The Wiener test for degenerate elliptic equations

E. B. Fabes; D. S. Jerison; C. E. Kenig

Annales de l'institut Fourier (1982)

  • Volume: 32, Issue: 3, page 151-182
  • ISSN: 0373-0956

Abstract

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We consider degenerated elliptic equations of the form ∑ i , j D x i ( a i j ( x ) D x j ) , where λ w ( x ) | ξ | 2 ≤ ∑ i , j a i j ( x ) ξ i ξ j ≤ Λ w ( x ) | ξ | 2 . Under suitable assumptions on w , we obtain a characterization of Wiener type (involving weighted capacities) for the set of regular points for these operators. The set of regular points is shown to depend only on w . The main tool we use is an estimate for the Green function in terms of w .

How to cite

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Fabes, E. B., Jerison, D. S., and Kenig, C. E.. "The Wiener test for degenerate elliptic equations." Annales de l'institut Fourier 32.3 (1982): 151-182. <http://eudml.org/doc/74544>.

@article{Fabes1982,
abstract = {We consider degenerated elliptic equations of the form\begin\{\}\sum \_\{i,j\} D\_\{x\_i\}(a\_\{ij\}(x) D\_\{x\_j\}), \text\{where\} \lambda w(x) |\xi |^2 \le \sum \_\{i,j\} a\_\{ij\} (x) \xi \_i\xi \_j \le \Lambda w(x) |\xi |^2.\end\{\}Under suitable assumptions on $w$, we obtain a characterization of Wiener type (involving weighted capacities) for the set of regular points for these operators. The set of regular points is shown to depend only on $w$. The main tool we use is an estimate for the Green function in terms of $w$.},
author = {Fabes, E. B., Jerison, D. S., Kenig, C. E.},
journal = {Annales de l'institut Fourier},
keywords = {weighted capacities; set of regular points; Green function},
language = {eng},
number = {3},
pages = {151-182},
publisher = {Association des Annales de l'Institut Fourier},
title = {The Wiener test for degenerate elliptic equations},
url = {http://eudml.org/doc/74544},
volume = {32},
year = {1982},
}

TY - JOUR
AU - Fabes, E. B.
AU - Jerison, D. S.
AU - Kenig, C. E.
TI - The Wiener test for degenerate elliptic equations
JO - Annales de l'institut Fourier
PY - 1982
PB - Association des Annales de l'Institut Fourier
VL - 32
IS - 3
SP - 151
EP - 182
AB - We consider degenerated elliptic equations of the form\begin{}\sum _{i,j} D_{x_i}(a_{ij}(x) D_{x_j}), \text{where} \lambda w(x) |\xi |^2 \le \sum _{i,j} a_{ij} (x) \xi _i\xi _j \le \Lambda w(x) |\xi |^2.\end{}Under suitable assumptions on $w$, we obtain a characterization of Wiener type (involving weighted capacities) for the set of regular points for these operators. The set of regular points is shown to depend only on $w$. The main tool we use is an estimate for the Green function in terms of $w$.
LA - eng
KW - weighted capacities; set of regular points; Green function
UR - http://eudml.org/doc/74544
ER -

References

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  1. [1] L. CARLESON, Selected Problems on Exceptional Sets, 1967, Van Nostrand. Zbl0189.10903MR37 #1576
  2. [2] R. COIFMAN and C. FEFFERMAN, Weighted norm inequalities for maximal functions and singular integrals, Studia Math., 51 (1974), 241-250. Zbl0291.44007MR50 #10670
  3. [3] J. DENY, Théorie de la capacité dans les espaces fonctionnels, Séminaire Brelot-Choquet-Deny (Théorie du Potentiel). no. 1, (1964-1965), 1-13. Zbl0138.36605
  4. [4] E.B. FABES, D.S. JERISON and C.E. KENIG, Boundary behavior of solutions of degenerate elliptic equations, preprint. Zbl0488.35034
  5. [5] E.B. FABES, C.E. KENIG, and R.P. SERAPIONI, The local regularity of solutions of degenerate elliptic equations, Comm. in P.D.E., 7(1) (1982), 77-116. Zbl0498.35042MR84i:35070
  6. [6] F. GEHRING, The Lp integrability of the partial derivatives of a quasi conformal mapping, Acta Math., 130 (1973), 266-277. Zbl0258.30021MR53 #5861
  7. [7] D. KINDERLEHRER and G. STAMPACCHIA, An Introduction to Variational Inequalities and their Applications, 1980, Academic Press, N.Y., N.Y. Zbl0457.35001MR81g:49013
  8. [8] W. LITTMAN, G. STAMPACCHIA and H. WEINBERGER, Regular points for elliptic equations with discontinuous coefficients, Ann. della Scuola Normale Sup. di Pisa, S. 3, vol. 17 (1963), 45-79. Zbl0116.30302MR28 #4228

Citations in EuDML Documents

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  1. Marco Biroli, Umberto Mosco, Wiener criterion for degenerate elliptic obstacle problem
  2. Marco Biroli, Umberto Mosco, Wiener criterion for degenerate elliptic obstacle problem
  3. Albo Carlos Cavalheiro, Regularity of weak solutions to certain degenerate elliptic equations
  4. Maz'ya, Vladimir, On Wiener's type regularity of a boundary point for higher order elliptic equations
  5. S. Chanillo, R. L. Wheeden, Existence and estimates of Green's function for degenerate elliptic equations
  6. Carmela Vitanza, Pietro Zamboni, A Variational Inequality for a Degenerate Elliptic Operator Under Minimal Assumptions on the Coefficients
  7. Kazuhiro Kurata, Satoko Sugano, Fundamental solution, eigenvalue asymptotics and eigenfunctions of degenerate elliptic operators with positive potentials
  8. Carlos E. Kenig, Wei-Ming Ni, On the elliptic equation L u - k + K exp [ 2 u ] = 0
  9. Erika Battaglia, Stefano Biagi, Andrea Bonfiglioli, [unknown]
  10. John L. Lewis, Kaj Nyström, Boundary behaviour for p harmonic functions in Lipschitz and starlike Lipschitz ring domains

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