Boundary regularity and compactness for overdetermined problems

Ivan Blank; Henrik Shahgholian

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2003)

  • Volume: 2, Issue: 4, page 787-802
  • ISSN: 0391-173X

Abstract

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Let D be either the unit ball B 1 ( 0 ) or the half ball B 1 + ( 0 ) , let f be a strictly positive and continuous function, and let u and Ω ⊂ D solve the following overdetermined problem: Δ u ( x ) = χ Ω ( x ) f ( x ) in D , 0 ∈ ∂ Ω , u = | ∇ u | = 0 in Ω c , where χ Ω denotes the characteristic function of Ω , Ω c denotes the set D ∖ Ω , and the equation is satisfied in the sense of distributions. When D = B 1 + ( 0 ) , then we impose in addition that u ( x ) ≡ 0 on { ( x ' , x n ) | x n = 0 } . We show that a fairly mild thickness assumption on Ω c will ensure enough compactness on u to give us “blow-up” limits, and we show how this compactness leads to regularity of ∂ Ω . In the case where f is positive and Lipschitz, the methods developed in Caffarelli, Karp, and Shahgholian (2000) lead to regularity of ∂ Ω under a weaker thickness assumption

How to cite

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Blank, Ivan, and Shahgholian, Henrik. "Boundary regularity and compactness for overdetermined problems." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 2.4 (2003): 787-802. <http://eudml.org/doc/84519>.

@article{Blank2003,
abstract = {Let $D$ be either the unit ball $B_1(0)$ or the half ball $B_1^+(0),$ let $f$ be a strictly positive and continuous function, and let $u$ and $\Omega \subset D$ solve the following overdetermined problem:\[ \Delta u(x) = \chi \_\{\_\Omega \}(x) f(x) \ \ \text\{in\} \ \ D, \ \ \ \ 0 \in \partial \Omega , \ \ \ \ u = |\nabla u| = 0 \ \ \text\{in\} \ \ \Omega ^c, \]where $\chi _\{_\Omega \}$ denotes the characteristic function of $\Omega ,$$\Omega ^c$ denotes the set $D \setminus \Omega ,$ and the equation is satisfied in the sense of distributions. When $D = B_1^+(0),$ then we impose in addition that\[ u(x) \equiv 0 \ \ \text\{on\} \ \ \lbrace \; (x^\{\prime \}, \; x\_n) \; | \; x\_n = 0 \; \rbrace \,. \]We show that a fairly mild thickness assumption on $\Omega ^c$ will ensure enough compactness on $u$ to give us “blow-up” limits, and we show how this compactness leads to regularity of $\partial \Omega .$ In the case where $f$ is positive and Lipschitz, the methods developed in Caffarelli, Karp, and Shahgholian (2000) lead to regularity of $\partial \Omega $ under a weaker thickness assumption},
author = {Blank, Ivan, Shahgholian, Henrik},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {4},
pages = {787-802},
publisher = {Scuola normale superiore},
title = {Boundary regularity and compactness for overdetermined problems},
url = {http://eudml.org/doc/84519},
volume = {2},
year = {2003},
}

