Principle of symmetric criticality and evolution equations

Pages: 1 - 10, Issue Special, July 2003

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Goro Akagi - Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan (email)
Jun Kobayashi - Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan (email)
Mitsuharu Ôtani - Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1, Okubo, Tokyo, 169-8555, Japan (email)

Abstract:    Let X be a Banach space on which a symmetry group G linearly acts and let J be a G-invariant functional defined on X. In 1979, R. Palais [6] gave some sufficient conditions to guarantee the so-called "Principle of Symmetric Criticality": every critical point of J restricted on the subspace of symmetric points becomes also a critical point of J on the whole space X. In [5], this principle was generalized to the case where J is non-smooth and the setting does not require the full variational structure when G is compact or isometric.
   The purpose of this paper is to combine this result with the abstract theory developed in [1] and [2] concerning the evolution equation: $du(t)$/$dt + \partial\upsilon^1(u(t))- \partial\upsilon^2(u(t))$ ∋ $f(t)$ in V*, where $\partial\upsilon^i$ is the so-called subdifferential operator from a Banach space X into its dual V*. It is assumed that there exists a Hilbert space H satisfying $V \subset H \subset V $ and that G acts on these spaces as isometries. In this setting, the existence of G-symmetric solution for above equation can be discussed.
   As an application, a parabolic problem with the p-Laplacian in unbounded domains is discussed.

Keywords:  symmetric criticality, group action, evolution equations, subdifferential, p-Laplacian, unbounded domain.
Mathematics Subject Classification:  Primary: 58E05, 58E40, 34G25; Secondary: 35K55.

Received: September 2002;      Revised: March 2003;      Published: July 2003.