Weighted Miranda-Talenti inequality and applications to equations with discontinuous coefficients

Salvatore Leonardi

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 1, page 43-59
  • ISSN: 0010-2628

Abstract

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Let Ω be an open bounded set in ℝ n ( n ≥ 2 ) , with C 2 boundary, and N p , λ ( Ω ) ( 1 < p < + ∞ , 0 ≤ λ < n ) be a weighted Morrey space. In this note we prove a weighted version of the Miranda-Talenti inequality and we exploit it to show that, under a suitable condition of Cordes type, the Dirichlet problem: ∑ i , j = 1 n a i j ( x ) ∂ 2 u ∂ x i ∂ x j = f ( x ) ∈ N p , λ ( Ω ) in Ω u = 0 on ∂ Ω has a unique strong solution in the functional space u ∈ W 2 , p ∩ W o 1 , p ( Ω ) : ∂ 2 u ∂ x i ∂ x j ∈ N p , λ ( Ω ) , i , j = 1 , 2 , ... , n .

How to cite

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Leonardi, Salvatore. "Weighted Miranda-Talenti inequality and applications to equations with discontinuous coefficients." Commentationes Mathematicae Universitatis Carolinae 43.1 (2002): 43-59. <http://eudml.org/doc/248992>.

@article{Leonardi2002,
abstract = {Let $\Omega $ be an open bounded set in $\mathbb \{R\}^\{n\}$$(n\ge 2)$, with $C^2$ boundary, and $N^\{p,\lambda \}(\Omega )$ ($1 < p < +\infty $, $0\le \lambda < n$) be a weighted Morrey space. In this note we prove a weighted version of the Miranda-Talenti inequality and we exploit it to show that, under a suitable condition of Cordes type, the Dirichlet problem: \[ \left\lbrace \begin\{array\}\{ll\}\sum \_\{i,j=1\}^n a\_\{ij\}(x) \frac\{\partial ^2 u\}\{\partial x\_i \partial x\_j\} = f(x) \in N^\{p,\lambda \}(\Omega ) \quad & \text\{ in \} \Omega \ u=0 & \text\{ on \} \partial \Omega \end\{array\}\right.\] has a unique strong solution in the functional space \[ \left\lbrace u \in W^\{2,p\} \cap W^\{1,p\}\_o(\Omega ) : \frac\{\partial ^2 u\}\{\partial x\_i \partial x\_j\} \in N^\{p,\lambda \}(\Omega ), i,j=1,2,\,\ldots , n\right\rbrace . \]},
author = {Leonardi, Salvatore},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Miranda-Talenti inequality; nonvariational elliptic equations; Hölder regularity; Miranda-Talenti inequality; Hölder regularity; nonvariational elliptic equations},
language = {eng},
number = {1},
pages = {43-59},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Weighted Miranda-Talenti inequality and applications to equations with discontinuous coefficients},
url = {http://eudml.org/doc/248992},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Leonardi, Salvatore
TI - Weighted Miranda-Talenti inequality and applications to equations with discontinuous coefficients
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 1
SP - 43
EP - 59
AB - Let $\Omega $ be an open bounded set in $\mathbb {R}^{n}$$(n\ge 2)$, with $C^2$ boundary, and $N^{p,\lambda }(\Omega )$ ($1 < p < +\infty $, $0\le \lambda < n$) be a weighted Morrey space. In this note we prove a weighted version of the Miranda-Talenti inequality and we exploit it to show that, under a suitable condition of Cordes type, the Dirichlet problem: \[ \left\lbrace \begin{array}{ll}\sum _{i,j=1}^n a_{ij}(x) \frac{\partial ^2 u}{\partial x_i \partial x_j} = f(x) \in N^{p,\lambda }(\Omega ) \quad & \text{ in } \Omega \ u=0 & \text{ on } \partial \Omega \end{array}\right.\] has a unique strong solution in the functional space \[ \left\lbrace u \in W^{2,p} \cap W^{1,p}_o(\Omega ) : \frac{\partial ^2 u}{\partial x_i \partial x_j} \in N^{p,\lambda }(\Omega ), i,j=1,2,\,\ldots , n\right\rbrace . \]
LA - eng
KW - Miranda-Talenti inequality; nonvariational elliptic equations; Hölder regularity; Miranda-Talenti inequality; Hölder regularity; nonvariational elliptic equations
UR - http://eudml.org/doc/248992
ER -

