The first initial boundary-value problem for evolution equations generated by symmetric functions of principal curvatures
We are interested in unique global classical solutions of the following problem: \[ -u_t+F_m[u] =g\quad \text{in} \quad Q= \Omega\times (0,T), \tag{1} \] \[ u-\varphi=0 \quad \text{on} \quad\partial'Q= \partial''Q \cup\Omega(0), \tag{2} \] where \(\partial''Q= \partial \Omega \times(0,T)\), \(\Omega(0)= \{(x,t)\mid x\in\Omega\), \(t=0\}\), and \(\Omega\) is a region with a smooth boundary in \(\mathbb{R}^n\), which, for brevity, is assumed to be bounded. In (2), \(\varphi\) is a smooth function of \((x,t)\), specified in \(\overline Q\); in (1), \(F_m[u] =S_m^{1/m} (\mu(u))\), where \(\mu(u)= (\mu_1(u), \dots, \mu_n(u))\), \(\mu_i(u)= k_i(u) \sqrt{1+u^2_x}\), \(i=1, \dots, n\), \(k_i(u)\) are the principal curvatures of the graph of the function \(x_n=u(x,t)\), \(x\in \overline\Omega\) (where \(t\) is regarded as a parameter), and \[ S_m^{1/m} (\mu)= \sum_{i_1< \cdots<i_m}\mu_{i_1}\dots\mu_{i_m} \] is an elementary symmetric function of the arguments \(\mu_1, \dots, \mu_n\). Finally, \(g=\widetilde g\sqrt {1+u^2_x}\), where \(\widetilde g:\overline Q\to\mathbb{R}^1\) is a known function. We consider equation (1) on the function set \(\mathbb{K}_m\), which is a connected region of parabolicity of (1), containing functions that are convex in \(x\). This set is defined as \[ \mathbb{K}_m= \biggl\{v:v\in C^2 (\overline \Omega),\;S_l\bigl(\mu(u) \bigr)(x)>0, \quad l=1, \dots, m,\;\forall x\in \overline \Omega\biggr\}. \] Our main result states sufficient conditions on \(\partial\Omega\), \(\tilde g\), and \(\varphi\) such that (1), (2) has a unique classical solution in \(\mathbb{K}_m\) and describes the smoothness of the solution.
scientific article; zbMATH DE number 1050666