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Large H-surfaces via the mountain-pass-lemma. (English) Zbl 0582.58010
L’A. considère le système \(\Delta u=2Hu_ x\wedge u_ y\) où \(B=disque\) unité de \({\mathbb{R}}^ 2\) avec condition de Dirichlet ou condition de Plateau. L’énergie associée à ce système est \(E(u)=(1/2)\int | \nabla u|^ 2+(2H/3)\int u\circ u_ x\bigwedge u_ x\). L’A. montre que si E admet un minimum local \bu, alors il existe une deuxième solution \(\bar u\neq \underline u\). Il retrouve en particulier les résultats du rapporteur et de J. M. Coron [C. R. Acad. Sci., Paris, Sér. I 295, 615-618 (1982; Zbl 0505.49019); Commun. Pure Appl. Math. 37, 149-187 (1984; Zbl 0537.49022)], de M. Struve [”Nonuniqueness in the Plateau problem”, Arch. Rat. Mech. Anal. (to appear)] et de K. Steffen [”On the nonuniqueness of surfaces with prescribed constant mean curvature” (to appear)]. La démonstration repose sur une méthode de min-max et une analyse du défaut de la condition de Palais-Smale comme dans l’article de J. Sacks et K. Uhlenbeck [Ann. Math., II. Ser. 113, 1-24 (1981; Zbl 0462.58014)] et du rapporteur et J. M. Coron [Arch. Rat. Mech. Anal. 89, 21-56 (1985); C. R. Acad. Sci., Paris, Sér. I 298, 389-392 (1984)].
Reviewer: H.Brezis

MSC:
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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References:
[1] Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Functional Analysis14, 349-381 (1973) · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[2] Brezis, H., Coron, J.-M.: Sur la conjecture de Rellich pour les surfaces à courbure moyenne prescrite. C.R. Acad. Sci. Paris Ser. I.295, 615-618 (1982) · Zbl 0505.49019
[3] Brezis, H., Coron, J.-M.: Multiple solutions ofH-systems and Rellich’s conjecture. Comm. Pure Appl. Math.37, 149-187 (1984) · Zbl 0537.49022 · doi:10.1002/cpa.3160370202
[4] Brezis, H., Coron, J.-M.: Convergence of solutions ofH-systems and applications to surfaces of constant mean curvature (preprint)
[5] Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math.36, 437-477 (1983) · Zbl 0541.35029 · doi:10.1002/cpa.3160360405
[6] Cerami, G., Fortunato, D., Struwe, M.: Bifurcation and multiplicity results for Yamabe’s equation (to appear) · Zbl 0568.35039
[7] Courant, R.: Dirichlet’s principle, conformal mapping and minimal surfaces. New York: Interscience 1950 · Zbl 0040.34603
[8] Donaldson, S.K.: An application of gauge theory to four dimensional topology. J. Differential Geometry18, 279-315 (1983) · Zbl 0507.57010
[9] Eisen, G.: A selection lemma for measurable sets, and lower semicontinuity of multiple integrals. Manuscripta Math.27, 73-79 (1979) · Zbl 0404.28004 · doi:10.1007/BF01297738
[10] Heinz, E.: Über die Existenz einer Fläche konstanter mitlerer Krümmung bei vorgegebener Berandung. Math. Ann.127, 258-287 (1954) · Zbl 0055.15303 · doi:10.1007/BF01361126
[11] Heinz, E.: On the nonexistence of a surface of constant mean curvature with finite area and prescribed rectifiable boundary. Arch. Rat. Mech. Anal.35, 249-252 (1969) · Zbl 0184.32802 · doi:10.1007/BF00248159
[12] Hildebrandt, S.: On the Plateau problem for surfaces of constant mean curvature. Comm. Pure Appl. Math.23, 97-114 (1970) · Zbl 0181.38703 · doi:10.1002/cpa.3160230105
[13] Lions, P.-L.: Applications de la méthode de concentration-compacité. C.R. Acad. Sci. Paris296, 645-648 (1983) · Zbl 0522.49008
[14] Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. Math.113, 1-24 (1981) · Zbl 0462.58014 · doi:10.2307/1971131
[15] Steffen, K.: Flächen konstanter mittlerer Krümmung mit vorgegebenem Volumen oder Flächeninhalt. Arch. Rat. Mech. Anal.49, 99-128 (1972) · Zbl 0259.53043 · doi:10.1007/BF00281413
[16] Steffen, K.: Ein verbesserter Existenzstz für Flächen konstanter mittlerer Krümmung. Manuscripta Math.6, 105-139 (1972) · Zbl 0229.53011 · doi:10.1007/BF01369709
[17] Steffen, K.: On the existence of surfaces with prescribed mean curvature and boundary. Math. Z.146, 113-135 (1976) · Zbl 0343.49016 · doi:10.1007/BF01187700
[18] Steffen, K.: On the nonuniqueness of surfaces with prescribed constant mean curvature spanning a given contour (to appear) · Zbl 0678.49036
[19] Struwe, M.: Nonuniqueness in the Plateau problem for surfaces of constant mean curvature. Arch. Rat. Mech. Anal. (to appear) · Zbl 0603.49027
[20] Struwe, M.: On a critical point theory for minimal surfaces spanning a wire in ? N . J. Reine Angew. Math.349, 1-23 (1984) · Zbl 0521.49028 · doi:10.1515/crll.1984.349.1
[21] Struwe, M.: A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z.187, 511-517 (1984) · Zbl 0545.35034 · doi:10.1007/BF01174186
[22] Wente, H.C.: An existence theorem for surfaces of constant mean curvature. J. Math. Anal. Appl.26, 318-344 (1969) · Zbl 0181.11501 · doi:10.1016/0022-247X(69)90156-5
[23] Wente, H.C.: A general existence theorem for surfaces of constant mean curvature. Math. Z.120, 277-288 (1971) · Zbl 0214.11101 · doi:10.1007/BF01117500
[24] Wente, H.C.: The differential equation ?x=2Hx u?xv with vanishing boundary values. Proc. A.M.S.50, 59-77 (1980) · Zbl 0473.49029
[25] Wente, H.C.: Large solutions to the volume constrained Plateau problem. Arch. Rat. Mech. Anal.75, 59-77 (1980) · Zbl 0473.49029 · doi:10.1007/BF00284621
[26] Wente, H.C.: Private communication
[27] Werner, H.: Das Problem von Douglas für Flächen konstanter mittlerer Krümmung. Math. Ann.133, 303-319 (1957) · Zbl 0077.34901 · doi:10.1007/BF01342884
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