PDE's on low dimensional sets

elective seminars

We will discuss two approaches to partial differential equations on low dimensional sets in R^n. In the first place we will introduce a

construction of a Sobolev-type space based on the notion of a tangent space to a measure. A closely related notion is the tangent gradient. We will present simple examples of elliptic problems exploiting the tools we have just indicated. We will deal with the existence and regularity of solutions. The dependence of the Sobolev space on the underlying measure will be discussed. Another tool which will be presented is the notion of the Gamma-convergence of variational functionals and we will see its

application to studying PDE's.

Partial Differential Equations considered on smooth manifolds are natural generalizations of problems studied on open subsets of the Euclidean spaces. In applications we may need to consider equations on sets without any structure of a topological manifold e.g. on one-dimensional sets like letter X or Y.

There are different ways to deal with such problems. We wish to pursue two approaches to this issue. The first one is based on measure theoretic tools, we consider the low dimensional set as a support of a measure defined over R^n. The second approach employs the fattening of the set and uses the Gamma-convergence technique.

It is well-known that it is convenient to consider a weak formulation of a PDE in a suitable Sobolev space. We will present constructions of Sobolev-type spaces which are based on the notion of a tangent space to a measure and the tangent gradient. We will present examples of simple elliptic problems where the apparatus we have just presented is applicable. We will also study the questions of existence and regularity of solutions.

The Sobolev spaces with respect to measure look like their classical counterpart. However, even in simple situations the we may encounter unexpected difficulties e.g. with the construction of higher order Sobolev spaces.

We will also show that the Sobolev spaces continuously depend upon measure. This is particularly interesting when we consider a sequence of such spaces and the dimension of the measure support drops in the limit.

The second approach to studying PDE's on low dimensional sets is based on the notion of the Gamma-limit of variational functionals. One of the conclusions following from this type of convergence is convergence of minimizers to a minimizer of the limiting functional. Here we use the fact that the minimizers are solutions to the Euler-Lagrange equations. We will present examples how this technique works.

Apart from this we will address the question if it is possible to the

derive known solution formulas for equations on a graph. We have in mind e.g. the d'Alembert formula for the wave equation on a graph.

1. G.Bouchitté, G.Buttazzo, I.Fragalà, Convergence of Sobolev spaces on varying manifolds. J. Geom. Anal. 11 (2001), no. 3, 399–422.

2. G.Bouchitte, G.Buttazzo, P.Seppecher, Energies with respect to a measure and applications to low-dimensional structures, Calc. Var. Partial Differential Equations, 5 (1997), no. 1, 37--54.

3. C. Cattaneo, L. Fontana, D’Alembert formula on finite one-dimensional networks, J. Math. Anal. Appl. 284 (2003) 403–424

4. X.-F.Chen, M.Kowalczyk, Michał Existence of equilibria for the Cahn-Hilliard equation via local minimizers of the perimeter. Comm. Partial Differential Equations 21 (1996), no. 7-8, 1207–1233.

another one presented in class

1. Students know and understand the notion of a tangent space to a measure and its possible different constructions.

2. Students know and understand the notion of a Sobolev space with respect to a measure.

3. Students know and understand the notion of the Gamma-convergence of functionals.