ERA-MS
Wiener's criterion at $\infty$ for the heat equation and its measure-theoretical counterpart
Ugur G. Abdulla
keywords: super- or subtemperatures PWB solution regularity (or irregularity) of $\infty$ heat equation Wiener's criterion parabolic measure thermal capacity metric compactification of $\Bbb{R}^{N+1}$ parabolic Dirichlet problem unbounded domains
IPI
On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines
Ugur G. Abdulla
We develop a new variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundary. We employ optimal control framework, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consists of the minimization of the sum of $L_2$-norm declinations from the available measurement of the temperature flux on the fixed boundary and available information on the phase transition temperature on the free boundary. This approach allows one to tackle situations when the phase transition temperature is not known explicitly, and is available through measurement with possible error. It also allows for the development of iterative numerical methods of least computational cost due to the fact that for every given control vector, the parabolic PDE is solved in a fixed region instead of full free boundary problem. We prove well-posedness in Sobolev spaces framework and convergence of discrete optimal control problems to the original problem both with respect to cost functional and control.
keywords: second order parabolic PDE convergence in functional optimal control convergence in control. energy estimate Inverse Stefan problem method of lines Sobolev spaces traces of Sobolev functions embedding theorems discrete optimal control problem
DCDS
Regularity of $\infty$ for elliptic equations with measurable coefficients and its consequences
Ugur G. Abdulla
This paper introduces a notion of regularity (or irregularity) of the point at infinity ($\infty$) for the unbounded open set $\Omega\subset {\mathbb R}^{N}$ concerning second order uniformly elliptic equations with bounded and measurable coefficients, according as whether the ${\mathcal A}$- harmonic measure of $\infty$ is zero (or positive). A necessary and sufficient condition for the existence of a unique bounded solution to the Dirichlet problem in an arbitrary open set of ${\mathbb R}^{N}, N\ge 3$ is established in terms of the Wiener test for the regularity of $\infty$. It coincides with the Wiener test for the regularity of $\infty$ in the case of Laplace equation. From the topological point of view, the Wiener test at $\infty$ presents thinness criteria of sets near $\infty$ in fine topology. Precisely, the open set is a deleted neigborhood of $\infty$ in fine topology if and only if $\infty$ is irregular.
keywords: Dirichlet problem fine topology measurable coefficients Uniformly elliptic equations regularity (or irregularity) of $\infty$ differential generator. ${\mathcal A}$-thinness characteristic Markov process ${\mathcal A}$-harmonic measure ${\mathcal A}$-super- or subharmonicity Brownian motion Newtonian capacity PWB solution Wiener test metric compactification of $R^{N+1}$
IPI
On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences
Ugur G. Abdulla
We consider a variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundary. We employ optimal control framework, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consist of the minimization of the sum of $L_2$-norm declinations from the available measurement of the temperature flux on the fixed boundary and available information on the phase transition temperature on the free boundary. This approach allows one to tackle situations when the phase transition temperature is not known explicitly, and is available through measurement with possible error. It also allows for the development of iterative numerical methods of least computational cost due to the fact that for every given control vector, the parabolic PDE is solved in a fixed region instead of full free boundary problem. In Inverse Problems and Imaging, 7, 2(2013), 307-340 we proved well-posedness in Sobolev spaces framework and convergence of time-discretized optimal control problems. In this paper we perform full discretization and prove convergence of the discrete optimal control problems to the original problem both with respect to cost functional and control.
keywords: embedding theorems Inverse Stefan problem traces of Sobolev functions Sobolev spaces second order parabolic PDE convergence in control. convergence in functional optimal control discrete optimal control problem method of finite differences energy estimate

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