The parabolic mixed Cauchy-Dirichlet problem in spaces of functions which are hölder continuous with respect to space variables

Davide Guidetti

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1996)

  • Volume: 7, Issue: 3, page 161-168
  • ISSN: 1120-6330

Abstract

top
We give a new proof, based on analytic semigroup methods, of a maximal regularity result concerning the classical Cauchy-Dirichlet's boundary value problem for second order parabolic equations. More specifically, we find necessary and sufficient conditions on the data in order to have a strict solution u which is bounded with values in C 2 + θ ⁢ Ω ¯ (0 < < 1), with ∂ t ⁡ u bounded with values in C θ ⁢ Ω ¯ .

How to cite

top

Guidetti, Davide. "The parabolic mixed Cauchy-Dirichlet problem in spaces of functions which are hölder continuous with respect to space variables." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 7.3 (1996): 161-168. <http://eudml.org/doc/244293>.

@article{Guidetti1996,
abstract = {We give a new proof, based on analytic semigroup methods, of a maximal regularity result concerning the classical Cauchy-Dirichlet's boundary value problem for second order parabolic equations. More specifically, we find necessary and sufficient conditions on the data in order to have a strict solution \( u \) which is bounded with values in \( C^\{2 + \theta\} (\overline\{\Omega\}) \)(0 < < 1), with\( \partial\_\{t\} u \) bounded with values in \( C^\{\theta\} (\overline\{\Omega\}) \).},
author = {Guidetti, Davide},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Parabolic equations; Cauchy-Dirichlet problem; Maximal regularity; Analytic semigroups; analytic semigroup; maximal regularity; strict solution},
language = {eng},
month = {12},
number = {3},
pages = {161-168},
publisher = {Accademia Nazionale dei Lincei},
title = {The parabolic mixed Cauchy-Dirichlet problem in spaces of functions which are hölder continuous with respect to space variables},
url = {http://eudml.org/doc/244293},
volume = {7},
year = {1996},
}

TY - JOUR
AU - Guidetti, Davide
TI - The parabolic mixed Cauchy-Dirichlet problem in spaces of functions which are hölder continuous with respect to space variables
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1996/12//
PB - Accademia Nazionale dei Lincei
VL - 7
IS - 3
SP - 161
EP - 168
AB - We give a new proof, based on analytic semigroup methods, of a maximal regularity result concerning the classical Cauchy-Dirichlet's boundary value problem for second order parabolic equations. More specifically, we find necessary and sufficient conditions on the data in order to have a strict solution \( u \) which is bounded with values in \( C^{2 + \theta} (\overline{\Omega}) \)(0 < < 1), with\( \partial_{t} u \) bounded with values in \( C^{\theta} (\overline{\Omega}) \).
LA - eng
KW - Parabolic equations; Cauchy-Dirichlet problem; Maximal regularity; Analytic semigroups; analytic semigroup; maximal regularity; strict solution
UR - http://eudml.org/doc/244293
ER -

References

top
  1. ACQUISTAPACE, P. - TERRENI, B., Hölder classes with boundary conditions as interpolation spaces. Math. Zeit., 195, 1987, 451-471. Zbl0635.35024MR900340DOI10.1007/BF01166699
  2. BOLLEY, P. - CAMUS, J. - THE LAI, P., Estimation de la résolvante du problème de Dirichlet dans les espaces de Hölder. C. R. Acad. Sci. Paris, 305, Serie I, 1987, 253-256. Zbl0634.35055MR907955
  3. GUIDETTI, D., On elliptic problems in Besov spaces. Math. Nachr., 152, 1991, 247-275. Zbl0767.46027MR1121237DOI10.1002/mana.19911520120
  4. KNERR, B., Parabolic interior Schauder estimates by the maximum principle. Arch. Rat. Mech. Anal., 75, 1980, 51-58. Zbl0468.35014MR592103DOI10.1007/BF00284620
  5. KRUZHKOV, S. - CASTRO, A. - LOPEZ, M., Mayoraciones de Schauder y theorema de existencia de las soluciónes del problema de Cauchy para ecuaciones parabolicas lineales y no lineales (I). Revista Ciencias Matemáticas, vol. 1, n. 1, 1980, 55-76. Zbl0572.35046MR622442
  6. LOPEZ MORALES, M., Primer problema de contorno para ecuaciones parabolicas lineales y no lineales. Revista Ciencias Matemáticas, vol. 13, n. 1, 1992, 3-20. MR1206783
  7. LUNARDI, A., Analytic semigroups and optimal regularity in the parabolic problems. Progress in Nonlinear Differential Equations and Their Applications, vol. 16, Birkhäuser, 1995. Zbl0816.35001MR1329547DOI10.1007/978-3-0348-9234-6
  8. LUNARDI, A. - SINESTRARI, E. - VON WAHL, W., A semigroup approach to the time dependent parabolic initial boundary value problem. Diff. Int. Equations, 63, 1992, 88-116. Zbl0596.45019
  9. SINESTRARI, E. - VON WAHL, W., On the solutions of the first boundary value problem for the linear parabolic problem. Proc. Royal Soc. Edinburgh, 108A, 1988, 339-355. Zbl0664.35041MR943808DOI10.1017/S0308210500014712
  10. SOLONNIKOV, V. A., On the boundary value problems for linear parabolic systems of differential equations of general form. Proc. Steklov Inst. Math., 83 (1965), (ed. O. A. Ladyzenskaya); Amer. Math. Soc, 1967. Zbl0164.12502MR211083
  11. STEWART, H. B., Generation of analytic semigroups by strongly elliptic operators under general boundary conditions. Trans. Amer. Math. Soc., 259, 1980, 299-310. Zbl0451.35033MR561838DOI10.2307/1998159

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.