^{0,\,\lambda-\,\epsilon}(\overline{\Omega})\,, $$ for $\,\alpha>\,0\,$ ($\epsilon >\,0\,$ arbitrarily small). Roughly speaking, for each fixed $\,\lambda\,,$ the family $\, C^{0,\,\lambda}_\alpha(\overline{\Omega}) \,$ is a refinement of the single Hölder classical space $\, C^{0,\,\lambda}(\overline{\Omega})=\,C^{0,\,\lambda}_0(\overline{\Omega})\,.$ On the other hand, for $\,\lambda=\,0\,$ and $\,\alpha>\,0\,,$ $\,C^{0,\,0}_\alpha(\overline{\Omega})=\,\, D^{0,\,\alpha}(\overline{\Omega})\,$ is a Log space. The more interesting feature is that, as for classical Hölder (and Sobolev) spaces, full regularity occurs. namely, for each $\,\lambda>\,0\,$ and arbitrary real $\,\alpha\,,$ $\,\nabla^2\,u $ and $\,f\,$ enjoy the same $\, C^{0,\,\lambda}_\alpha(\overline{\Omega}) \,$ regularity. All the above setup is presented as part of a more general picture.

Keywords: data spaces of continuous functions, full regularity., Linear elliptic boundary value problems, continuity properties of higher order derivatives, classical solutions.

Mathematics Subject Classification: Primary: 31B10; Secondary: 31B35, 33E30, 35A09, 35B65, 35J25, 58F15, 58F1.

Citation: Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43

References:

[1]	H. Beirão da Veiga, On the solutions in the large of the two-dimensional flow of a nonviscous incompressible fluid, J. Diff. Eq., 54 (1984), 373-389. doi: 10.1016/0022-0396(84)90149-9.
[2]	H. Beirão da Veiga, Concerning the existence of classical solutions to the Stokes system. On the minimal assumptions problem, J. Math. Fluid Mech., 16 (2014), 539-550. doi: 10.1007/s00021-014-0170-9.
[3]	H. Beirão da Veiga, An overview on classical solutions to $2-D$ Euler equations and to elliptic boundary value problems, in Recent Progress in the Theory of the Euler and Navier-Stokes Equations (eds. J. C. Robinson, J. L. Rodrigo, W. Sadowski and A. V. López), London Math. Soc. Lecture Notes, (forthcoming).
[4]	H. Beirão da Veiga, On some regularity results for the stationary Stokes system and the $2-D$ Euler equations, Portugaliae Math., 72 (2015), 285-307. doi: 10.4171/PM/1969.
[5]	H. Beirão da Veiga, H-log spaces of continuous functions, potentials, and elliptic boundary value problems, arXiv:1503.04173, 2015.
[6]	L. Bers, F. John and M. Schechter, Partial Differential Equations, John Wiley and Sons, Inc., New-York, 1964.
[7]	C. C. Burch, The Dini condition and regularity of weak solutions of elliptic equations, J. Diff. Eq., 30 (1978), 308-323. doi: 10.1016/0022-0396(78)90003-7.
[8]	D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces Foundations and Harmonic Analysis, Springer, Basel 2013. doi: 10.1007/978-3-0348-0548-3.
[9]	O. A. Ladyzenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New-York, 1969.
[10]	V. L. Shapiro, Generalized and classical solutions of the nonlinear stationary Navier-Stokes equations, Trans. Amer. Math. Soc., 216 (1976), 61-79. doi: 10.1090/S0002-9947-1976-0390550-X.
[11]	I. I. Sharapudinov, The basis property of the Haar system in the space $L^{p(t)}[0,1]$, and the principle of localization in the mean, Mat. Sb. (N.S.), 130 (1986), 275-283, 286.
[12]	V. A. Solonnikov, On estimates of Green's tensors for certain boundary value problems, Doklady Akad. Nauk., 130 (1960), 128-131.

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References:

[1]	H. Beirão da Veiga, On the solutions in the large of the two-dimensional flow of a nonviscous incompressible fluid, J. Diff. Eq., 54 (1984), 373-389. doi: 10.1016/0022-0396(84)90149-9.
[2]	H. Beirão da Veiga, Concerning the existence of classical solutions to the Stokes system. On the minimal assumptions problem, J. Math. Fluid Mech., 16 (2014), 539-550. doi: 10.1007/s00021-014-0170-9.
[3]	H. Beirão da Veiga, An overview on classical solutions to $2-D$ Euler equations and to elliptic boundary value problems, in Recent Progress in the Theory of the Euler and Navier-Stokes Equations (eds. J. C. Robinson, J. L. Rodrigo, W. Sadowski and A. V. López), London Math. Soc. Lecture Notes, (forthcoming).
[4]	H. Beirão da Veiga, On some regularity results for the stationary Stokes system and the $2-D$ Euler equations, Portugaliae Math., 72 (2015), 285-307. doi: 10.4171/PM/1969.
[5]	H. Beirão da Veiga, H-log spaces of continuous functions, potentials, and elliptic boundary value problems, arXiv:1503.04173, 2015.
[6]	L. Bers, F. John and M. Schechter, Partial Differential Equations, John Wiley and Sons, Inc., New-York, 1964.
[7]	C. C. Burch, The Dini condition and regularity of weak solutions of elliptic equations, J. Diff. Eq., 30 (1978), 308-323. doi: 10.1016/0022-0396(78)90003-7.
[8]	D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces Foundations and Harmonic Analysis, Springer, Basel 2013. doi: 10.1007/978-3-0348-0548-3.
[9]	O. A. Ladyzenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New-York, 1969.
[10]	V. L. Shapiro, Generalized and classical solutions of the nonlinear stationary Navier-Stokes equations, Trans. Amer. Math. Soc., 216 (1976), 61-79. doi: 10.1090/S0002-9947-1976-0390550-X.
[11]	I. I. Sharapudinov, The basis property of the Haar system in the space $L^{p(t)}[0,1]$, and the principle of localization in the mean, Mat. Sb. (N.S.), 130 (1986), 275-283, 286.
[12]	V. A. Solonnikov, On estimates of Green's tensors for certain boundary value problems, Doklady Akad. Nauk., 130 (1960), 128-131.

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