# Boundary regularity and compactness for overdetermined problems

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

top

## Abstract

top
Let $D$ be either the unit ball ${B}_{1}\left(0\right)$ or the half ball ${B}_{1}^{+}\left(0\right),$ let $f$ be a strictly positive and continuous function, and let $u$ and $\Omega \subset D$ solve the following overdetermined problem:$\Delta u\left(x\right)={\chi }_{{}_{\Omega }}\left(x\right)f\left(x\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}D,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0\in \partial \Omega ,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}u=|\nabla u|=0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\Omega }^{c},$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 $D={B}_{1}^{+}\left(0\right),$ then we impose in addition that$u\left(x\right)\equiv 0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{on}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\left\{\phantom{\rule{0.277778em}{0ex}}\left({x}^{\text{'}},\phantom{\rule{0.277778em}{0ex}}{x}_{n}\right)\phantom{\rule{0.277778em}{0ex}}|\phantom{\rule{0.277778em}{0ex}}{x}_{n}=0\phantom{\rule{0.277778em}{0ex}}\right\}\phantom{\rule{0.166667em}{0ex}}.$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

## How to cite

top

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

top
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

top

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.