References

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  1. Adams R.A., Sobolev Spaces, Academic Press Inc., Orlando, 1977. Zbl1098.46001
  2. Brezis H., Analisi Funzionale, Liguori Editore, Napoli, 1986. 
  3. Campanato S., Maggiorazioni interpolatorie negli spazi H λ m , p ( Ø m e g a ) , Ann. Mat. Pura Appl. Ser. IV LXXV (1967), 261-276. (1967) MR0216284
  4. Campanato S., Un risultato relativo ad equazioni ellittiche del secondo ordine di tipo non variazionale, Ann. Scuola. Norm. Sup. Pisa (III) XXI Fasc. IV (1967), 701-707. (1967) Zbl0157.42202MR0224996
  5. Campanato S., Non variational differential systems. A condition for local existence and uniqueness, Proceedings of the Caccioppoli Conference (1989), Ricerche di Matem., Suppl., XL (1991), 129-140. (1991) Zbl0796.35052MR1306303
  6. Campanato S., On the condition of nearness between operators, Ann. Mat. Pura Appl. Ser. IV CLXVII (1994), 243-256. (1994) Zbl0820.47050MR1313557
  7. Campanato S., Attuale formulazione della teoria degli operatori vicini e attuale definizione di operatore ellittico, Le Matematiche LI 2 (1996), 291-298. (1996) 
  8. Campanato S., Cannarsa P., Second order nonvariational elliptic systems, Bollettino U.M.I. (5) 17-B (1980), 1365-1394. (1980) Zbl0449.35033MR0770854
  9. Coifman R.R., Fefferman C., Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. (1974) Zbl0291.44007MR0358205
  10. Cordes H.O., Über die erste Rundwertaufgabe bei quasi linearen Differential gleichungen zweiter Ordunung in mehr als zwei Variablen Math. Ann., 131 (1956), 278-312. (1956) MR0091400
  11. Cordes H.O., Vereinfachter Beweis der Existenz einer Apriori-Holderkonstanten, Math. Ann. 138 (1959), 155-178. (1959) MR0109256
  12. Cordes H.O., Zero order a priori estimates for solution of elliptic differential equations, Proc. of Symposia in Pure Math. IV, pp.157-166, 1961. MR0146511
  13. García-Cuerva J., Rubio de Francia J.L., Weighted Norm Inequalities and Related Topics, North Holland Math. Studies 116, Amsterdam, 1985. MR0848147
  14. Giaquinta M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Math. Studies, vol. 105 Princeton University Press, Princeton, 1983. Zbl0516.49003MR0717034
  15. Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Second Edit., Springer Verlag, 1983. Zbl1042.35002MR0737190
  16. Giusti E., Sulla regolarità delle soluzioni di una classe di equazioni ellittiche, Rend. Sem. Matem. Univ. Padova XXXIX (1967), 362-375. (1967) Zbl0164.13302MR0226173
  17. Giusti E., Equazioni Ellittiche del Secondo Ordine, Pitagora Editrice, Bologna, 1978. 
  18. Grisvard P., Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics 24, Pitman (Advanced Publishing Program), Boston, 1985. Zbl0695.35060MR0775683
  19. Guglielmino F., Nuovi contributi allo studio delle equazioni paraboliche del secondo ordine di tipo non variazionale, Ricerche di Matem. 14 (1965), 124-144. (1965) Zbl0141.29301MR0201832
  20. Koshelev A.I., Regularity Problem for Quasilinear Elliptic and Parabolic Systems, Springer Verlag, 1995. Zbl0847.35023MR1442954
  21. Leonardi S., On the Campanato nearness condition, Le Matematiche 48.1 (1993), 179-181. (1993) Zbl0805.47067MR1283760
  22. Leonardi S., Remarks on the Regularity of Solutions of Elliptic Systems, Kluwer Academic/Plenum Publishers, pp.325-344, New York, 1999. Zbl0952.35034MR1727457
  23. Miranda C., Istituzioni di Analisi Funzionale Lineare, U.M.I. 1978. Zbl0697.46002
  24. Muckenhoupt B., Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. (1972) Zbl0236.26016MR0293384
  25. Muckenhoupt B., Whedeen R.L., Weighted norm inequalities for singular and fractional integrals, Trans. Amer. Math. Soc. 161 (1971), 249-258. (1971) MR0285938
  26. Murthy M.K.V., Stampacchia G., Boundary value problems for some degenerate-elliptic operators, Ann. Mat. Pura Appl. Ser. IV 80 (1968), 1-122. (1968) Zbl0185.19201MR0249828
  27. Nečas J., On the Regularity of Weak Solutions to Nonlinear Elliptic Systems of Partial Differential Equations, Lectures at Scuola Normale Sup. Pisa, 1979. 
  28. Nicolosi F., Problemi parabolici in più variabili, Le Matematiche XXVII 1 (1972), 153-166. (1972) Zbl0264.35041MR0326173
  29. Pucci C., Equazioni ellittiche con soluzioni in W 2 , p , p < 2 , Convegno sulle Equaz. alle Derivate Parziali, pp.145-148, Bologna, 1967. MR0271530
  30. Pucci C., Talenti G., Elliptic (second order) partial differential equations with measurable coefficients and approximating integral equations, Adv. Math. 19 1 (1976), 48-105. (1976) MR0419989
  31. Stein E.M., Note on singular integrals, Proc. Amer. Math. Soc. 8 (1957), 250-254. (1957) Zbl0077.27301MR0088606
  32. Stein E.M., Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, New Jersey, 1970. Zbl0281.44003MR0290095
  33. Stein E.M., Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, 1993. Zbl0821.42001MR1232192
  34. Stein E.M., Weiss G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, New Jersey, 1971. Zbl0232.42007MR0304972
  35. Talenti G., Sopra una classe di equazioni ellittiche a coefficienti misurabili, Ann. Mat. Pura Appl. LXIX (1965), 285-304. (1965) Zbl0145.36602MR0201816
  36. Talenti G., Equazioni lineari ellittiche in due variabili, Le Matematiche XXI (1966), 339-376. (1966) Zbl0149.07402MR0204845
  37. Torchinsky A., Real-variable Methods in Harmonic Analysis, Academic Press Inc., Orlando, 1986. Zbl1097.42002MR0869816

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