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.

MSC:

 35J60 Nonlinear elliptic equations 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations

NKA
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References:

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