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Weak minima of variational integrals. (English) Zbl 0802.35016
Consider the variational equation on \(\Omega \subseteq \mathbb{R}^ n\) \[ \text{div} A (x,\nabla u) = 0 \tag{1} \] where \(A:\Omega \times \mathbb{R}^{m \times n} \to \mathbb{R}^{m \times n}\), \(A(x,\xi) \cong | \xi |^ p\), \(p>1\). The definition of a very weak solution of (1) is given as follows. Let \(\max \{p - 1,1\} < r < p\) and \(u \in W^{1,r}_{\text{loc}} (\Omega, \mathbb{R}^ m)\) satisfy the integral identity \[ \int_ \Omega \bigl \langle A(x, \nabla u) |\nabla\varphi \bigr \rangle dx = 0 \] for any \(\varphi \in W_ 0^{1,{r/r - p + 1}} (\Omega, \mathbb{R}^ m)\). Then \(u\) is called a very weak solution of (1). One of the results proved is the following regularity theorem.
Theorem: There exists \(r_ 1 = r_ 1 (p,u)\), \(p-1<r_ 1<p\), such that every very weak solution \(u \in W^{1,r_ 1}_{\text{loc}} (\Omega, \mathbb{R}^ m)\), actually belongs to \(W^{1,p}_{\text{loc}} (\Omega, \mathbb{R}^ m)\).
Reviewer: T.Iwaniec

35D10 Regularity of generalized solutions of PDE (MSC2000)
35J60 Nonlinear elliptic equations
35J50 Variational methods for elliptic systems
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