PARTIAL REGULARITY RESULTS FOR EVOLUTIONAL P −LAPLACIAN SYSTEMS WITH NATURAL GROWTH

Masashi Misawa Dept. of Computer Science, The Univ. of Electro-Communications

ABSTRACT

Let Ω be a domain in an Euclidean space R^{m} for m ≥ 2, T be a positive number and set Q = (0, T) × Ω. Let p, 1 < p < ∞. For a map u : Q ⟼ R^{n}, u = (u^{1}, ⋯, u^{n}) , consider the evolutional p−Laplacian system

∂_{t}u^{i} − ∑^{m}_{α,β=1} D_{α} (|Du|_{g}^{p−2}g^{αβ}(z, u)D_{β}u^{i}) = f^{i}(z, u, Du) i = 1, ⋯, n, (1.1)

where D_{α} = \frac{∂}{∂x^{α}} , α = 1, ⋯, m, Du is the spatial gradient of a map u, Du = (D_{α}u^{i}) , |Du|_{g}^{2} = ∑_{i=1}^{n} ∑_{α,β=1}^{m} g^{αβ}(z, u)D_{α}u^{i}D_{β}u^{i}, f is a Carathéodory function defined on Q × R^{n} × R^{mn} with values into R^{n} and (g^{αβ}) is a symmetric matrix with measurable coefficients g^{αβ}(z, u) defined on Q. We assume that there exist positive constants λ, Λ and a such that

λ|ξ|^{2} ≤ g^{αβ}(z, u)ξ_{α}ξ_{β} ≤ Λ|ξ|^{2}  (1.2)

holds for all ξ = (ξ_{α}) ∈ R^{m}, almost every z ∈ Q and all u ∈ R^{n} and that

|f(z, u, P)| ≤ a|P |^{p}  (1.3)

is satisfied for almost every z ∈ Q, all u ∈ R^{n} and P = (P_{α}^{i}) ∈ R^{mn}. Moreover, we suppose that the coefficients g^{αβ} are uniformly Hölder continuous functions in Q×R^{n} with a Hölder exponet β, 0 < β < 1, and a Hölder constant C > 0, that is,

|g^{αβ}(z_{1}, u_{1}) − g^{αβ}(z_{2}, u_{2})| ≤ C (|t_{1} − t_{2}|^{\frac{1}{2}} + |x_{1} − x_{2}|)^{β} + C |u_{1} − u_{2}|^{β}  (1.4)

holds for all z_{i} = (t_{i}, x_{i}) ∈ R^{m+1} and u_{i} ∈ R^{n}, i = 1, 2. Here and in what follows, the notation |ξ|^{2} = ξ · ξ = ξ_{α}^{i}ξ_{α}^{i} and |ξ|_{g}^{2} = g^{αβ}ξ_{α} · ξ_{β} is used and the summation notation over repeated indices is adopted.

The typical example of such evolution systems as (1.1) is the gradient flow for the p−energy functional, defined for functions u belonging to the Sobolev space W^{1,p} (Ω, R^{n}) ,

E_{p}(u) = ∫_{Ω} \frac{1}{p} (g^{αβ}(x, u)D_{α}u · D_{β}u)^{\frac{p}{2}} dx.  (1.5)

Then the equation, which describes the gradient flow for (1.5), is (1.1) with the coefficients g^{αβ}(x, u) and the lower order term

f(x, u, Du) = − \frac{1}{2} |Du|_{g}^{p−2} \frac{dg^{αβ}}{du} (z, u)D_{α}u · D_{β}u.  (1.6)