Parabolic problems with mixed variable lateral conditions: an abstract approach
The author presents a new approach to initial-boundary value problems for linear parabolic differential equations of second order with mixed and time-dependent boundary conditions. In detail, the following boundary conditions are considered: If \(\Omega\) is a uniformly \(C^{1,1}\) open set of \(\mathbb{R}^N\) with boundary \(\Gamma\) and \(\Gamma^t_0\) is a uniform family of \(C^{1,1}\) submanifolds of \(\Gamma\) \((t\in [0,T])\), then \(u(x,t)= g_0(x, t)\) on \(\Sigma_0:= \bigcup_{t\in(0, T)}\Gamma^t_0\times \{t\}\) and \((\partial u/\partial\nu_A)(x,t)= g_1(x, t)\) on \((\Gamma\times (0,T))\backslash\overline\Sigma_0\). Here \(\nu_A\) is the conormal vector related to the elliptic operator \(A\) in \(\partial_tu+ Au=f\). The author is interested in sufficiently wide conditions on the family \((\Gamma^t_0)_{t\in[0, T]}\) in order to obtain regularity of the type: If \(f\in L^2(\Omega\times (0,T))\) and \(u(x,0)\), \(g_0\), \(g_1\) are in suitable trace spaces, then \(\partial_tu\), \(Au\in L^2(\Omega\times (0,T))\), \(|\nabla u(.,t)|_{L^2(\Omega)}\in L^\infty(0, T)\). He proves a corresponding theorem if \(\sup_{x\in\Gamma^t_0} d(x,\Gamma^s_0)\leq \int^t_s\rho(\xi) d\xi\) for \(s<t\) (with \(\rho\in L^4(0,T)\)). This theorem follows from an abstract result on evolution equations on variable domains. The abstract theory is an essential part of the present paper. An application of the abstract theory to parabolic problems in non-cylindrical domains is also given.
mixed and time-dependent boundary conditions
scientific article; zbMATH DE number 1022296
scientific article