Principle of symmetric criticality and evolution equations
Goro Akagi - Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan (email) Abstract:
Let linearly acts and let G be a J-invariant functional defined on G. In 1979, R. Palais [6] gave some sufficient conditions to guarantee the so-called "Principle of Symmetric Criticality": every critical point of X restricted on the subspace of symmetric points becomes also a critical point of J on the whole space J. In [5], this principle was generalized to the case where X is non-smooth and the setting does not require the full variational structure when J is compact or isometric.
GThe 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 *, where $\partial\upsilon^i$ is the so-called subdifferential operator from a Banach space V into its dual X*. It is assumed that there exists a Hilbert space V satisfying $V \subset H \subset V $ and that H acts on these spaces as isometries. In this setting, the existence of G-symmetric solution for above equation can be discussed.
GAs 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.
Received: September 2002; Revised: March 2003; Published: July 2003. |