The approximation approach to construction of

E. A. Mazepa

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 5 (2015), pp. 25-35.

Résumé

This paper is devoted to the development of approximation approach to the construction of solutions of boundary value problems for semilinear elliptic equations on arbitrary non-compact Riemannian manifolds. Methods of studies are essentially based on approach based on the introduction of equivalence classes of functions on Riemannian manifold [5; 6; 9]. Also it summarizes the methodology for constructing a generalized solution of the Dirichlet problem for linear elliptic equations in bounded domains of $ R ^ n $ and arbitrary noncompact Riemannian manifolds (see: [2; 3, p. 237–240]). To date, there are several approaches to the introduction of the generalized solution of boundary value problems. One of them is based on Hilbert space methods and allows you to define the action elliptic operator on a much wider class functions, rather than class $ C ^ 2 $. Another approach to the construction of generalized solutions of elliptic equations originates in papers of Poincare late 19th century. He created a method allowing to consider solutions Dirichlet problem without any restrictions on the domains in which it can be solved, but remaining in the classical assumptions with respect to continuity of boundary data. Ideas of this method were embodied in papers of Perron, Vallee Poussin, M.V. Keldysh, A.A. Grigoryan. In this paper we study bounded solutions of the semilinear equation (2) $$ \Delta u = \phi (| u |) u, $$ where $ \phi (\xi) $ — nonnegative nondecreasing continuously differentiable function in the $ \xi \geq 0 $. We introduce the following notation. Let $M$ be an arbitrary smooth connected noncompact Riemannian manifold without boundary and let $\{B_k\}_{k=1}^{\infty}$ be an exhaustion of $M$, i.e., a sequence of precompact open subsets of $M$ such that $\overline{B_k}\subset B_{k+1}$ and $M=\bigcup_{k=1}^{\infty}B_k$. Throughout the sequel, we assume that boundaries $\partial B_k$ are $C^1$-smooth submanifolds. Let $f_1$ and $f_2$ be arbitrary continuous functions on $M$. Say that $f_1$ and $f_2$ are equivalent on $M$ and write $f_1\sim f_2$ if for some exhaustion $\{B_k\}_{k=1}^{\infty}$ of $M$ we have $$ \lim_{k\to\infty}\sup_{M\setminus B_k}|f_1-f_2|=0. $$ It is easy to verify that the relation "$\sim$" is an equivalence which does not depend on the choice of the exhaustion of the manifold and so partitions the set of all continuous functions on $M$ into equivalence classes. Denote the equivalence class of a function $f$ by $[f]$. A function $f$ is called asymptotically nonnegative whenever there exists a continuous function $w\geq 0$ on $M$ with $w\sim f$. Say that a boundary value problem for (2) is solvable on $M$ with boundary conditions of class $[f]$ whenever there exists a solution $u(x)$ to (2) on $M$ with $u\in [f]$. We introduce the concept of a generalized solution of the boundary value problem with the boundary the conditions of the class $ [f] $ for the equation (2) on manifold $ M $. To do this, we consider the sequence of decisions boundary value problems (3) in $ B_k $ $$ \left \{ \begin{array}{lcl} \Delta u_{k, f} = u_{k, f} \phi (| u_{k, f} |) \quad {\text in} \quad B_k {,} \\ u_{k, f} |_{\partial B_k} = f |_{\partial B_k}. \\ \end{array} \right. $$ It is shown that the above sequence $u_{k, f}$ of decisions has a convergent subsequence with limit function $u_f$, which is a solution of equation (2). In the case when there are no solutions of equation (2) with the boundary conditions of the class $ [f] $ on the manifold $ M $, the function $ u_f $, described above, call a generalized solution of equation (2) with the boundary conditions of the class $ [f] $. Remark. This approximate approach to the definition of a generalized solution of the boundary value problem (in particular, the Dirichlet problem) for harmonic functions in bounded domains of $\mathbb{R}^n $ goes back to the works of Wiener and Keldysh (see [3, p. 237–296]), and similar studies for linear elliptic equations on manifolds were held earlier in [2]. Proof of the main results is based on the principle of maximum, theorem of uniqueness for solutions of linear elliptic differential equations and similar results for solutions of quasilinear elliptic differential equations on precompact subset of $ M $ (see [5; 6]). The following theorem is the main result. Theorem 1. Let $ f $ — asymptotically non-negative continuous bounded on $ M $ function, then 1) the sequence of functions $ u_{1, f}, \ldots, u_{k, f}, \ldots $ is a solution of (3) converges uniformly on $ M $ to the limit function $ u_f $; 2) the function $ u_f $ does not depend on the exhaustion $ \{B_k \}_{k = 1}^\infty $ manifold $ M $ and $ f $ representative of the equivalence class; 3) if a class $ [f] $ is valid for the equation (2), then the function $ u_f $ is the only solution of the equation (2) on $M$ such that $ u_f \in [f] $, those $ u_f \equiv u $.

Détail

Keywords: semilinear elliptic equation, boundary value problem, approximation approach, generalized solutions, noncompact Riemannian manifolds, the Dirichlet problem.

Bibliographie

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