Structure of approximate solutions of variational problems with extended-valued convex integrands

Alexander J. Zaslavski

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In this paper we consider a nonlinear periodic system driven by the vector ordinary $p$ -Laplacian and having a nonsmooth locally Lipschitz potential, which is positively homogeneous. Using a variational approach which exploits the homogeneity of the potential, we establish the existence of a nonconstant solution.

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Let $L : ℝ^{N} \times ℝ^{N} \to ℝ$ be a borelian function and consider the following problems $inf \{F (y) = \int_{a}^{b} L (y (t), y^{'} (t)) d t : y \in A C ([a, b], ℝ^{N}), y (a) = A, y (b) = B\} (P)$ $inf \{F^{* *} (y) = \int_{a}^{b} Ł (y (t), y^{'} (t)) d t : y \in A C ([a, b], ℝ^{N}), y (a) = A, y (b) = B\} \cdot (P^{* *})$ We give a sufficient condition, weaker then superlinearity, under which $inf F = inf F^{* *}$ if $L$ is just continuous in $x$ . We then extend a result of Cellina on the Lipschitz regularity of the minima of $(P)$ when $L$ is not superlinear.

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Convergence of approximate solutions of variational problems

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Periodic solutions for systems with nonsmooth and partially coercive potential

Michael E. Filippakis (2006)

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In this paper we consider nonlinear periodic systems driven by the one-dimensional $p$ -Laplacian and having a nonsmooth locally Lipschitz potential. Using a variational approach based on the nonsmooth Critical Point Theory, we establish the existence of a solution. We also prove a multiplicity result based on a nonsmooth extension of the result of Brezis-Nirenberg (Brezis, H., Nirenberg, L., Remarks on finding critical points, Comm. Pure Appl. Math. 44 (1991), 939–963.) due to Kandilakis-Kourogenis-Papageorgiou...