Neumann boundary value problem for general curvature flow with forcing term
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Neumann boundary value problem for general curvature flow with forcing term Ling Xiao1 Received: 3 May 2020 / Accepted: 30 October 2020 © Springer Nature B.V. 2020
Abstract We consider the evolution of a strictly convex hypersurface by a class of general curvature. We prove that given some Neumann boundary condition, the flow exists for all time and converges to a solution with prescribed general curvature that satisfies the Neumann boundary condition. Our method also works for the corresponding elliptic setting. Keywords Neumann boundary value problem · General curvature flow · Convergence analysis Mathematics Subject Classification Primary 53C44; Secondary 35K20 · 53C42
1 Introduction In this paper, we study the deformation of a strictly convex graph over a bounded, convex domain ⊂ Rn , n ≥ 2, to a convex graph with prescribed general curvature and Neumann boundary condition. More precisely, let (t) = {X := (x, u(x, t))|(x, t) ∈ × [0, T )}, we study the long time existence and convergence of the following flow problem ⎧ in × [0, T ) ⎪ ⎨ u˙ = w ( f (κ[(t)]) − (x, u)) u ν = ϕ(x, u) on ∂ × [0, T ) (1.1) ⎪ ⎩ u|t=0 = u 0 in , ¯ × R → R are smooth functions, ν denotes the outer unit normal to ∂, where, ϕ : w = 1 + |Du|2 , κ[(t)] = (κ1 , . . . , κn ) denotes the principal curvatures of (t), and ¯ → R, the initial hypersurface, is a smooth, strictly convex function over . u0 : To guarantee that as long as the flow exits, (t) stays convex, the curvature function f has to satisfy some structure conditions. Accordingly, the function f is assumed to be defined in the convex cone n+ ≡ {λ ∈ Rn : each component λi > 0} in Rn and satisfying the following conditions:
B 1
Ling Xiao [email protected] Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA
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Geometriae Dedicata
f i (λ) ≡
∂ f (λ) > 0 in n+ , 1 ≤ i ≤ n, ∂λi
(1.2)
f is a concave function.
(1.3)
and In addition, f will be assumed to satisfy some more technical assumptions. These include f > 0 in n+ , f = 0
on ∂ n+ ,
(1.4)
f (1, . . . , 1) = 1,
(1.5)
and f is homogeneous of degree one.
(1.6)
Moreover, for any C > 0 and every compact set E ⊂ n+ , there is R = R(E, C) > 0 such that f (λ1 , . . . , λn−1 , λn + R) ≥ C, ∀λ ∈ E. (1.7) An example of functions satisfying all assumptions above is given by 1 1 1 f = 2 Hnn + (Hn /Hl ) n−l , where Hl is the normalized l-th elementary symmetric poly1
nomial. However, we point out that the pure curvature quotient (Hn /Hl ) n−l does not satisfy (1.7). For a graph of u, the induced metric and its inverse matrix are given by gi j = δi j + u i u j and
g i j = δi j −
ui u j , w2
(1.8)
where w = 1 + |Du|2 . Following [2], the principle curvature of graph u are eigenvalues of the symmetric matrix A[u] = [ai j ] : ai j =
γ ik u kl γ l j ui uk , where γ ik = δi j − . w w(1 + w)
(1.9)
The inverse of γ i j is denoted by γi j , and γi j = δi j +
ui uk . 1+w
(1.10)
Geometrically [γi j ] is the square root of the metric, i.e. γik γk j = gi j . Now, for any positive
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