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

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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

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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

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