# American Institute of Mathematical Sciences

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

## Elliptic boundary value problems in spaces of continuous functions

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

Received  September 2014 Revised  February 2015 Published  December 2015

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.
Citation: Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete & 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.  Google Scholar [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.  Google Scholar [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, ().   Google Scholar [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.  Google Scholar [5] H. Beirão da Veiga, H-log spaces of continuous functions, potentials, and elliptic boundary value problems, arXiv:1503.04173, 2015. Google Scholar [6] L. Bers, F. John and M. Schechter, Partial Differential Equations, John Wiley and Sons, Inc., New-York, 1964.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] O. A. Ladyzenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New-York, 1969.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] V. A. Solonnikov, On estimates of Green's tensors for certain boundary value problems, Doklady Akad. Nauk., 130 (1960), 128-131. Google Scholar

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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.  Google Scholar [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.  Google Scholar [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, ().   Google Scholar [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.  Google Scholar [5] H. Beirão da Veiga, H-log spaces of continuous functions, potentials, and elliptic boundary value problems, arXiv:1503.04173, 2015. Google Scholar [6] L. Bers, F. John and M. Schechter, Partial Differential Equations, John Wiley and Sons, Inc., New-York, 1964.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] O. A. Ladyzenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New-York, 1969.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] V. A. Solonnikov, On estimates of Green's tensors for certain boundary value problems, Doklady Akad. Nauk., 130 (1960), 128-131. Google Scholar
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