Necessary conditions of solvability and isoperimetric estimates for some Monge-Ampère problems in the plane
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- by Cristian Enache
- Proc. Amer. Math. Soc. 143 (2015), 309-315
- DOI: https://doi.org/10.1090/S0002-9939-2014-12222-2
- Published electronically: September 24, 2014
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Abstract:
This article is mainly devoted to the solvability of the Monge-Ampère equation $\det \left ( D^{2}u\right ) =1,$ in a $C^{2}$ bounded strictly convex domain $\Omega \subset \mathbb {R}^{2}$, subject to a contact angle boundary condition. A necessary condition for the solvability of this problem, involving the maximal value of the curvature $k\left ( s\right )$ of $\partial \Omega$ and the contact angle, was derived by X.-N. Ma in 1999, making use of a maximum principle for an appropriate P-function. Our main goal here is to prove a complementary result. More precisely, we will derive a new necessary condition of solvability, involving the minimal value of the curvature $k\left ( s\right )$ of $\partial \Omega$ and the contact angle. The main ingredients of our proof are the derivation of a minimum principle for the P-function employed by X.-N. Ma in his proof, respectively, the use of some computations in normal coordinates with respect to the boundary $\partial \Omega$. Finally, a similar minimum principle will be employed to derive some isoperimetric estimates for the classical convex solution of the Monge-Ampère equation, subject to the homogeneous Dirichlet boundary condition.References
- Catherine Bandle, On isoperimetric gradient bounds for Poisson problems and problems of torsional creep, Z. Angew. Math. Phys. 30 (1979), no. 4, 713–715 (English, with German summary). MR 547661, DOI 10.1007/BF01590849
- Luminiţa Barbu, On some estimates for a fluid surface in a short capillary tube, Appl. Math. Comput. 219 (2013), no. 15, 8192–8197. MR 3037527, DOI 10.1016/j.amc.2013.02.016
- Luminita Barbu and Cristian Enache, A minimum principle for a soap film problem in $\Bbb R^2$, Z. Angew. Math. Phys. 64 (2013), no. 2, 321–328. MR 3041572, DOI 10.1007/s00033-012-0240-x
- Luminita Barbu and Cristian Enache, A maximum principle for some fully nonlinear elliptic equations with applications to Weingarten hypersurfaces, Complex Var. Elliptic Equ. 58 (2013), no. 12, 1725–1736. MR 3170732, DOI 10.1080/17476933.2012.712966
- L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation, Comm. Pure Appl. Math. 37 (1984), no. 3, 369–402. MR 739925, DOI 10.1002/cpa.3160370306
- Cristian Enache, Maximum principles and symmetry results for a class of fully nonlinear elliptic PDEs, NoDEA Nonlinear Differential Equations Appl. 17 (2010), no. 5, 591–600. MR 2728539, DOI 10.1007/s00030-010-0070-5
- E. Hopf, Elementare Bemerkung über die Lösung partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus, Berlin Sber. Preuss. Akad. Wiss 19 (1927), 147–152.
- Eberhard Hopf, A remark on linear elliptic differential equations of second order, Proc. Amer. Math. Soc. 3 (1952), 791–793. MR 50126, DOI 10.1090/S0002-9939-1952-0050126-X
- P.-L. Lions, N. S. Trudinger, and J. I. E. Urbas, The Neumann problem for equations of Monge-Ampère type, Comm. Pure Appl. Math. 39 (1986), no. 4, 539–563. MR 840340, DOI 10.1002/cpa.3160390405
- Xi-Nan Ma, A necessary condition of solvability for the capillarity boundary of Monge-Ampère equations in two dimensions, Proc. Amer. Math. Soc. 127 (1999), no. 3, 763–769. MR 1487323, DOI 10.1090/S0002-9939-99-04750-4
- Xi-nan Ma, A sharp minimum principle for the problem of torsional rigidity, J. Math. Anal. Appl. 233 (1999), no. 1, 257–265. MR 1684385, DOI 10.1006/jmaa.1999.6291
- Xi-Nan Ma, Sharp size estimates for capillary free surfaces without gravity, Pacific J. Math. 192 (2000), no. 1, 121–134. MR 1741026, DOI 10.2140/pjm.2000.192.121
- L. E. Payne and G. A. Philippin, Some remarks on the problems of elastic torsion and of torsional creep, in Some Aspects of Mechanics of Continua, Part I, Jadavpur Univ. Calcutta, India (1977), 32–40.
- Gérard A. Philippin, A minimum principle for the problem of torsional creep, J. Math. Anal. Appl. 68 (1979), no. 2, 526–535. MR 533510, DOI 10.1016/0022-247X(79)90133-1
- G. A. Philippin and V. Proytcheva, A minimum principle for the problem of St-Venant in $\Bbb R^N$, $N\ge 2$, Z. Angew. Math. Phys. 63 (2012), no. 6, 1085–1090. MR 3000716, DOI 10.1007/s00033-012-0217-9
- G. A. Philippin and A. Safoui, Some applications of the maximum principle to a variety of fully nonlinear elliptic PDE’s, Z. Angew. Math. Phys. 54 (2003), no. 5, 739–755. Special issue dedicated to Lawrence E. Payne. MR 2019177, DOI 10.1007/s00033-003-3200-7
- G. A. Philippin and A. Safoui, Some minimum principles for a class of elliptic boundary value problems, Appl. Anal. 83 (2004), no. 3, 231–241. MR 2033237, DOI 10.1080/00036810310001632754
- René P. Sperb, Maximum principles and their applications, Mathematics in Science and Engineering, vol. 157, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 615561
- John Urbas, Nonlinear oblique boundary value problems for Hessian equations in two dimensions, Ann. Inst. H. Poincaré C Anal. Non Linéaire 12 (1995), no. 5, 507–575. MR 1353259, DOI 10.1016/S0294-1449(16)30150-0
- John Urbas, A note on the contact angle boundary condition for Monge-Ampère equations, Proc. Amer. Math. Soc. 128 (2000), no. 3, 853–855. MR 1646210, DOI 10.1090/S0002-9939-99-05222-3
Bibliographic Information
- Cristian Enache
- Affiliation: Department of Mathematics and Informatics, Ovidius University of Constanta, Constanta, 900527, Romania
- Email: cenache@univ-ovidius.ro
- Received by editor(s): January 8, 2013
- Received by editor(s) in revised form: April 7, 2013
- Published electronically: September 24, 2014
- Additional Notes: The author was supported by the strategic grant POSDRU/88/1.5/S/49516 Project ID 49516 (2009), co-financed by the European Social Fund Investing in People, within the Sectorial Operational Programme Human Resources Development 2007–2013.
- Communicated by: Joachim Krieger
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 143 (2015), 309-315
- MSC (2010): Primary 35J60, 35J96, 35J25, 35B50; Secondary 53C45
- DOI: https://doi.org/10.1090/S0002-9939-2014-12222-2
- MathSciNet review: 3272756