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An Isaac Newton Institute Workshop

Quantum Graphs, their Spectra and Applications

Maximum principle for elliptic inequalities on the stratified sets

Author: Oleg M. Penkin (Belgorod State University, Russia)

Abstract

A concept of an elliptic equation (and inequality) on the stratified set was described in details in [P] (see also [NP]). Roughly speaking, a stratified set $\Omega$ is a connected set in $\mathbb{R}^n$, consisting of a finite number of smooth manifolds (strata). One can imagine a simplicial complex as an example. Using a special "stratified" measure we define an analogue of a divergence operator (acting on tangent vector fields) as a density of the "flow" of that field. Finally, we define an analogue of the Laplacian on the stratified set. Among other results we give a following strong maximum principle (jointly with S.N. Oshepkova). Theorem. A solution of inequality $\Delta u\geq 0$ cannot have a point of local nontrivial maximum on $\Omega_0$.

Here $\Omega_0$ is a connected part of $\Omega$, consisting of strata in them and such that $\overline\Omega_0=\Omega$. Our proof is based on the following lemma.

Lemma. Let $f_i$ be a continuous function $(i=0,\dots,k)$ on $[0;a]$ which is differentiable on $(0;a]$. Let us also assume $f_i$ be nonpositive and $f_i(0)=0$. Then an inequality $ r^kf_k'(r)+\dots+rf_1'(r)+f_0'(r)\geq 0\ (r\in(0;a]) $ follows $f_i\equiv 0$.

We give also some applications. For example, a following analogue of so called Bochner's lemma is an easy consequence of the strong maximum principle:

Theorem. Let $\Omega_0=\Omega$ and $\Delta u\geq 0$. Then $u\equiv{\rm const}$.

Bibliography [P] Penkin, O. M. About a geometrical approach to multistructures and some qualitative properties of solutions. Partial differential equations on multistructures (Luminy, 1999), 183--191, Lecture Notes in Pure and Appl. Math., 219, Dekker, New York, 2001

[NP] Nicaise, Serge; Penkin, Oleg M. Poincar\'e -- Perron's method for the Dirichlet problem on stratified sets. J. Math. Anal. Appl. 296, No.2, 504-520 (2004)