A Variational Approach to Lyapunov Type Inequalities: From by Antonio Cañada, Salvador Villegas
By Antonio Cañada, Salvador Villegas
This ebook highlights the present nation of Lyapunov-type inequalities via a close research. aimed at researchers and scholars operating in differential equations and people attracted to the functions of balance concept and resonant structures, the ebook starts with an summary Lyapunov’s unique effects and strikes ahead to incorporate regular effects bought long ago ten years. specific proofs and an emphasis on simple rules are supplied for various boundary stipulations for usual differential equations, together with Neumann, Dirichlet, periodic, and antiperiodic stipulations. Novel result of larger eigenvalues, structures of equations, partial differential equations in addition to variational methods are offered. To this recognize, a brand new and unified variational standpoint is brought for the remedy of such difficulties and a scientific dialogue of alternative kinds of boundary stipulations is featured.
Various difficulties make the learn of Lyapunov-type inequalities of curiosity to these in natural and utilized arithmetic. Originating with the learn of the soundness houses of the Hill equation, different questions arose for example in structures at resonance, crystallography, isoperimetric difficulties, Rayleigh variety quotients and oscillation and periods of disconjugacy and it bring about the research of Lyapunov-type inequalities for differential equations. This classical zone of
mathematics remains to be of significant curiosity and is still a resource of inspiration.
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Extra info for A Variational Approach to Lyapunov Type Inequalities: From ODEs to PDEs
116) be satisfied. 117). x0 ; L/. x0 ; L/. 20. 117) with non-strict ones. 116). 115) is the trivial one. However, in this last case, a more precise condition may be obtained. This is shown in the next proposition. 2. 112) has a unique solution. Proof. 0; d/ Therefore, problem PM(0,d) has only the trivial solution. x0 ; L/ a similar reasoning is valid and we obtain that problem PM(d,L) has only the trivial solution. L x0 /2 in the interval Œx0 ; L, we deduce that either problem PM(0,x0 ) or problem PM(x0 ,L) has only the trivial solution.
115) is the trivial one. 0; L/: In this case, both problems PM(0,d) and PM(d,L) have nontrivial solutions. d; L/; we have a contradiction. 0; L/: To this respect, the following proposition may be of interest. 1. 0; L/. 117) Proof. 116) be satisfied. 117). x0 ; L/. x0 ; L/. 20. 117) with non-strict ones. 116). 115) is the trivial one. However, in this last case, a more precise condition may be obtained. This is shown in the next proposition. 2. 112) has a unique solution. Proof. 0; d/ Therefore, problem PM(0,d) has only the trivial solution.
7. In the previous corollary, the conditions on the function a. 19) 16 2 A Variational Characterization of the Best Lyapunov Constants are given, respectively, in terms of the L1 norm kaC k1 and L1 norm kaC k1 : Clearly, they are not related in general, in the sense that none of them imply the other. 19) when one studies what happens for p ! 1C and p ! 1), u C k is not identically the zero function. 1 Neumann Boundary Conditions, As a Paradigm of Linear Resonant Problems 17 2 : To understand why this election p 1 is suitable, we must see in detail the proof of the next theorem, especially the part where the Lagrange multiplier Theorem is applied (see  for more details).