Parabolic Equations Minimizing Linear Growth Functionals: L²-Theory

Abstract

Let Ω be a bounded set in ℝ^N with boundary of class C¹. We are interested in the problem

$$ \left\{ \begin{gathered} \frac{{\partial u}} {{\partial t}} = diva\left( {x,Du} \right)in Q = \left( {0,\infty } \right) \times \Omega , \hfill \\ u\left( {t,x} \right) = \phi \left( x \right)on S = \left( {0,\infty } \right) \times \partial \Omega , \hfill \\ u\left( {0,x} \right) = u_0 \left( x \right)in x \in \Omega \hfill \\ \end{gathered} \right. $$

(1)

where ϕ ∈ L¹(∂Ω), u⁰ ∈ L²(Ω) and a(x, ξ) = ∇_ξ f(x, ξ, f being a function with linear growth in ‖ξ‖ as ‖ξ‖ → ∞. One of the classical examples is the nonparametric area integrand for which $f(x,\xi ) = \sqrt {1 + \left\| \xi \right\|^2 }$. Problem (6.1) for this particular f is the time-dependent minimal surface equation, and has been studied in [145] and [90]. Other examples of problems of type (6.1) are the following: The evolution problem for plastic antiplanar shear, studied in [208], which corresponds to the plasticity functional f given by

$$ f(\xi ) = \left\{ \begin{gathered} \tfrac{1} {2}\left\| \xi \right\|^2 if \left\| \xi \right\| \leqslant 1, \hfill \\ \left\| \xi \right\| - \tfrac{1} {2} if \left\| \xi \right\| \geqslant 1; \hfill \\ \end{gathered} \right. $$

(2)

evolution problems associated with Lagranggians

$$ f(x,\xi ) = \sqrt {1 + a_{ij} (x)\xi _i \xi _i } $$

where the functions a _ij are continous and satisfy a_ij(x) = a_ij(x), ||ξ||² ≤ a_ij(x) ξ_iξ_j ≤ C||ξ||² for all ξ ∈ ℝ^N; and the Lagrangian

$$ g(x,\xi ) = \sqrt {1 + x^2 + \left\| \xi \right\|^2 } $$

$$ g(x,\xi ) = \sqrt {1 + x^2 + \left\| \xi \right\|^2 } $$

which was considered by S. Bernstein ([46]). On the other hand, problem (6.1) was studied in [129] for some Lagrangians f, which do not include the nonparametric area integrand, but include instead the plasticity functional and the total variation flow for which f(ξ) = ‖ξ‖. An application of this type of equations to faceted crystal growth is studied in [140].