We study Lispchitz solutions of partial differential relations $\nabla u\in K$, where $u$ is a vector-valued function in an open subset of $R^n$. In some cases the set of solutions turns out to be surprisingly large. The general theory is then used to construct counter-examples to regularity of solutions of Euler-Lagrange systems satisfying classical ellipticity conditions.

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Convex integration for Lipschitz mappings and counterexamples to regularity

We consider variational integrals ∫ Ω F ( x , u , D u ) d x with integrands F(x, u, p) growing polynomially and of class C2 in p and Holder-continuous in (x, u). Under the main assumption that F(x, u, p) is strictly quasiconvex we prove that each minimizer is of Class C1,μ in an open set Ω0 ⊂ Ω with meas (Ω − Ω0) = 0.

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Partial regularity of minimizers of quasiconvex integrals

(1993). On very weak solutions of certain elliptic systems. Communications in Partial Differential Equations: Vol. 18, No. 9-10, pp. 1515-1537.

On very weak solutions of certain elliptic systems

We prove the Kato conjecture for elliptic operators and N×N-systems in divergence form of arbitrary order 2m on 
$ \Bbb R^n $
 . More precisely, we assume the coefficients to be bounded measurable and the ellipticity is taken in the sense of a Garding inequality. We identify the domain of their square roots as the natural Sobolev space 
$ H^m(\Bbb R^n,\Bbb C^N) $
 . We also make some remarks on the relation between various ellipticity conditions and Garding inequality.

https://link.springer.com/content/pdf/10.1007/PL00001377.pdf

The Kato square root problem for higher order elliptic operators and systems on $ \Bbb R^n $

We prove partial regularity for the vector-valued differential forms solving the system δ(A(x, ω))=0, dω=0, and for the gradient of the vector-valued functions solving the system div A(x, Du)=B(x, u, Du). Here the mapping A, with A(x, w) ≈ (1+ + ¦ω¦2)(p − 2)/2 ω (p⩾2), satisfies a quasimonotonicity condition which, when applied to the gradient A(x, ω)=Dωf(x, ω) of a real-valued functionf, is analogous to but stronger than quasiconvexity for f. The case 1<p<2 for monotone A is reduced to the case p⩾2 by a duality technique.

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Jiři Souček

Papers

Caccioppoli's inequality and Legendre-Hadamard condition