Minimal hypersurfaces over soft obstacles

O. V. Titov

Izvestiya. Mathematics , Tome 8 (1974) no. 2, pp. 379-421.

Résumé

In the present work the following variational problem is discussed: minimize the area functional $$ F(u)=\int_G\sqrt{1+|\nabla u|^2}\,dx $$ in the class of all functions $W_0^{1,1}(G)$ for which $\int_{D\Subset G}u\,dx\geqslant V=\mathrm{const}$. For small enough $V$ the existence of an extremal is proved, and it is shown that it belongs to $C^{1,\alpha}(\overline G)$ with a Hölder index $\alpha$, $0\alpha\leqslant1$.

Bibliographie

[1] Lewy H., Stampacchia G., “On the regularity of the solution of a variational inequality”, Comm. Pure Appl. Math., 22(2) (1969), 153–188 | DOI | MR | Zbl

[2] Lewy H., Stampacchia G., “On existence and smoothness of solutions of some non-coercive variational inequalities”, Arch. Rat. Mech. and Anal., 41(4) (1971), 241–253 | MR | Zbl

[3] Nitsche J. C. C., “Variational problems with inequalities as boundary conditions or how to fashion a cheap hat for Giacometti's brother”, Arch. Rat. Mech. and Anal., 35(2) (1969), 83–113 | MR | Zbl

[4] Miranda M., “Nouveaux résultats pour les hypersurfaces minimales”, Actes Congrès Intern. Math., v. 2, 1970, 853–858 | MR

[5] Giusti E., “Superfici minime cartesiane con ostacoli discontinue”, Arch. Rat. Mech. and Anal., 40(4) (1971), 231–267 | DOI | MR

[6] Giaquinta M., Pepe L., “Esistenza e regolarita per il problema dell'area minima con ostacoli in $n$ variabili”, Ann. Sc. Norm. Sup. di Pisa, 25(3) (1971), 481–507 | MR | Zbl

[7] Wente H. C., “A general existence theorem for surfaces of constant mean curvature”, Math. Z., 120(3) (1971), 277–288 | DOI | MR | Zbl

[8] de Giorgi E., “Sulla differenziabilitàe l'analiticità delle estremali degli integrali multipli regolari”, Mem. Acad. Sci. Torino, Ser. 3, 3:1 (1957), 25–43 | MR | Zbl

[9] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1964 | MR

[10] Ladyzhenskaya O. A., Uralt'seva N. N., “Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations”, Comm. Pure Appl. Math., 23(4) (1970), 677–703 | DOI | MR | Zbl

[11] Temam R., “Solutions generalisés de certaines équations du type hypersurfaces minima”, Arch. Rat. Mech. and Anal., 44(2) (1971), 121–156 | MR | Zbl

[12] Morrey C. B., Multiple integrals in the calculus of variations, Springer-Verlag, 1966 | MR | Zbl

[13] Serrin J., “On the definition and properties of certain variational integrals”, Trans. Amer. Math. Soc., 101(1) (1961), 139–167 | DOI | MR | Zbl

[14] Stampacchia G., “On some regular multiple integral problems in the calculus of variations”, Comm. Pure Appl. Math., 16(4) (1963), 383–421 | DOI | MR | Zbl

[15] Bakelman I. Ya., “Giperpoverkhnosti s dannoi srednei kriviznoi i kvazilineinye ellipticheskie uravneniya s silnymi nelineinostyami”, Matem. sb., 75 (1968), 604–638 | MR

[16] Hopf E., “Elementare Bemerkungen über die Lösungen partiellen Differentialgleichungen zweiter Ordnung von elliptischen Typen”, Sitz. Ber. Preuss. Akad. Wiss., 119 (1927), 147–152

[17] Schauder J., “Über lineare elliptische Differentialgleichungen zweiter Ordnung”, Math. Z., 38(2) (1934), 257–282 | DOI | MR | Zbl

[18] Landis E. M., Uravneniya vtorogo poryadka ellipticheskogo i parabolicheskogo tipov, Nauka, M., 1971 | MR

[19] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz v normirovannykh prostranstvakh, Fizmatgiz, M., 1959 | MR

[20] Cordes H. O., “Die erste Randuertaufgabe bei Differentialgleichungen zweiter Ordnung in mehr als zwei Variabeln”, Math. Ann., 131(3) (1956), 278–318 | DOI | MR