American Institute of Mathematical Sciences

January  2013, 33(1): 305-320. doi: 10.3934/dcds.2013.33.305

Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation

 1 Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, Via A. Valerio 12/1, 34127 Trieste, Italy, Italy

Received  August 2011 Published  September 2012

We discuss existence and regularity of bounded variation solutions of the Dirichlet problem for the one-dimensional capillarity-type equation \begin{equation*} \Big( u'/{ \sqrt{1+{u'}^2}}\Big)' = f(t,u) \quad \hbox{ in } {]-r,r[}, \qquad u(-r)=a, \, u(r) = b. \end{equation*} We prove interior regularity of solutions and we obtain a precise description of their boundary behaviour. This is achieved by a direct and elementary approach that exploits the properties of the zero set of the right-hand side $f$ of the equation.
Citation: Franco Obersnel, Pierpaolo Omari. Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 305-320. doi: 10.3934/dcds.2013.33.305
References:
 [1] G. Anzellotti, The Euler equation for functionals with linear growth, Trans. Amer. Math. Soc., 290 (1985), 483-501. doi: 10.1090/S0002-9947-1985-0792808-4.  Google Scholar [2] D. Bonheure, F. Obersnel and P. Omari, Heteroclinic solutions of the prescribed curvature equation with a double-well potential, preprint, (2011). Google Scholar [3] D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical solutions of a prescribed curvature equation, J. Differential Equations, 243 (2007), 208-237. doi: 10.1016/j.jde.2007.05.031.  Google Scholar [4] D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical positive solutions of a prescribed curvature equation with singularities, Rend. Istit. Mat. Univ. Trieste, 39 (2007), 63-85.  Google Scholar [5] G. Buttazzo, M. Giaquinta and S. Hildebrandt, "One-dimensional Variational Problems. An Introduction,'' Clarendon Press, Oxford, 1998.  Google Scholar [6] K. C. Chang, The spectrum of the 1-Laplace operator, Commun. Contemp. Math., 11 (2009), 865-894. doi: 10.1142/S0219199709003570.  Google Scholar [7] M. Emmer, Esistenza, unicità e regolarità nelle superfici di equilibrio nei capillari, Ann. Univ. Ferrara Sez. VII (N.S.), 18 (1973), 79-94.  Google Scholar [8] C. Gerhardt, Existence and regularity of capillary surfaces, Boll. Un. Mat. Ital. (4), 10 (1974), 317-335.  Google Scholar [9] C. Gerhardt, Existence, regularity, and boundary behavior of generalized surfaces of prescribed mean curvature, Math. Z., 139 (1974), 173-198. doi: 10.1007/BF01418314.  Google Scholar [10] M. Giaquinta, Regolarità delle superfici $BV$ con curvatura media assegnata, Boll. Un. Mat. Ital. (4), 8 (1973), 567-578.  Google Scholar [11] E. Giusti, "Minimal Surfaces and Functions of Bounded Variations," Birkhäuser, Basel, 1984.  Google Scholar [12] P. Habets and P. Omari, Multiple positive solutions of a one-dimensional prescribed mean curvature problem, Commun. Contemp. Math., 9 (2007), 701-730. doi: 10.1142/S0219199707002617.  Google Scholar [13] A. Hammerstein, Nichtlineare Integralgleichungen nebst Anwendungen, Acta Math., 54 (1930), 117-176. doi: 10.1007/BF02547519.  Google Scholar [14] V. K. Le, Some existence results on non-trivial solutions of the prescribed mean curvature equation, Adv. Nonlinear Stud., 5 (2005), 133-161. Google Scholar [15] V. K. Le, Variational method based on finite dimensional approximation in a generalized prescribed mean curvature problem, J. Differential Equations, 246 (2009), 3559-3578. doi: 10.1016/j.jde.2008.11.015.  Google Scholar [16] U. Massari, Esistenza e regolarità delle ipersuperficie di curvatura media assegnata in $\RR^n$, Arch. Rational Mech. Anal., 55 (1974), 357-382. doi: 10.1007/BF00250439.  Google Scholar [17] J. Mawhin, J. R. Ward Jr. and M. Willem, Variational methods and semilinear elliptic equations, Arch. Rational Mech. Anal., 95 (1986), 269-277. doi: 10.1007/BF00251362.  Google Scholar [18] A. Mellet and J. Vovelle, Existence and regularity of extremal solutions for a mean-curvature equation, J. Differential Equations, 249 (2010), 37-75. doi: 10.1016/j.jde.2010.03.026.  Google Scholar [19] M. Miranda, Dirichlet problem with L 1 data for the non-homogeneous minimal surface equation,, Indiana Univ. Math. J., 24 (): 227.  doi: 10.1512/iumj.1974.24.24020.  Google Scholar [20] F. Obersnel, Classical and non-classical sign changing solutions of a one-dimensional autonomous prescribed curvature equation, Adv. Nonlinear Stud., 7 (2007), 1-13.  Google Scholar [21] F. Obersnel and P. Omari, Existence and multiplicity results for the prescribed mean curvature equation via lower and upper solutions, Differential Integral Equations, 22 (2009), 853-880.  Google Scholar [22] F. Obersnel and P. Omari, Positive solutions of the Dirichlet problem for the prescribed mean curvature equation, J. Differential Equations, 249 (2010), 1674-1725. doi: 10.1016/j.jde.2010.07.001.  Google Scholar [23] F. Obersnel, P. Omari and S. Rivetti, Existence, regularity and stability properties of periodic solutions of a capillarity equation in the presence of lower and upper solutions, Nonlinear Anal. Real World Appl., 13 (2012), 2830-2852. doi: 10.1016/j.nonrwa.2012.04.012.  Google Scholar [24] L. Schwartz, Les théorèmes de Whitney sur les fonctions différentiables, Séminaire Bourbaki, Soc. Math. France, Paris, 1 (1995), 355-363.  Google Scholar [25] J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London Ser. A, 264 (1969), 413-496. doi: 10.1098/rsta.1969.0033.  Google Scholar

