Parabolic Equations Minimizing Linear Growth Functionals: L2-Theory

Part of the Progress in Mathematics book series (PM, volume 223)


Let Ω be a bounded set in ℝN with boundary of class C1. We are interested in the problem
$$ \left\{ \begin{gathered} \frac{{\partial u}} {{\partial t}} = diva\left( {x,Du} \right)in Q = \left( {0,\infty } \right) \times \Omega , \hfill \\ u\left( {t,x} \right) = \phi \left( x \right)on S = \left( {0,\infty } \right) \times \partial \Omega , \hfill \\ u\left( {0,x} \right) = u_0 \left( x \right)in x \in \Omega \hfill \\ \end{gathered} \right. $$
where ϕ ∈ L1(∂Ω), u0L2(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
$$ f(\xi ) = \left\{ \begin{gathered} \tfrac{1} {2}\left\| \xi \right\|^2 if \left\| \xi \right\| \leqslant 1, \hfill \\ \left\| \xi \right\| - \tfrac{1} {2} if \left\| \xi \right\| \geqslant 1; \hfill \\ \end{gathered} \right. $$
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 aij(x) = aij(x), ||ξ||2aij(x) ξiξjC||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].


Dirichlet Problem Strong Solution Evolution Problem Abstract Cauchy Problem Nonlinear Semigroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Basel AG 2004

Authors and Affiliations

  1. 1.Departamento de Análisis MatemáticoUniversitat de ValenciaBurjassot (ValenciaSpain
  2. 2.Departamento de TecnologíaUniversitat Pompeu FabraBarcelonaSpain

Personalised recommendations