##
**Elliptic versus parabolic regularization for the equation of prescribed mean curvature.**
*(English)*
Zbl 0890.35046

It is well-known that the Dirichlet problem for the equation of prescribed mean curvature
\[
\text{div}((1+|Du|^2)^{-1/2}Du)+ h(x)= 0\quad\text{in }\Omega\subset\mathbb{R}^n,\;u=\Phi\quad\text{on }\partial\Omega\tag{1}
\]
has no classical solution if the curvature function \(h\) is too big. The authors investigate for this case what happens to the parabolic and elliptic regularizations of (1):
\[
u_t= \text{div}((1+|Du|^2)^{-1/2} Du)+ h(x)\text{ in }\Omega\times (0,\infty),\tag{2}
\]

\[ u(x,t)= \Phi \text{ on } \partial\Omega\times (0,\infty),\;u(x,0)= u_0(x)\text{ on }\Omega \] and \[ \text{div}((1+ |Du^\varepsilon|^2)Du^\varepsilon+ \varepsilon Du^\varepsilon)+ h(x)= 0\quad\text{in }\Omega,\;u^\varepsilon= \Phi\quad\text{on }\partial\Omega.\tag{3} \] More precisely, they study the asymptotic behavior of the solutions of (2) and (3) for \(t\to\infty\) and \(\varepsilon\to 0\), respectively. It is shown that \(u(x,t)/t\) converges in \(L_2\) to a function \(v\) (called ‘parabolic growth function’) which is the unique minimizer in \(\text{BV}(\overline\Omega)\cap L^2(\Omega)\) of the functional \(F(u)= \int_\Omega|Du|+ \int_\Omega(\textstyle{{1\over 2}} u^2-hu)dx\), whereas \(\varepsilon u^\varepsilon\) converges in \(W^{1,2}_0(\Omega)\) (it is assumed that \(\Phi=0\)) to an “elliptic growth function” \(w\) which is the unique minimizer in \(W^{1,2}_0(\Omega)\) of \[ J(u)= \int_\Omega|Du|dx+\int_\Omega({1\over 2}|Du|^2- hu)dx. \] Furthermore, the authors analyze the subset \(\Omega^*\) of \(\Omega\) on which \(v\) takes on its maximal value \(\lambda^*= \text{ess sup }v\). It is expected that \(\Omega^*\) is the set on which the solution \(u\) of (2) grows fastest and at constant rate \(\lambda^*\). What they show under the assumption \(\lambda^*>0\) is that \(\Omega^*\) has positive Lebesgue measure and that \(\Omega^*\) maximizes \((-P(G)+ \int_G h(x)dx)/|G|\) among all measurable subsets \(G\) of \(\Omega\) with maximum value equal to \(\lambda^*\), where \(|G|\) and \(P(G)\) denote the Lebesgue measure and the perimeter of \(G\), respectively. Here one should remember that \(|\int_G h(x)dx|< P(G)\) is a necessary condition for the classical solvability of (1). Defining \(\widetilde\Omega\) and \(\widetilde\lambda\) for the elliptic growth function \(w\) in an analoguous manner as \(\Omega^*\) and \(\lambda^*\) before and assuming that \(\widetilde\lambda>0\) and \(\widetilde\Omega\) has a nonempty interior, the authors prove \(\int_{\widetilde\Omega} h(x)dx\geq P(\widetilde\Omega)\) and \(\int_G h(x)dx\leq P(G)\) for all \(G\) compactly contained in \(\text{int}(\widetilde\Omega)\). For \(n= 2\), \(\Omega\) a rectangle and \(h=\text{const.}\), the parabolic growth function \(v\) is explicitly calculated. In the last chapter, the authors describe a number of numerical experiments.

\[ u(x,t)= \Phi \text{ on } \partial\Omega\times (0,\infty),\;u(x,0)= u_0(x)\text{ on }\Omega \] and \[ \text{div}((1+ |Du^\varepsilon|^2)Du^\varepsilon+ \varepsilon Du^\varepsilon)+ h(x)= 0\quad\text{in }\Omega,\;u^\varepsilon= \Phi\quad\text{on }\partial\Omega.\tag{3} \] More precisely, they study the asymptotic behavior of the solutions of (2) and (3) for \(t\to\infty\) and \(\varepsilon\to 0\), respectively. It is shown that \(u(x,t)/t\) converges in \(L_2\) to a function \(v\) (called ‘parabolic growth function’) which is the unique minimizer in \(\text{BV}(\overline\Omega)\cap L^2(\Omega)\) of the functional \(F(u)= \int_\Omega|Du|+ \int_\Omega(\textstyle{{1\over 2}} u^2-hu)dx\), whereas \(\varepsilon u^\varepsilon\) converges in \(W^{1,2}_0(\Omega)\) (it is assumed that \(\Phi=0\)) to an “elliptic growth function” \(w\) which is the unique minimizer in \(W^{1,2}_0(\Omega)\) of \[ J(u)= \int_\Omega|Du|dx+\int_\Omega({1\over 2}|Du|^2- hu)dx. \] Furthermore, the authors analyze the subset \(\Omega^*\) of \(\Omega\) on which \(v\) takes on its maximal value \(\lambda^*= \text{ess sup }v\). It is expected that \(\Omega^*\) is the set on which the solution \(u\) of (2) grows fastest and at constant rate \(\lambda^*\). What they show under the assumption \(\lambda^*>0\) is that \(\Omega^*\) has positive Lebesgue measure and that \(\Omega^*\) maximizes \((-P(G)+ \int_G h(x)dx)/|G|\) among all measurable subsets \(G\) of \(\Omega\) with maximum value equal to \(\lambda^*\), where \(|G|\) and \(P(G)\) denote the Lebesgue measure and the perimeter of \(G\), respectively. Here one should remember that \(|\int_G h(x)dx|< P(G)\) is a necessary condition for the classical solvability of (1). Defining \(\widetilde\Omega\) and \(\widetilde\lambda\) for the elliptic growth function \(w\) in an analoguous manner as \(\Omega^*\) and \(\lambda^*\) before and assuming that \(\widetilde\lambda>0\) and \(\widetilde\Omega\) has a nonempty interior, the authors prove \(\int_{\widetilde\Omega} h(x)dx\geq P(\widetilde\Omega)\) and \(\int_G h(x)dx\leq P(G)\) for all \(G\) compactly contained in \(\text{int}(\widetilde\Omega)\). For \(n= 2\), \(\Omega\) a rectangle and \(h=\text{const.}\), the parabolic growth function \(v\) is explicitly calculated. In the last chapter, the authors describe a number of numerical experiments.

Reviewer: F.Tomi (Heidelberg)

### MSC:

35J60 | Nonlinear elliptic equations |

35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |

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\textit{P. Marcellini} and \textit{K. Miller}, J. Differ. Equations 137, No. 1, 1--53 (1997; Zbl 0890.35046)

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