American Institute of Mathematical Sciences

Advanced
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

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 & 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.  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.  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.  doi: 10.4171/PM/1969.  Google Scholar

[5]

H. Beirão da Veiga, H-log spaces of continuous functions, potentials, and elliptic boundary value problems,, , (2015).   Google Scholar

[6]

L. Bers, F. John and M. Schechter, Partial Differential Equations,, John Wiley and Sons, (1964).   Google Scholar

[7]

C. C. Burch, The Dini condition and regularity of weak solutions of elliptic equations,, J. Diff. Eq., 30 (1978), 308.  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, (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, (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.  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.   Google Scholar

[12]

V. A. Solonnikov, On estimates of Green's tensors for certain boundary value problems,, Doklady Akad. Nauk., 130 (1960), 128.   Google Scholar

show all references

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.  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.  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.  doi: 10.4171/PM/1969.  Google Scholar

[5]

H. Beirão da Veiga, H-log spaces of continuous functions, potentials, and elliptic boundary value problems,, , (2015).   Google Scholar

[6]

L. Bers, F. John and M. Schechter, Partial Differential Equations,, John Wiley and Sons, (1964).   Google Scholar

[7]

C. C. Burch, The Dini condition and regularity of weak solutions of elliptic equations,, J. Diff. Eq., 30 (1978), 308.  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, (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, (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.  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.   Google Scholar

[12]

V. A. Solonnikov, On estimates of Green's tensors for certain boundary value problems,, Doklady Akad. Nauk., 130 (1960), 128.   Google Scholar

[1]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248
[2]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442
[3]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052
[4]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340
[5]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267
[6]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247
[7]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320
[8]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272
[9]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103
[10]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164
[11]

Honglei Lang, Yunhe Sheng. Linearization of the higher analogue of Courant algebroids. Journal of Geometric Mechanics, 2020, 12 (4) : 585-606. doi: 10.3934/jgm.2020025
[12]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436
[13]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318
[14]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119
[15]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252
[16]

Nicolas Rougerie. On two properties of the Fisher information. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020049
[17]

Agnaldo José Ferrari, Tatiana Miguel Rodrigues de Souza. Rotated $ A_n $-lattice codes of full diversity. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020118
[18]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384
[19]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469
[20]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (57)
  • HTML views (1)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences