We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
71/1999
Optimal existence theorems for nonhomogeneous differential inclusions
Stefan Müller and Mikhail A. Sytchev
Abstract
In this paper we address the question of solvability of the differential inclusions $$Du(\cdot)\in K(\cdot ,u(\cdot)), u|_{\partial \Omega} =f,u\in W^{1,\infty} (\Omega ; R^m),$$ where $Df(\cdot) \in F(\cdot,f(\cdot))$ a.e. in $\Omega$, and where $F:\Omega \times R^m \to 2^{R^{m \times n}}$ is a multi-valued function. Our approach to these problems is based on the idea to construct a sequence of approximate solutions which converges strongly and makes use of Gromov\'s idea (following earlier work of Nash and Kuiper) to control convergence of the gradients by appropriate selection of the elements of the sequence. In this paper we identify an optimal setting of this method. We show that if for each $(x,u) \in \Omega \times R^m$, each $\varepsilon > 0$, and each $v \in F(x,u)$ we can find a piece-wise affine function $\phi \in l_v +W^{1,\infty}_0 (\Omega ;R^m )$ (here $l_v (y) =v \cdot y)$) with $||dist(D\phi , K(x,u))||_{L^1} \leq \varepsilon$ and $D \phi \in F(x',u')$ a.e. for all (x',u') sufficiently close to (x,u), then we can resolve the differential inclusions. The result holds provided $\{(x,u,v,):v \in K(x,u)\}$ is the zero set of a nonnegative upper semicontinuous function d such that for each (x,u) the set K(x,u) is compact and $d(x,u,v_j) \to 0$ if and only if $dist(v_j,K(x,u)) \to 0$. We also discuss some generalizations and applications of this result.
