Oblique derivative problem for elliptic equations in non-divergence form with V M O coefficients

Giuseppe di Fazio; Dian K. Palagachev

Commentationes Mathematicae Universitatis Carolinae (1996)

  • Volume: 37, Issue: 3, page 537-556
  • ISSN: 0010-2628

Abstract

top
A priori estimates and strong solvability results in Sobolev space W 2 , p ( Ω ) , 1 < p < ∞ are proved for the regular oblique derivative problem ∑ i , j = 1 n a i j ( x ) ∂ 2 u ∂ x i ∂ x j = f ( x ) a.e. Ω ∂ u ∂ ℓ + σ ( x ) u = ϕ ( x ) on ∂ Ω when the principal coefficients a i j are V M O ∩ L ∞ functions.

How to cite

top

di Fazio, Giuseppe, and Palagachev, Dian K.. "Oblique derivative problem for elliptic equations in non-divergence form with $VMO$ coefficients." Commentationes Mathematicae Universitatis Carolinae 37.3 (1996): 537-556. <http://eudml.org/doc/247884>.

@article{diFazio1996,
abstract = {A priori estimates and strong solvability results in Sobolev space $W^\{2,p\}(\Omega )$, $1<p<\infty $ are proved for the regular oblique derivative problem \[ \{\left\lbrace \begin\{array\}\{ll\} \sum \_\{i,j=1\}^n a^\{ij\}(x)\frac\{\partial ^2u\}\{\partial x\_i\partial x\_j\} =f(x) \text\{ a.e. \} \Omega \\ \frac\{\partial u\}\{\partial \ell \}+\sigma (x)u =\varphi (x) \text\{ on \} \partial \Omega \end\{array\}\right.\} \] when the principal coefficients $a^\{ij\}$ are $V\hspace\{-1.2pt\}MO\cap L^\infty $ functions.},
author = {di Fazio, Giuseppe, Palagachev, Dian K.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {oblique derivative; elliptic equation; non divergence form; $V\hspace\{-1.2pt\}MO$ coefficients; strong solution; oblique derivative; elliptic equation; non divergence form; VMO coefficients; strong solution; singular integral estimates},
language = {eng},
number = {3},
pages = {537-556},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Oblique derivative problem for elliptic equations in non-divergence form with $VMO$ coefficients},
url = {http://eudml.org/doc/247884},
volume = {37},
year = {1996},
}

TY - JOUR
AU - di Fazio, Giuseppe
AU - Palagachev, Dian K.
TI - Oblique derivative problem for elliptic equations in non-divergence form with $VMO$ coefficients
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1996
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 37
IS - 3
SP - 537
EP - 556
AB - A priori estimates and strong solvability results in Sobolev space $W^{2,p}(\Omega )$, $1<p<\infty $ are proved for the regular oblique derivative problem \[ {\left\lbrace \begin{array}{ll} \sum _{i,j=1}^n a^{ij}(x)\frac{\partial ^2u}{\partial x_i\partial x_j} =f(x) \text{ a.e. } \Omega \\ \frac{\partial u}{\partial \ell }+\sigma (x)u =\varphi (x) \text{ on } \partial \Omega \end{array}\right.} \] when the principal coefficients $a^{ij}$ are $V\hspace{-1.2pt}MO\cap L^\infty $ functions.
LA - eng
KW - oblique derivative; elliptic equation; non divergence form; $V\hspace{-1.2pt}MO$ coefficients; strong solution; oblique derivative; elliptic equation; non divergence form; VMO coefficients; strong solution; singular integral estimates
UR - http://eudml.org/doc/247884
ER -

References

top
  1. Acquistapace P., On B M O regularity for linear elliptic systems, Ann. Mat. Pura Appl. 161 (1992), 231-270. (1992) Zbl0802.35015MR1174819
  2. Agmon S., Douglis A., Nirenberg L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math. 12 (1959), 623-727. (1959) Zbl0093.10401MR0125307
  3. Adams R., Sobolev Spaces, Academic Press, New York, 1975. Zbl1098.46001MR0450957
  4. Bramanti M., Commutators of integral operators with positive kernels, Le Matematiche XLIX (1994), 149-168. (1994) Zbl0840.42009MR1386370
  5. Chicco M., Third boundary value problem in H 2 , p ( Ø m e g a ) for a class of linear second order elliptic partial differential equations, Rend. Ist. Mat. Univ. Trieste 4 (1972), 85-94. (1972) MR0348258
  6. Chicco M., Terzo problema al contorno per una classe di equazioni ellittiche del secondo ordine a coefficienti discontinui, Ann. Mat. Pura Appl. (4) 112 (1977), 241-259; errata, ibid. (4) 130 (1982), 399-401. (1977) MR0435582
  7. Chiarenza F., Frasca M., Longo P., Interior W 2 , p estimates for non divergence elliptic equations with discontinuous coefficients, Ricerche di Mat. 60 (1991), 149-168. (1991) MR1191890
  8. Chiarenza F., Frasca M., Longo P., W 2 , p -solvability of the Dirichlet problem for non divergence elliptic equations with V M O coefficients, Trans. Amer. Math. Soc. 336 (1993), 841-853. (1993) Zbl0818.35023MR1088476
  9. Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin, 1983. Zbl1042.35002MR0737190
  10. John F., Nirenberg L., On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426. (1961) Zbl0102.04302MR0131498
  11. Luo Y., An Aleksandrov-Bakelman type maximum principle and applications, J. Diff. Equations 101 (1993), 213-231. (1993) Zbl0812.35014MR1204327
  12. Luo Y., Trudinger N.S., Linear second order elliptic equations with Venttsel boundary conditions, Proc. Roy. Soc. Edinburgh 118A (1991), 193-207. (1991) Zbl0771.35014MR1121663
  13. Miranda C., Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui, Ann. Mat. Pura Appl. 63 (1963), 353-386. (1963) Zbl0156.34001MR0170090
  14. Nicolosi F., Il terzo problema al contorno per le equazioni lineari ellittiche a coefficienti discontinui, Rend. Circ. Mat. Palermo 33 (1984), 351-368. (1984) 
  15. Nirenberg L., On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115-162. (1959) Zbl0088.07601MR0109940
  16. Sarason D., Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405. (1975) Zbl0319.42006MR0377518
  17. Talenti G., Problemi di derivata obliqua per equazioni ellittiche in due variabili, Boll. Un. Mat. Ital. (3) 22 (1967), 505-526. (1967) Zbl0156.33803MR0231048
  18. Viola G., Una stima a priori per la soluzione del terzo problema al contorno associato ad una classe di equazioni ellittiche del secondo ordine a coefficienti non regolari, Boll. Un. Mat. Ital. (6) 3-B (1984), 397-411. (1984) 

NotesEmbed ?

top

You must be logged in to post comments.