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References:
 [1] G. Anzellotti, The Euler equation for functionals with linear growth, Trans. Amer. Math. Soc., 290 (1985), 483-501. doi: 10.1090/S0002-9947-1985-0792808-4.  Google Scholar [2] D. Bonheure, F. Obersnel and P. Omari, Heteroclinic solutions of the prescribed curvature equation with a double-well potential, preprint, (2011). Google Scholar [3] D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical solutions of a prescribed curvature equation, J. Differential Equations, 243 (2007), 208-237. doi: 10.1016/j.jde.2007.05.031.  Google Scholar [4] D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical positive solutions of a prescribed curvature equation with singularities, Rend. Istit. Mat. Univ. Trieste, 39 (2007), 63-85.  Google Scholar [5] G. Buttazzo, M. Giaquinta and S. Hildebrandt, "One-dimensional Variational Problems. An Introduction,'' Clarendon Press, Oxford, 1998.  Google Scholar [6] K. C. Chang, The spectrum of the 1-Laplace operator, Commun. Contemp. Math., 11 (2009), 865-894. doi: 10.1142/S0219199709003570.  Google Scholar [7] M. Emmer, Esistenza, unicità e regolarità nelle superfici di equilibrio nei capillari, Ann. Univ. Ferrara Sez. VII (N.S.), 18 (1973), 79-94.  Google Scholar [8] C. Gerhardt, Existence and regularity of capillary surfaces, Boll. Un. Mat. Ital. (4), 10 (1974), 317-335.  Google Scholar [9] C. Gerhardt, Existence, regularity, and boundary behavior of generalized surfaces of prescribed mean curvature, Math. Z., 139 (1974), 173-198. doi: 10.1007/BF01418314.  Google Scholar [10] M. Giaquinta, Regolarità delle superfici $BV$ con curvatura media assegnata, Boll. Un. Mat. Ital. (4), 8 (1973), 567-578.  Google Scholar [11] E. Giusti, "Minimal Surfaces and Functions of Bounded Variations," Birkhäuser, Basel, 1984.  Google Scholar [12] P. Habets and P. Omari, Multiple positive solutions of a one-dimensional prescribed mean curvature problem, Commun. Contemp. Math., 9 (2007), 701-730. doi: 10.1142/S0219199707002617.  Google Scholar [13] A. Hammerstein, Nichtlineare Integralgleichungen nebst Anwendungen, Acta Math., 54 (1930), 117-176. doi: 10.1007/BF02547519.  Google Scholar [14] V. K. Le, Some existence results on non-trivial solutions of the prescribed mean curvature equation, Adv. Nonlinear Stud., 5 (2005), 133-161. Google Scholar [15] V. K. Le, Variational method based on finite dimensional approximation in a generalized prescribed mean curvature problem, J. Differential Equations, 246 (2009), 3559-3578. doi: 10.1016/j.jde.2008.11.015.  Google Scholar [16] U. Massari, Esistenza e regolarità delle ipersuperficie di curvatura media assegnata in $\RR^n$, Arch. Rational Mech. Anal., 55 (1974), 357-382. doi: 10.1007/BF00250439.  Google Scholar [17] J. Mawhin, J. R. Ward Jr. and M. Willem, Variational methods and semilinear elliptic equations, Arch. Rational Mech. Anal., 95 (1986), 269-277. doi: 10.1007/BF00251362.  Google Scholar [18] A. Mellet and J. Vovelle, Existence and regularity of extremal solutions for a mean-curvature equation, J. Differential Equations, 249 (2010), 37-75. doi: 10.1016/j.jde.2010.03.026.  Google Scholar [19] M. Miranda, Dirichlet problem with L 1 data for the non-homogeneous minimal surface equation,, Indiana Univ. Math. J., 24 (): 227.  doi: 10.1512/iumj.1974.24.24020.  Google Scholar [20] F. Obersnel, Classical and non-classical sign changing solutions of a one-dimensional autonomous prescribed curvature equation, Adv. Nonlinear Stud., 7 (2007), 1-13.  Google Scholar [21] F. Obersnel and P. Omari, Existence and multiplicity results for the prescribed mean curvature equation via lower and upper solutions, Differential Integral Equations, 22 (2009), 853-880.  Google Scholar [22] F. Obersnel and P. Omari, Positive solutions of the Dirichlet problem for the prescribed mean curvature equation, J. Differential Equations, 249 (2010), 1674-1725. doi: 10.1016/j.jde.2010.07.001.  Google Scholar [23] F. Obersnel, P. Omari and S. Rivetti, Existence, regularity and stability properties of periodic solutions of a capillarity equation in the presence of lower and upper solutions, Nonlinear Anal. Real World Appl., 13 (2012), 2830-2852. doi: 10.1016/j.nonrwa.2012.04.012.  Google Scholar [24] L. Schwartz, Les théorèmes de Whitney sur les fonctions différentiables, Séminaire Bourbaki, Soc. Math. France, Paris, 1 (1995), 355-363.  Google Scholar [25] J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London Ser. A, 264 (1969), 413-496. doi: 10.1098/rsta.1969.0033.  Google Scholar
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