# American Institute of Mathematical Sciences

October  2008, 20(4): 877-888. doi: 10.3934/dcds.2008.20.877

## Optimal Lyapunov inequalities for disfocality and Neumann boundary conditions using $L^p$ norms

Received  January 2007 Revised  August 2007 Published  January 2008

Motivated by the applications to nonlinear resonant boundary value problems with Neumann boundary conditions, this paper is devoted to the study of $L^{p}$ Lyapunov-type inequalities ($1 \leq p \leq \infty$) with mixed boundary conditions. We carry out a complete treatment of the problem for any constant $p \geq 1.$ Our main result is derived from a detailed analysis of the relationship between the existence of nontrivial solutions of these two different boundary problems.
Citation: Antonio Cañada, Salvador Villegas. Optimal Lyapunov inequalities for disfocality and Neumann boundary conditions using $L^p$ norms. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 877-888. doi: 10.3934/dcds.2008.20.877
 [1] M.J. Lopez-Herrero. The existence of weak solutions for a general class of mixed boundary value problems. Conference Publications, 2011, 2011 (Special) : 1015-1024. doi: 10.3934/proc.2011.2011.1015 [2] J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177 [3] Shujie Li, Zhitao Zhang. Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 489-493. doi: 10.3934/dcds.1999.5.489 [4] Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control & Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018 [5] Santiago Cano-Casanova. Coercivity of elliptic mixed boundary value problems in annulus of $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3819-3839. doi: 10.3934/dcds.2012.32.3819 [6] Carmen Calvo-Jurado, Juan Casado-Díaz, Manuel Luna-Laynez. Parabolic problems with varying operators and Dirichlet and Neumann boundary conditions on varying sets. Conference Publications, 2007, 2007 (Special) : 181-190. doi: 10.3934/proc.2007.2007.181 [7] B. Abdellaoui, E. Colorado, I. Peral. Existence and nonexistence results for a class of parabolic equations with mixed boundary conditions. Communications on Pure & Applied Analysis, 2006, 5 (1) : 29-54. doi: 10.3934/cpaa.2006.5.29 [8] P. De Maesschalck. Ackerberg-O'Malley resonance in boundary value problems with a turning point of any order. Communications on Pure & Applied Analysis, 2007, 6 (2) : 311-333. doi: 10.3934/cpaa.2007.6.311 [9] Monica Motta, Caterina Sartori. Uniqueness of solutions for second order Bellman-Isaacs equations with mixed boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 739-765. doi: 10.3934/dcds.2008.20.739 [10] R. Kannan, S. Seikkala. Existence of solutions to some Phi-Laplacian boundary value problems. Conference Publications, 2001, 2001 (Special) : 211-217. doi: 10.3934/proc.2001.2001.211 [11] Patricia Bauman, Daniel Phillips, Jinhae Park. Existence of solutions to boundary value problems for smectic liquid crystals. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 243-257. doi: 10.3934/dcdss.2015.8.243 [12] Antonio Iannizzotto, Nikolaos S. Papageorgiou. Existence and multiplicity results for resonant fractional boundary value problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 511-532. doi: 10.3934/dcdss.2018028 [13] John R. Graef, Shapour Heidarkhani, Lingju Kong. Existence of nontrivial solutions to systems of multi-point boundary value problems. Conference Publications, 2013, 2013 (special) : 273-281. doi: 10.3934/proc.2013.2013.273 [14] Lingju Kong, Qingkai Kong. Existence of nodal solutions of multi-point boundary value problems. Conference Publications, 2009, 2009 (Special) : 457-465. doi: 10.3934/proc.2009.2009.457 [15] Mingxin Wang. Existence and uniqueness of solutions of free boundary problems in heterogeneous environments. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 415-421. doi: 10.3934/dcdsb.2018179 [16] T. J. Christiansen. Resonances and balls in obstacle scattering with Neumann boundary conditions. Inverse Problems & Imaging, 2008, 2 (3) : 335-340. doi: 10.3934/ipi.2008.2.335 [17] Eugenio Montefusco, Benedetta Pellacci, Gianmaria Verzini. Fractional diffusion with Neumann boundary conditions: The logistic equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2175-2202. doi: 10.3934/dcdsb.2013.18.2175 [18] Monica Motta, Caterina Sartori. Uniqueness results for boundary value problems arising from finite fuel and other singular and unbounded stochastic control problems. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 513-535. doi: 10.3934/dcds.2008.21.513 [19] Antonella Marini, Thomas H. Otway. Strong solutions to a class of boundary value problems on a mixed Riemannian--Lorentzian metric. Conference Publications, 2015, 2015 (special) : 801-808. doi: 10.3934/proc.2015.0801 [20] Daniel Franco, Donal O'Regan. Existence of solutions to second order problems with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 273-280. doi: 10.3934/proc.2003.2003.273

2017 Impact Factor: 1.179