Absolutely minimal extensions of functions on metric spaces

V. A. Milman

Sbornik. Mathematics, Tome 190 (1999) no. 6, pp. 859-885

Résumé

Extensions of a real-valued function from the boundary $\partial X_0$ of an open subset $X_0$ of a metric space ${(X,d)}$ to $X_0$ are discussed. For the broad class of initial data coming under discussion (linearly bounded functions) locally Lipschitz extensions to $X_0$ that preserve localized moduli of continuity are constructed. In the set of these extensions an absolutely minimal extension is selected, which was considered before by Aronsson for Lipschitz initial functions in the case $X_0\subset\mathbb R^n$. An absolutely minimal extension can be regarded as an $\infty$-harmonic function, that is, a limit of $p$-harmonic functions as $p\to+\infty$. The proof of the existence of absolutely minimal extensions in a metric space with intrinsic metric is carried out by the Perron method. To this end, $\infty$-subharmonic, $\infty$-superharmonic, and $\infty$-harmonic functions on a metric space are defined and their properties are established.

Bibliographie

[1] Buzeman G., Vypuklye poverkhnosti, Nauka, M., 1964 | MR

[2] McShane E., “Extension of range of function”, Bull. Amer. Math. Soc., 40:12 (1934), 837–842 | DOI | MR | Zbl

[3] Mustata C., Extension of Hölder functions and some related problems of best approximation, Research Seminars. Preprint No 7, Babes–Bolyai University. Faculty of Mathematics, 1991, p. 71–86 | MR | Zbl

[4] Aronsson G., “Extension of functions satisfying Lipschitz conditions”, Ark. Mat., 6:6 (1967), 551–561 | DOI | MR | Zbl

[5] Batthacharya T., Di Benedetto E., Manfredi J., “Limits as $p\to\infty$ of $\Delta_pu_p=f$ and related extremal problems”, Rend. Sem. Mat. Univ. Politec. Torino, 1989, 15–68

[6] Aronsson G., “On the partial differential equation $u^2_xu_{xx}+2u_xu_yu_{xy}+u^2_yu_{yy}=0$”, Ark. Mat., 7 (1968), 395–425 | DOI | MR | Zbl

[7] Jensen R., “Uniqueness of Lipschitz extensions: Minimizing the sup-norm of the gradient”, Arch. Rational Mech. Anal., 123:1 (1993), 51–74 | DOI | MR | Zbl

[8] Crandall M. G., Ishii H., Lions P.-L., “User's guide to viscosity solutions of second order partial differential equations”, Bull. Amer. Math. Soc., New Ser., 27:1 (1992), 1–67 | DOI | MR | Zbl

[9] Aronsson G., “Construction of singular solutions to the $p$-harmonic equations and its limit equation for $p=\infty$”, Manuscripta Math., 56:2 (1986), 135–158 | DOI | MR

[10] Evans L., “Estimates for smooth absolutely minimizing Lipschitz extensions”, Electron. J. Differential Equations, 1993, no. 3, 1–10 ; ; http://ejde.math.swt.eduhttp://ejde.math.unt.edu | MR

[11] Lindqvist P., Manfredi J., “The Harnack inequality for $\infty$-harmonic functions”, Electron. J. Differential Equations, 1995, no. 4, 1–5 ; ; http://ejde.math.swt.eduhttp://ejde.math.unt.edu | MR

[12] Janfalk U., “Behaviour in the limit, as $p\to\infty$, of minimizers of functionals involving $p$-Dirichlet integrals”, SIAM J. Math. Anal., 27:26 (1996), 341–360 | DOI | MR | Zbl

[13] Krcho J., “Morphometric analysis of relief on the basis of geometric aspects of field theory”, Acta Geographico-Physica (Bratislava), 1973, no. 1, 15–78

[14] Milman V. A., “Lipshitsevy prodolzheniya lineino ogranichennykh funktsii”, Matem. sb., 189:8 (1998), 67–92 | MR

[15] Milman V. A., “Prodolzhenie funktsii s sokhraneniem modulya nepreryvnosti”, Vestsi AN Belarusi. Ser. fiz.-mat. navuk, 1996, no. 4, 23–28 | MR | Zbl

[16] Pleschinskii N. B., “O postroenii funktsii, udovletvoryayuschikh usloviyu Geldera s zadannym pokazatelem”, Izv. vuzov. Ser. matem., 1984, no. 8, 74–77 | Zbl

[17] Czipcer J., Geher L., “Extension of functions satisfying a Lipschitz condition”, Acta Math. Hungar., 6:1–2 (1955), 213–220 | MR

[18] Le Gruyer E., Archer J. C., “Stability and convergence of extension schemes to continuous functions in general metric spaces”, SIAM J. Math. Anal., 27:1 (1996), 274–285 | DOI | MR | Zbl

[19] Le Gruyer E., Archer J. C., “Harmonious extensions”, SIAM J. Math. Anal., 29:1 (1998), 279–292 | DOI | MR | Zbl

[20] Dzyadyk V. K., Vvedenie v teoriyu ravnomernogo priblizheniya funktsii polinomami, Nauka, M., 1977 | MR | Zbl

[21] Shvarts L., Analiz, T. 1, Mir, M., 1972

[22] Brelo M., Osnovy klassicheskoi teorii potentsiala, Mir, M., 1964 | MR | Zbl

[23] Dedonne Zh., Osnovy sovremennogo analiza, Mir, M., 1964

[24] Sleźak B., “A mean value theorem in metric spaces”, Constructive Theory of Functions, Publ. House Bulgarian Acad. Sci., Sofia, 1988, 47–49 | MR

[25] Milman V. A., “Prodolzhenie funktsii, sokhranyayuschee modul nepreryvnosti”, Matem. zametki, 61:2 (1997), 236–245 | MR | Zbl