TY - JOUR
AU - Blank, Ivan
AU - Shahgholian, Henrik
TI - Boundary regularity and compactness for overdetermined problems
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2003
PB - Scuola normale superiore
VL - 2
IS - 4
SP - 787
EP - 802
AB - Let $D$ be either the unit ball $B_1(0)$ or the half ball $B_1^+(0),$ let $f$ be a strictly positive and continuous function, and let $u$ and $\Omega \subset D$ solve the following overdetermined problem:\[ \Delta u(x) = \chi _{_\Omega }(x) f(x) \ \ \text{in} \ \ D, \ \ \ \ 0 \in \partial \Omega , \ \ \ \ u = |\nabla u| = 0 \ \ \text{in} \ \ \Omega ^c, \]where $\chi _{_\Omega }$ denotes the characteristic function of $\Omega ,$$\Omega ^c$ denotes the set $D \setminus \Omega ,$ and the equation is satisfied in the sense of distributions. When $D = B_1^+(0),$ then we impose in addition that\[ u(x) \equiv 0 \ \ \text{on} \ \ \lbrace \; (x^{\prime }, \; x_n) \; | \; x_n = 0 \; \rbrace \,. \]We show that a fairly mild thickness assumption on $\Omega ^c$ will ensure enough compactness on $u$ to give us “blow-up” limits, and we show how this compactness leads to regularity of $\partial \Omega .$ In the case where $f$ is positive and Lipschitz, the methods developed in Caffarelli, Karp, and Shahgholian (2000) lead to regularity of $\partial \Omega $ under a weaker thickness assumption
LA - eng
UR - http://eudml.org/doc/84519
ER -

References

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  1. [B] I. Blank, Sharp results for the regularity and stability of the free boundary in the obstacle problem, Indiana Univ. Math. J. 50 (2001), 1077–1112. Zbl1032.35170MR1871348
  2. [C1] L. A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (1977), 155–184. Zbl0386.35046MR454350
  3. [C2] L. A. Caffarelli, Compactness methods in free boundary problems, Comm. Partial Differential Equations 5 (1980), 427–448. Zbl0437.35070MR567780
  4. [C3] L. A. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl. 4 (1998), 383–402. Zbl0928.49030MR1658612
  5. [CKS] L. A. Caffarelli – L. Karp – H. Shahgholian, Regularity of a free boundary with application to the Pompeiu problem, Ann. of Math. 151 (2000), 269–292. Zbl0960.35112MR1745013
  6. [F] A. Friedman, “Variational Principles and Free Boundary Problems”, Wiley, 1982. Zbl0564.49002MR679313
  7. [GT] D. Gilbarg – N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, 2nd ed., Springer-Verlag, 1983. Zbl0562.35001MR737190
  8. [I] V. Isakov, “Inverse Source Problems”, AMS Math. Surveys and Monographs 34, Providence, Rhode Island, 1990. Zbl0721.31002MR1071181
  9. [KN] D. Kinderlehrer – L. Nirenberg, Regularity in free boundary value problems, Ann. Scuola Norm. Sup. Pisa 4 (1977), 373–391. Zbl0352.35023MR440187
  10. [KS] L. Karp – H. Shahgholian, On the optimal growth of functions with bounded Laplacian, Electron. J. Differential Equations 2000 (2000), 1–9. Zbl0937.35029MR1735060
  11. [KT] C.E. Kenig – T. Toro, Free boundary regularity for harmonic measures and Poisson Kernels, Ann. of Math. 150 (1999), 369–454. Zbl0946.31001MR1726699
  12. [M] A. S. Margulis, Potential theory for L p -densities and its applications to inverse problems of gravimetry, Theory and Practice of Gravitational and Magnetic Fields Interpretation in USSR, Naukova Dumka Press, Kiev, 1983, 188–197 (Russian). 
  13. [R] E. R. Reifenberg, Solution of the Plateau Problem for m -dimensional surfaces of varying topological type, Acta Math. 104 (1960), 1–92. Zbl0099.08503MR114145
  14. [Sc] D. G. Schaeffer, Some examples of singularities in a free boundary, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1977), 133–144. Zbl0354.35033MR516201
  15. [St] V. N. Strakhov, The inverse logarithmic potential problem for contact surface, Physics of the Solid Earth 10 (1974), 104–114 [translated from Russian]. 
  16. [SU] H. Shahgholian – N. Uraltseva, Regularity properties of a free boundary near contact points with the fixed boundary, Duke Univ. Math. J. 116 (2003), 1–34. Zbl1050.35157MR1950478
  17. [T] T. Toro, Doubling and flatness: geometry of measures, Notices Amer. Math. Soc. 44 (1997), 1087–1094. Zbl0909.31006MR1470167

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