A new approach to the existence of almost everywhere solutions of nonlinear PDE’s. (English) Zbl 0911.35029

In this survey paper the author presents results of his joint investigation with P. Marcellini concerning solvability of certain first and second order partial differential equations, both in scalar and vectorial case. The motivation for this investigation comes from the calculus of variations, nonlinear elasticity and optimal design theory. A typical result (in the scalar case) reads as follows.
Consider the Dirichlet problem \[ \begin{cases} F(Du(x))=0,\quad &\text{a.e. }x\in \Omega \\ u(x)=\varphi (x), & x\in \partial \Omega , \end{cases} \tag{\(*\)} \] \(\Omega \subset \mathbb R^n\) being a bounded (or unbounded) open domain, \(F^n:\mathbb R\to \mathbb R\) and \(\varphi \in W^{1,\infty }(\Omega)\). If \(E=\{\xi \in \mathbb R^n:\;F(\xi)=0\}\) and \[ D\varphi (x) \text{ is compactly contained in \text{int conv }} E \text{ a.e. in }\Omega , \tag{\(**\)} \] where int conv stands for the interior of the convex hull of \(E\). Then there exists (a dense set of) \(u\in W^{1,\infty }(\Omega)\) that satisfies (\(*\)). If, in addition, \(\varphi \in C^1(\Omega)\) and \(E\) is closed then (\(**\)) can be replaced by \(D\varphi (x)\in \text{int conv }E\) in \(\Omega\).
A vector analogue as well as a second order extension of this theorem are also considered.
Reviewer: O.Došlý (Brno)


35D05 Existence of generalized solutions of PDE (MSC2000)
35F30 Boundary value problems for nonlinear first-order PDEs
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