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)

**The video for this talk should appear here if JavaScript is enabled.**

If it doesn't, something may have gone wrong with our embedded player.

We'll get it fixed as soon as possible.

If it doesn't, something may have gone wrong with our embedded player.

We'll get it fixed as soon as possible.