Elliptic boundary value problem in Vanishing Mean Oscillation hypothesis

Maria Alessandra Ragusa

Commentationes Mathematicae Universitatis Carolinae (1999)

  • Volume: 40, Issue: 4, page 651-663
  • ISSN: 0010-2628

Abstract

top
In this note the well-posedness of the Dirichlet problem (1.2) below is proved in the class H 0 1 , p ( Ω ) for all 1 < p < ∞ and, as a consequence, the Hölder regularity of the solution u . ℒ is an elliptic second order operator with discontinuous coefficients ( V M O ) and the lower order terms belong to suitable Lebesgue spaces.

How to cite

top

Ragusa, Maria Alessandra. "Elliptic boundary value problem in Vanishing Mean Oscillation hypothesis." Commentationes Mathematicae Universitatis Carolinae 40.4 (1999): 651-663. <http://eudml.org/doc/248421>.

@article{Ragusa1999,
abstract = {In this note the well-posedness of the Dirichlet problem (1.2) below is proved in the class $H^\{1,p\}_0(\Omega )$ for all $1<p<\infty $ and, as a consequence, the Hölder regularity of the solution $u$. $\mathcal \{L\}$ is an elliptic second order operator with discontinuous coefficients $(VMO)$ and the lower order terms belong to suitable Lebesgue spaces.},
author = {Ragusa, Maria Alessandra},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {elliptic equations; Morrey spaces; elliptic equations; Morrey spaces},
language = {eng},
number = {4},
pages = {651-663},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Elliptic boundary value problem in Vanishing Mean Oscillation hypothesis},
url = {http://eudml.org/doc/248421},
volume = {40},
year = {1999},
}

TY - JOUR
AU - Ragusa, Maria Alessandra
TI - Elliptic boundary value problem in Vanishing Mean Oscillation hypothesis
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 4
SP - 651
EP - 663
AB - In this note the well-posedness of the Dirichlet problem (1.2) below is proved in the class $H^{1,p}_0(\Omega )$ for all $1<p<\infty $ and, as a consequence, the Hölder regularity of the solution $u$. $\mathcal {L}$ is an elliptic second order operator with discontinuous coefficients $(VMO)$ and the lower order terms belong to suitable Lebesgue spaces.
LA - eng
KW - elliptic equations; Morrey spaces; elliptic equations; Morrey spaces
UR - http://eudml.org/doc/248421
ER -

References

top
  1. Agmon S., Lectures on Elliptic boundary Value Problems, Van Nostrand, N.J., 1965. Zbl0142.37401MR0178246
  2. Browder F.E., Strongly elliptic systems of differential equations, in: Contributions to the Theory of Partial Differential Equations, Princeton University Press, Princeton N.J., 1954, pp.15-51. Zbl0057.32901MR0067306
  3. Bers L., Schechter M., Elliptic Equations In Partial Differential Equations, Interscience, New York, 1964, pp.131-299. MR0165224
  4. Chiarenza F., Frasca M., Longo P., Interior W 2 , p estimates for non-divergence elliptic equations with discontinuous coefficients, Ricerche Mat. 40 (1991), 149-168. (1991) MR1191890
  5. Chiarenza F., Frasca M., Longo P., W 2 , p -solvability of the Dirichlet problem for nondivergence elliptic equations with V M O coefficients, Trans. Amer. Math. Soc. 336 (1993), 841-853. (1993) MR1088476
  6. Coifman R., Rochberg R., Weiss G., Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635. (1976) Zbl0326.32011MR0412721
  7. De Giorgi E., Sulla differenziabilita' e l'analiticita' delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. 3 (1957), 25-43. (1957) Zbl0084.31901MR0093649
  8. Di Fazio G., L p Estimates for divergence form elliptic equations with discontinuous coefficients, Boll. U.M.I. (7) 10-A (1996), 409-420. (1996) MR1405255
  9. Friedman A., Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. Zbl1121.46001MR0445088
  10. Friedrichs K.O., The identity of weak and strong extensions of differential operators, Trans. Amer. Math. Soc. 55 (1944), 132-151. (1944) Zbl0061.26201MR0009701
  11. Friedrichs K.O., On the differentiability of the solutions of linear elliptic equations, Comm. Pure Appl. Math. 6 (1953), 299-326. (1953) MR0058828
  12. Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, 1983. Zbl1042.35002MR0737190
  13. Garding L., Dirichlet's problem for linear elliptic partial differential equations, Math. Scand. 1 (1953), 55-72. (1953) Zbl0053.39101MR0064979
  14. John F., Nirenberg L., On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426. (1961) Zbl0102.04302MR0131498
  15. Ladyzenskaya O.A., Uraltśeva N.N., Linear and Quasilinear Elliptic Equations, English Translation: Academic Press, New York, 2nd Russian ed., 1973. 
  16. Lax P.D., On Cauchy's problem for hyperbolic equations and the differentiability of solutions of elliptic equations, Comm. Pure Appl. Math. 8 (1955), 615-633. (1955) Zbl0067.07502MR0078558
  17. Miranda C., Partial Differential Equations of Elliptic Type, second revised edition, Springer Verlag, 1970. Zbl0198.14101MR0284700
  18. Miranda C., Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui, Annali di Matematica 63 (1963), 353-386. (1963) Zbl0156.34001MR0170090
  19. Miranda C., Alcune osservazioni sulla maggiorazione in L ν delle soluzioni deboli delle equazioni ellittiche del secondo ordine, Annali di Matematica 61 (1963), 151-170. (1963) MR0177187
  20. Neri U., Some properties of bounded mean oscillation, Studia Math. 41 (1977), 63-75. (1977) MR0445210
  21. Nirenberg L., On nonlinear elliptic partial differential equations and Hölder continuity, Comm. Pure Appl. Math. 6 (1953), 103-156. (1953) Zbl0050.09801MR0064986
  22. Nirenberg L., Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math. 5 (1955), 649-675. (1955) Zbl0067.07602MR0075415
  23. Sarason D., On functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405. (1975) MR0377518
  24. Stampacchia G., Contributi alla regolarizzazione delle soluzioni dei problemi al contorno per equazioni del secondo ordine ellittiche, Ann. Scuola Norm. Sup. Pisa (3) 12 (1958), 223-245. (1958) Zbl0082.09701MR0125313
  25. Spampacchia G., I problemi al contorno per le equazioni differentiali di tipo ellittico, Atti VI Congr. Un. Mat. Ital., Napoli Edizioni Cremonese, 1960, pp.21-44. MR0123807
  26. Stampacchia G., Le probléme de Dirichlet pour les équations elliptiques du second ordre á coefficients discontinuous, Ann. Inst. Fourier 15 (1965), 189-258. (1965) MR0192177
  27. Stampacchia G., Equations elliptiques du second ordre á coefficients discontinues, Séminaire de Mathématiques Supérieures 16, Les Presses de l'Université de Montréal, 1966. MR0251373
  28. Trudinger N.S., Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa (3) 27 (1973), 265-308. (1973) Zbl0279.35025MR0369884

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.